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today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size . and let 's talk about how it works . so first , you need to have the gel . this can be made out of different kinds of substances , such as agarose and polyacrylamide , both of which i 'll discuss later . and the electrophoresis part of it means that you need to have an electrical field passing through the gel to get the bands to move . so to create an electrical field , we have to have a cathode and an anode . the cathode is on this side . remember from general chemistry that at the cathode , reduction takes place , so you 'll have a negative charge . but on the other side , where the anode is , you 'll have a positive charge , because this is where oxidation is taking place . and to power this , you 'll need some kind of battery to connect them . and these are both usually also connected to a big box that contains the gel electrophoresis apparatus . in order for charge to flow across , you need something that will conduct electricity , so you have a buffer that 's usually composed of different ions . so this is completely covering this entire gel . and remember that it 's always there , but i 'm going to erase it just because it 's going to get a little messy for what i need to show you next . so next , you 'll load a sample of dna . in order to load your sample , you 'll need to take a pipette and put it into one if these wells . so usually you 'll want to make sure that you have a dye mixed in with your sample so that you can see it as it 's running . so i 'll show that as a pink line here , so i 've filled that well , followed by yellow here , and green here . so that 's just the color of the dye . it can really be any color you want . this will just help you track the movement of the bands later on . once you have these bands , what will happen once you turn on the electrical field ? remember that dna has a negatively charged backbone because of all the phosphate groups , so what you 'll actually observe is that they 'll travel towards the positive end . so maybe after a short period of time , what you 'll see is that -- it 'll look something like this . maybe the pink will have split up into a few bands , indicating that there 's a few different sized fragments in there . with the yellow , you might only see one band . and with the green , you might actually have two bands , but they 're so close together still that it 's kind of hard to tell . so how can you really see what 's going on ? the solution is to let this run for a longer period of time . and eventually , what you 'll see is shown here at the bottom . you 'll see that whatever sized fragments were in your original samples have effectively split up by their size . and note that in pink , you 'll always want to load something known as a dna ladder . a dna ladder is like a standard . this is something that you buy from a company , and they tell you exactly what sizes their fragments so that you can match them up to your unknown samples . so this could be 400 base pairs , 200 base pairs , and 100 base pairs . and as you 'll note , the smallest fragments travel the furthest . this is because the smallest things are really easy to push with the electrical field . but when you have such a big molecule , or rather a big dna fragment , it can be hard to move . so you can see that the 400 base pair does n't move too fast or too far . and what 's this tell us about our unknown yellow and green samples ? this shows us that the yellow sample has a band that 's 200 base pairs long . and the green sample actually was composed of two different fragments , one that was 100 base pairs and another that was 200 base pairs . so what would we do with this information ? if you needed a particular size of dna , say for the next step of your experiment , if you wanted to insert it into a plasmid or a vector , you could cut this out of the gel and use it for that . now let 's talk about the two kinds of gels that are most commonly used . the first is agarose , and the second is sds-page . so agarose is a gel that 's usually used for separating big pieces of dna . so if you think about the pore size in the agarose , it has pretty big pores , so imagine it looking kind of like this . the gel is pretty big . there 's big holes here , so that you 'll be able to separate out the big pieces of dna that come through . however , if you 're trying to separate out little pieces , it wo n't be that obvious , because they 'll all just race through these giant holes . so remember that this is for big dna fragments . usually , this is for dna that 's bigger than 50 base pairs . sds-page , on the other hand , can be used for very small things . so imagine that being a much finer weaving with smaller pores . although this can be used for small pieces of dna , it can also be used for proteins . you might be wondering , what does sds-page even stand for . the sds part is sodium dodecyl sulfate . this is a chemical agent that denatures proteins , disrupting any non-covalent interactions they may have . this makes it so that the charge of the proteins is n't a factor when they 're separating out onto the gel , and they 're only being separated strictly by size . the page part is polyacrylamide gel electrophoresis , or we 'll just leave it at ge . so polyacrylamide is the substance that gel 's made out . so how can we remember the difference between these two types of gels ? remember that sds-page is for small dna or protein cells . s for small , and s for sds . and agarose is for bigger fragments of dna . so today we 've talked about how you would setup a gel electrophoresis , why it works , and how you would want to pick the substance but your gel 's made of . if you were really doing this in the lab , now that you have your fragments of known size of dna or protein , you could either sequence them or use them in other molecular techniques .
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so to create an electrical field , we have to have a cathode and an anode . the cathode is on this side . remember from general chemistry that at the cathode , reduction takes place , so you 'll have a negative charge .
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the cathode is positive in a galvanic cell ... or is this instance different ?
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today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size . and let 's talk about how it works . so first , you need to have the gel . this can be made out of different kinds of substances , such as agarose and polyacrylamide , both of which i 'll discuss later . and the electrophoresis part of it means that you need to have an electrical field passing through the gel to get the bands to move . so to create an electrical field , we have to have a cathode and an anode . the cathode is on this side . remember from general chemistry that at the cathode , reduction takes place , so you 'll have a negative charge . but on the other side , where the anode is , you 'll have a positive charge , because this is where oxidation is taking place . and to power this , you 'll need some kind of battery to connect them . and these are both usually also connected to a big box that contains the gel electrophoresis apparatus . in order for charge to flow across , you need something that will conduct electricity , so you have a buffer that 's usually composed of different ions . so this is completely covering this entire gel . and remember that it 's always there , but i 'm going to erase it just because it 's going to get a little messy for what i need to show you next . so next , you 'll load a sample of dna . in order to load your sample , you 'll need to take a pipette and put it into one if these wells . so usually you 'll want to make sure that you have a dye mixed in with your sample so that you can see it as it 's running . so i 'll show that as a pink line here , so i 've filled that well , followed by yellow here , and green here . so that 's just the color of the dye . it can really be any color you want . this will just help you track the movement of the bands later on . once you have these bands , what will happen once you turn on the electrical field ? remember that dna has a negatively charged backbone because of all the phosphate groups , so what you 'll actually observe is that they 'll travel towards the positive end . so maybe after a short period of time , what you 'll see is that -- it 'll look something like this . maybe the pink will have split up into a few bands , indicating that there 's a few different sized fragments in there . with the yellow , you might only see one band . and with the green , you might actually have two bands , but they 're so close together still that it 's kind of hard to tell . so how can you really see what 's going on ? the solution is to let this run for a longer period of time . and eventually , what you 'll see is shown here at the bottom . you 'll see that whatever sized fragments were in your original samples have effectively split up by their size . and note that in pink , you 'll always want to load something known as a dna ladder . a dna ladder is like a standard . this is something that you buy from a company , and they tell you exactly what sizes their fragments so that you can match them up to your unknown samples . so this could be 400 base pairs , 200 base pairs , and 100 base pairs . and as you 'll note , the smallest fragments travel the furthest . this is because the smallest things are really easy to push with the electrical field . but when you have such a big molecule , or rather a big dna fragment , it can be hard to move . so you can see that the 400 base pair does n't move too fast or too far . and what 's this tell us about our unknown yellow and green samples ? this shows us that the yellow sample has a band that 's 200 base pairs long . and the green sample actually was composed of two different fragments , one that was 100 base pairs and another that was 200 base pairs . so what would we do with this information ? if you needed a particular size of dna , say for the next step of your experiment , if you wanted to insert it into a plasmid or a vector , you could cut this out of the gel and use it for that . now let 's talk about the two kinds of gels that are most commonly used . the first is agarose , and the second is sds-page . so agarose is a gel that 's usually used for separating big pieces of dna . so if you think about the pore size in the agarose , it has pretty big pores , so imagine it looking kind of like this . the gel is pretty big . there 's big holes here , so that you 'll be able to separate out the big pieces of dna that come through . however , if you 're trying to separate out little pieces , it wo n't be that obvious , because they 'll all just race through these giant holes . so remember that this is for big dna fragments . usually , this is for dna that 's bigger than 50 base pairs . sds-page , on the other hand , can be used for very small things . so imagine that being a much finer weaving with smaller pores . although this can be used for small pieces of dna , it can also be used for proteins . you might be wondering , what does sds-page even stand for . the sds part is sodium dodecyl sulfate . this is a chemical agent that denatures proteins , disrupting any non-covalent interactions they may have . this makes it so that the charge of the proteins is n't a factor when they 're separating out onto the gel , and they 're only being separated strictly by size . the page part is polyacrylamide gel electrophoresis , or we 'll just leave it at ge . so polyacrylamide is the substance that gel 's made out . so how can we remember the difference between these two types of gels ? remember that sds-page is for small dna or protein cells . s for small , and s for sds . and agarose is for bigger fragments of dna . so today we 've talked about how you would setup a gel electrophoresis , why it works , and how you would want to pick the substance but your gel 's made of . if you were really doing this in the lab , now that you have your fragments of known size of dna or protein , you could either sequence them or use them in other molecular techniques .
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today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size .
|
however , is it possible that the fragments might carry different charges and thus the forces applied might vary , and then that will affect the kinetics ?
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today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size . and let 's talk about how it works . so first , you need to have the gel . this can be made out of different kinds of substances , such as agarose and polyacrylamide , both of which i 'll discuss later . and the electrophoresis part of it means that you need to have an electrical field passing through the gel to get the bands to move . so to create an electrical field , we have to have a cathode and an anode . the cathode is on this side . remember from general chemistry that at the cathode , reduction takes place , so you 'll have a negative charge . but on the other side , where the anode is , you 'll have a positive charge , because this is where oxidation is taking place . and to power this , you 'll need some kind of battery to connect them . and these are both usually also connected to a big box that contains the gel electrophoresis apparatus . in order for charge to flow across , you need something that will conduct electricity , so you have a buffer that 's usually composed of different ions . so this is completely covering this entire gel . and remember that it 's always there , but i 'm going to erase it just because it 's going to get a little messy for what i need to show you next . so next , you 'll load a sample of dna . in order to load your sample , you 'll need to take a pipette and put it into one if these wells . so usually you 'll want to make sure that you have a dye mixed in with your sample so that you can see it as it 's running . so i 'll show that as a pink line here , so i 've filled that well , followed by yellow here , and green here . so that 's just the color of the dye . it can really be any color you want . this will just help you track the movement of the bands later on . once you have these bands , what will happen once you turn on the electrical field ? remember that dna has a negatively charged backbone because of all the phosphate groups , so what you 'll actually observe is that they 'll travel towards the positive end . so maybe after a short period of time , what you 'll see is that -- it 'll look something like this . maybe the pink will have split up into a few bands , indicating that there 's a few different sized fragments in there . with the yellow , you might only see one band . and with the green , you might actually have two bands , but they 're so close together still that it 's kind of hard to tell . so how can you really see what 's going on ? the solution is to let this run for a longer period of time . and eventually , what you 'll see is shown here at the bottom . you 'll see that whatever sized fragments were in your original samples have effectively split up by their size . and note that in pink , you 'll always want to load something known as a dna ladder . a dna ladder is like a standard . this is something that you buy from a company , and they tell you exactly what sizes their fragments so that you can match them up to your unknown samples . so this could be 400 base pairs , 200 base pairs , and 100 base pairs . and as you 'll note , the smallest fragments travel the furthest . this is because the smallest things are really easy to push with the electrical field . but when you have such a big molecule , or rather a big dna fragment , it can be hard to move . so you can see that the 400 base pair does n't move too fast or too far . and what 's this tell us about our unknown yellow and green samples ? this shows us that the yellow sample has a band that 's 200 base pairs long . and the green sample actually was composed of two different fragments , one that was 100 base pairs and another that was 200 base pairs . so what would we do with this information ? if you needed a particular size of dna , say for the next step of your experiment , if you wanted to insert it into a plasmid or a vector , you could cut this out of the gel and use it for that . now let 's talk about the two kinds of gels that are most commonly used . the first is agarose , and the second is sds-page . so agarose is a gel that 's usually used for separating big pieces of dna . so if you think about the pore size in the agarose , it has pretty big pores , so imagine it looking kind of like this . the gel is pretty big . there 's big holes here , so that you 'll be able to separate out the big pieces of dna that come through . however , if you 're trying to separate out little pieces , it wo n't be that obvious , because they 'll all just race through these giant holes . so remember that this is for big dna fragments . usually , this is for dna that 's bigger than 50 base pairs . sds-page , on the other hand , can be used for very small things . so imagine that being a much finer weaving with smaller pores . although this can be used for small pieces of dna , it can also be used for proteins . you might be wondering , what does sds-page even stand for . the sds part is sodium dodecyl sulfate . this is a chemical agent that denatures proteins , disrupting any non-covalent interactions they may have . this makes it so that the charge of the proteins is n't a factor when they 're separating out onto the gel , and they 're only being separated strictly by size . the page part is polyacrylamide gel electrophoresis , or we 'll just leave it at ge . so polyacrylamide is the substance that gel 's made out . so how can we remember the difference between these two types of gels ? remember that sds-page is for small dna or protein cells . s for small , and s for sds . and agarose is for bigger fragments of dna . so today we 've talked about how you would setup a gel electrophoresis , why it works , and how you would want to pick the substance but your gel 's made of . if you were really doing this in the lab , now that you have your fragments of known size of dna or protein , you could either sequence them or use them in other molecular techniques .
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although this can be used for small pieces of dna , it can also be used for proteins . you might be wondering , what does sds-page even stand for . the sds part is sodium dodecyl sulfate .
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what is the difference between sds page under reducing and nonreducing conditions ?
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today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size . and let 's talk about how it works . so first , you need to have the gel . this can be made out of different kinds of substances , such as agarose and polyacrylamide , both of which i 'll discuss later . and the electrophoresis part of it means that you need to have an electrical field passing through the gel to get the bands to move . so to create an electrical field , we have to have a cathode and an anode . the cathode is on this side . remember from general chemistry that at the cathode , reduction takes place , so you 'll have a negative charge . but on the other side , where the anode is , you 'll have a positive charge , because this is where oxidation is taking place . and to power this , you 'll need some kind of battery to connect them . and these are both usually also connected to a big box that contains the gel electrophoresis apparatus . in order for charge to flow across , you need something that will conduct electricity , so you have a buffer that 's usually composed of different ions . so this is completely covering this entire gel . and remember that it 's always there , but i 'm going to erase it just because it 's going to get a little messy for what i need to show you next . so next , you 'll load a sample of dna . in order to load your sample , you 'll need to take a pipette and put it into one if these wells . so usually you 'll want to make sure that you have a dye mixed in with your sample so that you can see it as it 's running . so i 'll show that as a pink line here , so i 've filled that well , followed by yellow here , and green here . so that 's just the color of the dye . it can really be any color you want . this will just help you track the movement of the bands later on . once you have these bands , what will happen once you turn on the electrical field ? remember that dna has a negatively charged backbone because of all the phosphate groups , so what you 'll actually observe is that they 'll travel towards the positive end . so maybe after a short period of time , what you 'll see is that -- it 'll look something like this . maybe the pink will have split up into a few bands , indicating that there 's a few different sized fragments in there . with the yellow , you might only see one band . and with the green , you might actually have two bands , but they 're so close together still that it 's kind of hard to tell . so how can you really see what 's going on ? the solution is to let this run for a longer period of time . and eventually , what you 'll see is shown here at the bottom . you 'll see that whatever sized fragments were in your original samples have effectively split up by their size . and note that in pink , you 'll always want to load something known as a dna ladder . a dna ladder is like a standard . this is something that you buy from a company , and they tell you exactly what sizes their fragments so that you can match them up to your unknown samples . so this could be 400 base pairs , 200 base pairs , and 100 base pairs . and as you 'll note , the smallest fragments travel the furthest . this is because the smallest things are really easy to push with the electrical field . but when you have such a big molecule , or rather a big dna fragment , it can be hard to move . so you can see that the 400 base pair does n't move too fast or too far . and what 's this tell us about our unknown yellow and green samples ? this shows us that the yellow sample has a band that 's 200 base pairs long . and the green sample actually was composed of two different fragments , one that was 100 base pairs and another that was 200 base pairs . so what would we do with this information ? if you needed a particular size of dna , say for the next step of your experiment , if you wanted to insert it into a plasmid or a vector , you could cut this out of the gel and use it for that . now let 's talk about the two kinds of gels that are most commonly used . the first is agarose , and the second is sds-page . so agarose is a gel that 's usually used for separating big pieces of dna . so if you think about the pore size in the agarose , it has pretty big pores , so imagine it looking kind of like this . the gel is pretty big . there 's big holes here , so that you 'll be able to separate out the big pieces of dna that come through . however , if you 're trying to separate out little pieces , it wo n't be that obvious , because they 'll all just race through these giant holes . so remember that this is for big dna fragments . usually , this is for dna that 's bigger than 50 base pairs . sds-page , on the other hand , can be used for very small things . so imagine that being a much finer weaving with smaller pores . although this can be used for small pieces of dna , it can also be used for proteins . you might be wondering , what does sds-page even stand for . the sds part is sodium dodecyl sulfate . this is a chemical agent that denatures proteins , disrupting any non-covalent interactions they may have . this makes it so that the charge of the proteins is n't a factor when they 're separating out onto the gel , and they 're only being separated strictly by size . the page part is polyacrylamide gel electrophoresis , or we 'll just leave it at ge . so polyacrylamide is the substance that gel 's made out . so how can we remember the difference between these two types of gels ? remember that sds-page is for small dna or protein cells . s for small , and s for sds . and agarose is for bigger fragments of dna . so today we 've talked about how you would setup a gel electrophoresis , why it works , and how you would want to pick the substance but your gel 's made of . if you were really doing this in the lab , now that you have your fragments of known size of dna or protein , you could either sequence them or use them in other molecular techniques .
|
you 'll see that whatever sized fragments were in your original samples have effectively split up by their size . and note that in pink , you 'll always want to load something known as a dna ladder . a dna ladder is like a standard . this is something that you buy from a company , and they tell you exactly what sizes their fragments so that you can match them up to your unknown samples .
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and what does the ladder go up by ?
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today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size . and let 's talk about how it works . so first , you need to have the gel . this can be made out of different kinds of substances , such as agarose and polyacrylamide , both of which i 'll discuss later . and the electrophoresis part of it means that you need to have an electrical field passing through the gel to get the bands to move . so to create an electrical field , we have to have a cathode and an anode . the cathode is on this side . remember from general chemistry that at the cathode , reduction takes place , so you 'll have a negative charge . but on the other side , where the anode is , you 'll have a positive charge , because this is where oxidation is taking place . and to power this , you 'll need some kind of battery to connect them . and these are both usually also connected to a big box that contains the gel electrophoresis apparatus . in order for charge to flow across , you need something that will conduct electricity , so you have a buffer that 's usually composed of different ions . so this is completely covering this entire gel . and remember that it 's always there , but i 'm going to erase it just because it 's going to get a little messy for what i need to show you next . so next , you 'll load a sample of dna . in order to load your sample , you 'll need to take a pipette and put it into one if these wells . so usually you 'll want to make sure that you have a dye mixed in with your sample so that you can see it as it 's running . so i 'll show that as a pink line here , so i 've filled that well , followed by yellow here , and green here . so that 's just the color of the dye . it can really be any color you want . this will just help you track the movement of the bands later on . once you have these bands , what will happen once you turn on the electrical field ? remember that dna has a negatively charged backbone because of all the phosphate groups , so what you 'll actually observe is that they 'll travel towards the positive end . so maybe after a short period of time , what you 'll see is that -- it 'll look something like this . maybe the pink will have split up into a few bands , indicating that there 's a few different sized fragments in there . with the yellow , you might only see one band . and with the green , you might actually have two bands , but they 're so close together still that it 's kind of hard to tell . so how can you really see what 's going on ? the solution is to let this run for a longer period of time . and eventually , what you 'll see is shown here at the bottom . you 'll see that whatever sized fragments were in your original samples have effectively split up by their size . and note that in pink , you 'll always want to load something known as a dna ladder . a dna ladder is like a standard . this is something that you buy from a company , and they tell you exactly what sizes their fragments so that you can match them up to your unknown samples . so this could be 400 base pairs , 200 base pairs , and 100 base pairs . and as you 'll note , the smallest fragments travel the furthest . this is because the smallest things are really easy to push with the electrical field . but when you have such a big molecule , or rather a big dna fragment , it can be hard to move . so you can see that the 400 base pair does n't move too fast or too far . and what 's this tell us about our unknown yellow and green samples ? this shows us that the yellow sample has a band that 's 200 base pairs long . and the green sample actually was composed of two different fragments , one that was 100 base pairs and another that was 200 base pairs . so what would we do with this information ? if you needed a particular size of dna , say for the next step of your experiment , if you wanted to insert it into a plasmid or a vector , you could cut this out of the gel and use it for that . now let 's talk about the two kinds of gels that are most commonly used . the first is agarose , and the second is sds-page . so agarose is a gel that 's usually used for separating big pieces of dna . so if you think about the pore size in the agarose , it has pretty big pores , so imagine it looking kind of like this . the gel is pretty big . there 's big holes here , so that you 'll be able to separate out the big pieces of dna that come through . however , if you 're trying to separate out little pieces , it wo n't be that obvious , because they 'll all just race through these giant holes . so remember that this is for big dna fragments . usually , this is for dna that 's bigger than 50 base pairs . sds-page , on the other hand , can be used for very small things . so imagine that being a much finer weaving with smaller pores . although this can be used for small pieces of dna , it can also be used for proteins . you might be wondering , what does sds-page even stand for . the sds part is sodium dodecyl sulfate . this is a chemical agent that denatures proteins , disrupting any non-covalent interactions they may have . this makes it so that the charge of the proteins is n't a factor when they 're separating out onto the gel , and they 're only being separated strictly by size . the page part is polyacrylamide gel electrophoresis , or we 'll just leave it at ge . so polyacrylamide is the substance that gel 's made out . so how can we remember the difference between these two types of gels ? remember that sds-page is for small dna or protein cells . s for small , and s for sds . and agarose is for bigger fragments of dna . so today we 've talked about how you would setup a gel electrophoresis , why it works , and how you would want to pick the substance but your gel 's made of . if you were really doing this in the lab , now that you have your fragments of known size of dna or protein , you could either sequence them or use them in other molecular techniques .
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so how can we remember the difference between these two types of gels ? remember that sds-page is for small dna or protein cells . s for small , and s for sds .
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by any chance , do you know the difference between native page and sds-page for protein electrophoresis ?
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today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size . and let 's talk about how it works . so first , you need to have the gel . this can be made out of different kinds of substances , such as agarose and polyacrylamide , both of which i 'll discuss later . and the electrophoresis part of it means that you need to have an electrical field passing through the gel to get the bands to move . so to create an electrical field , we have to have a cathode and an anode . the cathode is on this side . remember from general chemistry that at the cathode , reduction takes place , so you 'll have a negative charge . but on the other side , where the anode is , you 'll have a positive charge , because this is where oxidation is taking place . and to power this , you 'll need some kind of battery to connect them . and these are both usually also connected to a big box that contains the gel electrophoresis apparatus . in order for charge to flow across , you need something that will conduct electricity , so you have a buffer that 's usually composed of different ions . so this is completely covering this entire gel . and remember that it 's always there , but i 'm going to erase it just because it 's going to get a little messy for what i need to show you next . so next , you 'll load a sample of dna . in order to load your sample , you 'll need to take a pipette and put it into one if these wells . so usually you 'll want to make sure that you have a dye mixed in with your sample so that you can see it as it 's running . so i 'll show that as a pink line here , so i 've filled that well , followed by yellow here , and green here . so that 's just the color of the dye . it can really be any color you want . this will just help you track the movement of the bands later on . once you have these bands , what will happen once you turn on the electrical field ? remember that dna has a negatively charged backbone because of all the phosphate groups , so what you 'll actually observe is that they 'll travel towards the positive end . so maybe after a short period of time , what you 'll see is that -- it 'll look something like this . maybe the pink will have split up into a few bands , indicating that there 's a few different sized fragments in there . with the yellow , you might only see one band . and with the green , you might actually have two bands , but they 're so close together still that it 's kind of hard to tell . so how can you really see what 's going on ? the solution is to let this run for a longer period of time . and eventually , what you 'll see is shown here at the bottom . you 'll see that whatever sized fragments were in your original samples have effectively split up by their size . and note that in pink , you 'll always want to load something known as a dna ladder . a dna ladder is like a standard . this is something that you buy from a company , and they tell you exactly what sizes their fragments so that you can match them up to your unknown samples . so this could be 400 base pairs , 200 base pairs , and 100 base pairs . and as you 'll note , the smallest fragments travel the furthest . this is because the smallest things are really easy to push with the electrical field . but when you have such a big molecule , or rather a big dna fragment , it can be hard to move . so you can see that the 400 base pair does n't move too fast or too far . and what 's this tell us about our unknown yellow and green samples ? this shows us that the yellow sample has a band that 's 200 base pairs long . and the green sample actually was composed of two different fragments , one that was 100 base pairs and another that was 200 base pairs . so what would we do with this information ? if you needed a particular size of dna , say for the next step of your experiment , if you wanted to insert it into a plasmid or a vector , you could cut this out of the gel and use it for that . now let 's talk about the two kinds of gels that are most commonly used . the first is agarose , and the second is sds-page . so agarose is a gel that 's usually used for separating big pieces of dna . so if you think about the pore size in the agarose , it has pretty big pores , so imagine it looking kind of like this . the gel is pretty big . there 's big holes here , so that you 'll be able to separate out the big pieces of dna that come through . however , if you 're trying to separate out little pieces , it wo n't be that obvious , because they 'll all just race through these giant holes . so remember that this is for big dna fragments . usually , this is for dna that 's bigger than 50 base pairs . sds-page , on the other hand , can be used for very small things . so imagine that being a much finer weaving with smaller pores . although this can be used for small pieces of dna , it can also be used for proteins . you might be wondering , what does sds-page even stand for . the sds part is sodium dodecyl sulfate . this is a chemical agent that denatures proteins , disrupting any non-covalent interactions they may have . this makes it so that the charge of the proteins is n't a factor when they 're separating out onto the gel , and they 're only being separated strictly by size . the page part is polyacrylamide gel electrophoresis , or we 'll just leave it at ge . so polyacrylamide is the substance that gel 's made out . so how can we remember the difference between these two types of gels ? remember that sds-page is for small dna or protein cells . s for small , and s for sds . and agarose is for bigger fragments of dna . so today we 've talked about how you would setup a gel electrophoresis , why it works , and how you would want to pick the substance but your gel 's made of . if you were really doing this in the lab , now that you have your fragments of known size of dna or protein , you could either sequence them or use them in other molecular techniques .
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although this can be used for small pieces of dna , it can also be used for proteins . you might be wondering , what does sds-page even stand for . the sds part is sodium dodecyl sulfate .
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what is the difference between sds page under reducing and non reducing conditions ?
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today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size . and let 's talk about how it works . so first , you need to have the gel . this can be made out of different kinds of substances , such as agarose and polyacrylamide , both of which i 'll discuss later . and the electrophoresis part of it means that you need to have an electrical field passing through the gel to get the bands to move . so to create an electrical field , we have to have a cathode and an anode . the cathode is on this side . remember from general chemistry that at the cathode , reduction takes place , so you 'll have a negative charge . but on the other side , where the anode is , you 'll have a positive charge , because this is where oxidation is taking place . and to power this , you 'll need some kind of battery to connect them . and these are both usually also connected to a big box that contains the gel electrophoresis apparatus . in order for charge to flow across , you need something that will conduct electricity , so you have a buffer that 's usually composed of different ions . so this is completely covering this entire gel . and remember that it 's always there , but i 'm going to erase it just because it 's going to get a little messy for what i need to show you next . so next , you 'll load a sample of dna . in order to load your sample , you 'll need to take a pipette and put it into one if these wells . so usually you 'll want to make sure that you have a dye mixed in with your sample so that you can see it as it 's running . so i 'll show that as a pink line here , so i 've filled that well , followed by yellow here , and green here . so that 's just the color of the dye . it can really be any color you want . this will just help you track the movement of the bands later on . once you have these bands , what will happen once you turn on the electrical field ? remember that dna has a negatively charged backbone because of all the phosphate groups , so what you 'll actually observe is that they 'll travel towards the positive end . so maybe after a short period of time , what you 'll see is that -- it 'll look something like this . maybe the pink will have split up into a few bands , indicating that there 's a few different sized fragments in there . with the yellow , you might only see one band . and with the green , you might actually have two bands , but they 're so close together still that it 's kind of hard to tell . so how can you really see what 's going on ? the solution is to let this run for a longer period of time . and eventually , what you 'll see is shown here at the bottom . you 'll see that whatever sized fragments were in your original samples have effectively split up by their size . and note that in pink , you 'll always want to load something known as a dna ladder . a dna ladder is like a standard . this is something that you buy from a company , and they tell you exactly what sizes their fragments so that you can match them up to your unknown samples . so this could be 400 base pairs , 200 base pairs , and 100 base pairs . and as you 'll note , the smallest fragments travel the furthest . this is because the smallest things are really easy to push with the electrical field . but when you have such a big molecule , or rather a big dna fragment , it can be hard to move . so you can see that the 400 base pair does n't move too fast or too far . and what 's this tell us about our unknown yellow and green samples ? this shows us that the yellow sample has a band that 's 200 base pairs long . and the green sample actually was composed of two different fragments , one that was 100 base pairs and another that was 200 base pairs . so what would we do with this information ? if you needed a particular size of dna , say for the next step of your experiment , if you wanted to insert it into a plasmid or a vector , you could cut this out of the gel and use it for that . now let 's talk about the two kinds of gels that are most commonly used . the first is agarose , and the second is sds-page . so agarose is a gel that 's usually used for separating big pieces of dna . so if you think about the pore size in the agarose , it has pretty big pores , so imagine it looking kind of like this . the gel is pretty big . there 's big holes here , so that you 'll be able to separate out the big pieces of dna that come through . however , if you 're trying to separate out little pieces , it wo n't be that obvious , because they 'll all just race through these giant holes . so remember that this is for big dna fragments . usually , this is for dna that 's bigger than 50 base pairs . sds-page , on the other hand , can be used for very small things . so imagine that being a much finer weaving with smaller pores . although this can be used for small pieces of dna , it can also be used for proteins . you might be wondering , what does sds-page even stand for . the sds part is sodium dodecyl sulfate . this is a chemical agent that denatures proteins , disrupting any non-covalent interactions they may have . this makes it so that the charge of the proteins is n't a factor when they 're separating out onto the gel , and they 're only being separated strictly by size . the page part is polyacrylamide gel electrophoresis , or we 'll just leave it at ge . so polyacrylamide is the substance that gel 's made out . so how can we remember the difference between these two types of gels ? remember that sds-page is for small dna or protein cells . s for small , and s for sds . and agarose is for bigger fragments of dna . so today we 've talked about how you would setup a gel electrophoresis , why it works , and how you would want to pick the substance but your gel 's made of . if you were really doing this in the lab , now that you have your fragments of known size of dna or protein , you could either sequence them or use them in other molecular techniques .
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s for small , and s for sds . and agarose is for bigger fragments of dna . so today we 've talked about how you would setup a gel electrophoresis , why it works , and how you would want to pick the substance but your gel 's made of .
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bacterial dna appear as faint bands , what can i do for this ?
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today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size . and let 's talk about how it works . so first , you need to have the gel . this can be made out of different kinds of substances , such as agarose and polyacrylamide , both of which i 'll discuss later . and the electrophoresis part of it means that you need to have an electrical field passing through the gel to get the bands to move . so to create an electrical field , we have to have a cathode and an anode . the cathode is on this side . remember from general chemistry that at the cathode , reduction takes place , so you 'll have a negative charge . but on the other side , where the anode is , you 'll have a positive charge , because this is where oxidation is taking place . and to power this , you 'll need some kind of battery to connect them . and these are both usually also connected to a big box that contains the gel electrophoresis apparatus . in order for charge to flow across , you need something that will conduct electricity , so you have a buffer that 's usually composed of different ions . so this is completely covering this entire gel . and remember that it 's always there , but i 'm going to erase it just because it 's going to get a little messy for what i need to show you next . so next , you 'll load a sample of dna . in order to load your sample , you 'll need to take a pipette and put it into one if these wells . so usually you 'll want to make sure that you have a dye mixed in with your sample so that you can see it as it 's running . so i 'll show that as a pink line here , so i 've filled that well , followed by yellow here , and green here . so that 's just the color of the dye . it can really be any color you want . this will just help you track the movement of the bands later on . once you have these bands , what will happen once you turn on the electrical field ? remember that dna has a negatively charged backbone because of all the phosphate groups , so what you 'll actually observe is that they 'll travel towards the positive end . so maybe after a short period of time , what you 'll see is that -- it 'll look something like this . maybe the pink will have split up into a few bands , indicating that there 's a few different sized fragments in there . with the yellow , you might only see one band . and with the green , you might actually have two bands , but they 're so close together still that it 's kind of hard to tell . so how can you really see what 's going on ? the solution is to let this run for a longer period of time . and eventually , what you 'll see is shown here at the bottom . you 'll see that whatever sized fragments were in your original samples have effectively split up by their size . and note that in pink , you 'll always want to load something known as a dna ladder . a dna ladder is like a standard . this is something that you buy from a company , and they tell you exactly what sizes their fragments so that you can match them up to your unknown samples . so this could be 400 base pairs , 200 base pairs , and 100 base pairs . and as you 'll note , the smallest fragments travel the furthest . this is because the smallest things are really easy to push with the electrical field . but when you have such a big molecule , or rather a big dna fragment , it can be hard to move . so you can see that the 400 base pair does n't move too fast or too far . and what 's this tell us about our unknown yellow and green samples ? this shows us that the yellow sample has a band that 's 200 base pairs long . and the green sample actually was composed of two different fragments , one that was 100 base pairs and another that was 200 base pairs . so what would we do with this information ? if you needed a particular size of dna , say for the next step of your experiment , if you wanted to insert it into a plasmid or a vector , you could cut this out of the gel and use it for that . now let 's talk about the two kinds of gels that are most commonly used . the first is agarose , and the second is sds-page . so agarose is a gel that 's usually used for separating big pieces of dna . so if you think about the pore size in the agarose , it has pretty big pores , so imagine it looking kind of like this . the gel is pretty big . there 's big holes here , so that you 'll be able to separate out the big pieces of dna that come through . however , if you 're trying to separate out little pieces , it wo n't be that obvious , because they 'll all just race through these giant holes . so remember that this is for big dna fragments . usually , this is for dna that 's bigger than 50 base pairs . sds-page , on the other hand , can be used for very small things . so imagine that being a much finer weaving with smaller pores . although this can be used for small pieces of dna , it can also be used for proteins . you might be wondering , what does sds-page even stand for . the sds part is sodium dodecyl sulfate . this is a chemical agent that denatures proteins , disrupting any non-covalent interactions they may have . this makes it so that the charge of the proteins is n't a factor when they 're separating out onto the gel , and they 're only being separated strictly by size . the page part is polyacrylamide gel electrophoresis , or we 'll just leave it at ge . so polyacrylamide is the substance that gel 's made out . so how can we remember the difference between these two types of gels ? remember that sds-page is for small dna or protein cells . s for small , and s for sds . and agarose is for bigger fragments of dna . so today we 've talked about how you would setup a gel electrophoresis , why it works , and how you would want to pick the substance but your gel 's made of . if you were really doing this in the lab , now that you have your fragments of known size of dna or protein , you could either sequence them or use them in other molecular techniques .
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today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size .
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what is purpose by using dna as the gel electrophoresis ?
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today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size . and let 's talk about how it works . so first , you need to have the gel . this can be made out of different kinds of substances , such as agarose and polyacrylamide , both of which i 'll discuss later . and the electrophoresis part of it means that you need to have an electrical field passing through the gel to get the bands to move . so to create an electrical field , we have to have a cathode and an anode . the cathode is on this side . remember from general chemistry that at the cathode , reduction takes place , so you 'll have a negative charge . but on the other side , where the anode is , you 'll have a positive charge , because this is where oxidation is taking place . and to power this , you 'll need some kind of battery to connect them . and these are both usually also connected to a big box that contains the gel electrophoresis apparatus . in order for charge to flow across , you need something that will conduct electricity , so you have a buffer that 's usually composed of different ions . so this is completely covering this entire gel . and remember that it 's always there , but i 'm going to erase it just because it 's going to get a little messy for what i need to show you next . so next , you 'll load a sample of dna . in order to load your sample , you 'll need to take a pipette and put it into one if these wells . so usually you 'll want to make sure that you have a dye mixed in with your sample so that you can see it as it 's running . so i 'll show that as a pink line here , so i 've filled that well , followed by yellow here , and green here . so that 's just the color of the dye . it can really be any color you want . this will just help you track the movement of the bands later on . once you have these bands , what will happen once you turn on the electrical field ? remember that dna has a negatively charged backbone because of all the phosphate groups , so what you 'll actually observe is that they 'll travel towards the positive end . so maybe after a short period of time , what you 'll see is that -- it 'll look something like this . maybe the pink will have split up into a few bands , indicating that there 's a few different sized fragments in there . with the yellow , you might only see one band . and with the green , you might actually have two bands , but they 're so close together still that it 's kind of hard to tell . so how can you really see what 's going on ? the solution is to let this run for a longer period of time . and eventually , what you 'll see is shown here at the bottom . you 'll see that whatever sized fragments were in your original samples have effectively split up by their size . and note that in pink , you 'll always want to load something known as a dna ladder . a dna ladder is like a standard . this is something that you buy from a company , and they tell you exactly what sizes their fragments so that you can match them up to your unknown samples . so this could be 400 base pairs , 200 base pairs , and 100 base pairs . and as you 'll note , the smallest fragments travel the furthest . this is because the smallest things are really easy to push with the electrical field . but when you have such a big molecule , or rather a big dna fragment , it can be hard to move . so you can see that the 400 base pair does n't move too fast or too far . and what 's this tell us about our unknown yellow and green samples ? this shows us that the yellow sample has a band that 's 200 base pairs long . and the green sample actually was composed of two different fragments , one that was 100 base pairs and another that was 200 base pairs . so what would we do with this information ? if you needed a particular size of dna , say for the next step of your experiment , if you wanted to insert it into a plasmid or a vector , you could cut this out of the gel and use it for that . now let 's talk about the two kinds of gels that are most commonly used . the first is agarose , and the second is sds-page . so agarose is a gel that 's usually used for separating big pieces of dna . so if you think about the pore size in the agarose , it has pretty big pores , so imagine it looking kind of like this . the gel is pretty big . there 's big holes here , so that you 'll be able to separate out the big pieces of dna that come through . however , if you 're trying to separate out little pieces , it wo n't be that obvious , because they 'll all just race through these giant holes . so remember that this is for big dna fragments . usually , this is for dna that 's bigger than 50 base pairs . sds-page , on the other hand , can be used for very small things . so imagine that being a much finer weaving with smaller pores . although this can be used for small pieces of dna , it can also be used for proteins . you might be wondering , what does sds-page even stand for . the sds part is sodium dodecyl sulfate . this is a chemical agent that denatures proteins , disrupting any non-covalent interactions they may have . this makes it so that the charge of the proteins is n't a factor when they 're separating out onto the gel , and they 're only being separated strictly by size . the page part is polyacrylamide gel electrophoresis , or we 'll just leave it at ge . so polyacrylamide is the substance that gel 's made out . so how can we remember the difference between these two types of gels ? remember that sds-page is for small dna or protein cells . s for small , and s for sds . and agarose is for bigger fragments of dna . so today we 've talked about how you would setup a gel electrophoresis , why it works , and how you would want to pick the substance but your gel 's made of . if you were really doing this in the lab , now that you have your fragments of known size of dna or protein , you could either sequence them or use them in other molecular techniques .
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so imagine that being a much finer weaving with smaller pores . although this can be used for small pieces of dna , it can also be used for proteins . you might be wondering , what does sds-page even stand for .
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can polyacrylamide gels be used for the analysis of plasmid dna ?
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today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size . and let 's talk about how it works . so first , you need to have the gel . this can be made out of different kinds of substances , such as agarose and polyacrylamide , both of which i 'll discuss later . and the electrophoresis part of it means that you need to have an electrical field passing through the gel to get the bands to move . so to create an electrical field , we have to have a cathode and an anode . the cathode is on this side . remember from general chemistry that at the cathode , reduction takes place , so you 'll have a negative charge . but on the other side , where the anode is , you 'll have a positive charge , because this is where oxidation is taking place . and to power this , you 'll need some kind of battery to connect them . and these are both usually also connected to a big box that contains the gel electrophoresis apparatus . in order for charge to flow across , you need something that will conduct electricity , so you have a buffer that 's usually composed of different ions . so this is completely covering this entire gel . and remember that it 's always there , but i 'm going to erase it just because it 's going to get a little messy for what i need to show you next . so next , you 'll load a sample of dna . in order to load your sample , you 'll need to take a pipette and put it into one if these wells . so usually you 'll want to make sure that you have a dye mixed in with your sample so that you can see it as it 's running . so i 'll show that as a pink line here , so i 've filled that well , followed by yellow here , and green here . so that 's just the color of the dye . it can really be any color you want . this will just help you track the movement of the bands later on . once you have these bands , what will happen once you turn on the electrical field ? remember that dna has a negatively charged backbone because of all the phosphate groups , so what you 'll actually observe is that they 'll travel towards the positive end . so maybe after a short period of time , what you 'll see is that -- it 'll look something like this . maybe the pink will have split up into a few bands , indicating that there 's a few different sized fragments in there . with the yellow , you might only see one band . and with the green , you might actually have two bands , but they 're so close together still that it 's kind of hard to tell . so how can you really see what 's going on ? the solution is to let this run for a longer period of time . and eventually , what you 'll see is shown here at the bottom . you 'll see that whatever sized fragments were in your original samples have effectively split up by their size . and note that in pink , you 'll always want to load something known as a dna ladder . a dna ladder is like a standard . this is something that you buy from a company , and they tell you exactly what sizes their fragments so that you can match them up to your unknown samples . so this could be 400 base pairs , 200 base pairs , and 100 base pairs . and as you 'll note , the smallest fragments travel the furthest . this is because the smallest things are really easy to push with the electrical field . but when you have such a big molecule , or rather a big dna fragment , it can be hard to move . so you can see that the 400 base pair does n't move too fast or too far . and what 's this tell us about our unknown yellow and green samples ? this shows us that the yellow sample has a band that 's 200 base pairs long . and the green sample actually was composed of two different fragments , one that was 100 base pairs and another that was 200 base pairs . so what would we do with this information ? if you needed a particular size of dna , say for the next step of your experiment , if you wanted to insert it into a plasmid or a vector , you could cut this out of the gel and use it for that . now let 's talk about the two kinds of gels that are most commonly used . the first is agarose , and the second is sds-page . so agarose is a gel that 's usually used for separating big pieces of dna . so if you think about the pore size in the agarose , it has pretty big pores , so imagine it looking kind of like this . the gel is pretty big . there 's big holes here , so that you 'll be able to separate out the big pieces of dna that come through . however , if you 're trying to separate out little pieces , it wo n't be that obvious , because they 'll all just race through these giant holes . so remember that this is for big dna fragments . usually , this is for dna that 's bigger than 50 base pairs . sds-page , on the other hand , can be used for very small things . so imagine that being a much finer weaving with smaller pores . although this can be used for small pieces of dna , it can also be used for proteins . you might be wondering , what does sds-page even stand for . the sds part is sodium dodecyl sulfate . this is a chemical agent that denatures proteins , disrupting any non-covalent interactions they may have . this makes it so that the charge of the proteins is n't a factor when they 're separating out onto the gel , and they 're only being separated strictly by size . the page part is polyacrylamide gel electrophoresis , or we 'll just leave it at ge . so polyacrylamide is the substance that gel 's made out . so how can we remember the difference between these two types of gels ? remember that sds-page is for small dna or protein cells . s for small , and s for sds . and agarose is for bigger fragments of dna . so today we 've talked about how you would setup a gel electrophoresis , why it works , and how you would want to pick the substance but your gel 's made of . if you were really doing this in the lab , now that you have your fragments of known size of dna or protein , you could either sequence them or use them in other molecular techniques .
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the sds part is sodium dodecyl sulfate . this is a chemical agent that denatures proteins , disrupting any non-covalent interactions they may have . this makes it so that the charge of the proteins is n't a factor when they 're separating out onto the gel , and they 're only being separated strictly by size .
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what is the chemical formula for barium chloride enneahydrate ?
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today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size . and let 's talk about how it works . so first , you need to have the gel . this can be made out of different kinds of substances , such as agarose and polyacrylamide , both of which i 'll discuss later . and the electrophoresis part of it means that you need to have an electrical field passing through the gel to get the bands to move . so to create an electrical field , we have to have a cathode and an anode . the cathode is on this side . remember from general chemistry that at the cathode , reduction takes place , so you 'll have a negative charge . but on the other side , where the anode is , you 'll have a positive charge , because this is where oxidation is taking place . and to power this , you 'll need some kind of battery to connect them . and these are both usually also connected to a big box that contains the gel electrophoresis apparatus . in order for charge to flow across , you need something that will conduct electricity , so you have a buffer that 's usually composed of different ions . so this is completely covering this entire gel . and remember that it 's always there , but i 'm going to erase it just because it 's going to get a little messy for what i need to show you next . so next , you 'll load a sample of dna . in order to load your sample , you 'll need to take a pipette and put it into one if these wells . so usually you 'll want to make sure that you have a dye mixed in with your sample so that you can see it as it 's running . so i 'll show that as a pink line here , so i 've filled that well , followed by yellow here , and green here . so that 's just the color of the dye . it can really be any color you want . this will just help you track the movement of the bands later on . once you have these bands , what will happen once you turn on the electrical field ? remember that dna has a negatively charged backbone because of all the phosphate groups , so what you 'll actually observe is that they 'll travel towards the positive end . so maybe after a short period of time , what you 'll see is that -- it 'll look something like this . maybe the pink will have split up into a few bands , indicating that there 's a few different sized fragments in there . with the yellow , you might only see one band . and with the green , you might actually have two bands , but they 're so close together still that it 's kind of hard to tell . so how can you really see what 's going on ? the solution is to let this run for a longer period of time . and eventually , what you 'll see is shown here at the bottom . you 'll see that whatever sized fragments were in your original samples have effectively split up by their size . and note that in pink , you 'll always want to load something known as a dna ladder . a dna ladder is like a standard . this is something that you buy from a company , and they tell you exactly what sizes their fragments so that you can match them up to your unknown samples . so this could be 400 base pairs , 200 base pairs , and 100 base pairs . and as you 'll note , the smallest fragments travel the furthest . this is because the smallest things are really easy to push with the electrical field . but when you have such a big molecule , or rather a big dna fragment , it can be hard to move . so you can see that the 400 base pair does n't move too fast or too far . and what 's this tell us about our unknown yellow and green samples ? this shows us that the yellow sample has a band that 's 200 base pairs long . and the green sample actually was composed of two different fragments , one that was 100 base pairs and another that was 200 base pairs . so what would we do with this information ? if you needed a particular size of dna , say for the next step of your experiment , if you wanted to insert it into a plasmid or a vector , you could cut this out of the gel and use it for that . now let 's talk about the two kinds of gels that are most commonly used . the first is agarose , and the second is sds-page . so agarose is a gel that 's usually used for separating big pieces of dna . so if you think about the pore size in the agarose , it has pretty big pores , so imagine it looking kind of like this . the gel is pretty big . there 's big holes here , so that you 'll be able to separate out the big pieces of dna that come through . however , if you 're trying to separate out little pieces , it wo n't be that obvious , because they 'll all just race through these giant holes . so remember that this is for big dna fragments . usually , this is for dna that 's bigger than 50 base pairs . sds-page , on the other hand , can be used for very small things . so imagine that being a much finer weaving with smaller pores . although this can be used for small pieces of dna , it can also be used for proteins . you might be wondering , what does sds-page even stand for . the sds part is sodium dodecyl sulfate . this is a chemical agent that denatures proteins , disrupting any non-covalent interactions they may have . this makes it so that the charge of the proteins is n't a factor when they 're separating out onto the gel , and they 're only being separated strictly by size . the page part is polyacrylamide gel electrophoresis , or we 'll just leave it at ge . so polyacrylamide is the substance that gel 's made out . so how can we remember the difference between these two types of gels ? remember that sds-page is for small dna or protein cells . s for small , and s for sds . and agarose is for bigger fragments of dna . so today we 've talked about how you would setup a gel electrophoresis , why it works , and how you would want to pick the substance but your gel 's made of . if you were really doing this in the lab , now that you have your fragments of known size of dna or protein , you could either sequence them or use them in other molecular techniques .
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the gel is pretty big . there 's big holes here , so that you 'll be able to separate out the big pieces of dna that come through . however , if you 're trying to separate out little pieces , it wo n't be that obvious , because they 'll all just race through these giant holes . so remember that this is for big dna fragments .
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if i separate the nucleotides : g , t , a , c of a gene , how to read the sequences ?
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today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size . and let 's talk about how it works . so first , you need to have the gel . this can be made out of different kinds of substances , such as agarose and polyacrylamide , both of which i 'll discuss later . and the electrophoresis part of it means that you need to have an electrical field passing through the gel to get the bands to move . so to create an electrical field , we have to have a cathode and an anode . the cathode is on this side . remember from general chemistry that at the cathode , reduction takes place , so you 'll have a negative charge . but on the other side , where the anode is , you 'll have a positive charge , because this is where oxidation is taking place . and to power this , you 'll need some kind of battery to connect them . and these are both usually also connected to a big box that contains the gel electrophoresis apparatus . in order for charge to flow across , you need something that will conduct electricity , so you have a buffer that 's usually composed of different ions . so this is completely covering this entire gel . and remember that it 's always there , but i 'm going to erase it just because it 's going to get a little messy for what i need to show you next . so next , you 'll load a sample of dna . in order to load your sample , you 'll need to take a pipette and put it into one if these wells . so usually you 'll want to make sure that you have a dye mixed in with your sample so that you can see it as it 's running . so i 'll show that as a pink line here , so i 've filled that well , followed by yellow here , and green here . so that 's just the color of the dye . it can really be any color you want . this will just help you track the movement of the bands later on . once you have these bands , what will happen once you turn on the electrical field ? remember that dna has a negatively charged backbone because of all the phosphate groups , so what you 'll actually observe is that they 'll travel towards the positive end . so maybe after a short period of time , what you 'll see is that -- it 'll look something like this . maybe the pink will have split up into a few bands , indicating that there 's a few different sized fragments in there . with the yellow , you might only see one band . and with the green , you might actually have two bands , but they 're so close together still that it 's kind of hard to tell . so how can you really see what 's going on ? the solution is to let this run for a longer period of time . and eventually , what you 'll see is shown here at the bottom . you 'll see that whatever sized fragments were in your original samples have effectively split up by their size . and note that in pink , you 'll always want to load something known as a dna ladder . a dna ladder is like a standard . this is something that you buy from a company , and they tell you exactly what sizes their fragments so that you can match them up to your unknown samples . so this could be 400 base pairs , 200 base pairs , and 100 base pairs . and as you 'll note , the smallest fragments travel the furthest . this is because the smallest things are really easy to push with the electrical field . but when you have such a big molecule , or rather a big dna fragment , it can be hard to move . so you can see that the 400 base pair does n't move too fast or too far . and what 's this tell us about our unknown yellow and green samples ? this shows us that the yellow sample has a band that 's 200 base pairs long . and the green sample actually was composed of two different fragments , one that was 100 base pairs and another that was 200 base pairs . so what would we do with this information ? if you needed a particular size of dna , say for the next step of your experiment , if you wanted to insert it into a plasmid or a vector , you could cut this out of the gel and use it for that . now let 's talk about the two kinds of gels that are most commonly used . the first is agarose , and the second is sds-page . so agarose is a gel that 's usually used for separating big pieces of dna . so if you think about the pore size in the agarose , it has pretty big pores , so imagine it looking kind of like this . the gel is pretty big . there 's big holes here , so that you 'll be able to separate out the big pieces of dna that come through . however , if you 're trying to separate out little pieces , it wo n't be that obvious , because they 'll all just race through these giant holes . so remember that this is for big dna fragments . usually , this is for dna that 's bigger than 50 base pairs . sds-page , on the other hand , can be used for very small things . so imagine that being a much finer weaving with smaller pores . although this can be used for small pieces of dna , it can also be used for proteins . you might be wondering , what does sds-page even stand for . the sds part is sodium dodecyl sulfate . this is a chemical agent that denatures proteins , disrupting any non-covalent interactions they may have . this makes it so that the charge of the proteins is n't a factor when they 're separating out onto the gel , and they 're only being separated strictly by size . the page part is polyacrylamide gel electrophoresis , or we 'll just leave it at ge . so polyacrylamide is the substance that gel 's made out . so how can we remember the difference between these two types of gels ? remember that sds-page is for small dna or protein cells . s for small , and s for sds . and agarose is for bigger fragments of dna . so today we 've talked about how you would setup a gel electrophoresis , why it works , and how you would want to pick the substance but your gel 's made of . if you were really doing this in the lab , now that you have your fragments of known size of dna or protein , you could either sequence them or use them in other molecular techniques .
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today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size .
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is gel electrophoresis and simple electrophoresis same ?
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today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size . and let 's talk about how it works . so first , you need to have the gel . this can be made out of different kinds of substances , such as agarose and polyacrylamide , both of which i 'll discuss later . and the electrophoresis part of it means that you need to have an electrical field passing through the gel to get the bands to move . so to create an electrical field , we have to have a cathode and an anode . the cathode is on this side . remember from general chemistry that at the cathode , reduction takes place , so you 'll have a negative charge . but on the other side , where the anode is , you 'll have a positive charge , because this is where oxidation is taking place . and to power this , you 'll need some kind of battery to connect them . and these are both usually also connected to a big box that contains the gel electrophoresis apparatus . in order for charge to flow across , you need something that will conduct electricity , so you have a buffer that 's usually composed of different ions . so this is completely covering this entire gel . and remember that it 's always there , but i 'm going to erase it just because it 's going to get a little messy for what i need to show you next . so next , you 'll load a sample of dna . in order to load your sample , you 'll need to take a pipette and put it into one if these wells . so usually you 'll want to make sure that you have a dye mixed in with your sample so that you can see it as it 's running . so i 'll show that as a pink line here , so i 've filled that well , followed by yellow here , and green here . so that 's just the color of the dye . it can really be any color you want . this will just help you track the movement of the bands later on . once you have these bands , what will happen once you turn on the electrical field ? remember that dna has a negatively charged backbone because of all the phosphate groups , so what you 'll actually observe is that they 'll travel towards the positive end . so maybe after a short period of time , what you 'll see is that -- it 'll look something like this . maybe the pink will have split up into a few bands , indicating that there 's a few different sized fragments in there . with the yellow , you might only see one band . and with the green , you might actually have two bands , but they 're so close together still that it 's kind of hard to tell . so how can you really see what 's going on ? the solution is to let this run for a longer period of time . and eventually , what you 'll see is shown here at the bottom . you 'll see that whatever sized fragments were in your original samples have effectively split up by their size . and note that in pink , you 'll always want to load something known as a dna ladder . a dna ladder is like a standard . this is something that you buy from a company , and they tell you exactly what sizes their fragments so that you can match them up to your unknown samples . so this could be 400 base pairs , 200 base pairs , and 100 base pairs . and as you 'll note , the smallest fragments travel the furthest . this is because the smallest things are really easy to push with the electrical field . but when you have such a big molecule , or rather a big dna fragment , it can be hard to move . so you can see that the 400 base pair does n't move too fast or too far . and what 's this tell us about our unknown yellow and green samples ? this shows us that the yellow sample has a band that 's 200 base pairs long . and the green sample actually was composed of two different fragments , one that was 100 base pairs and another that was 200 base pairs . so what would we do with this information ? if you needed a particular size of dna , say for the next step of your experiment , if you wanted to insert it into a plasmid or a vector , you could cut this out of the gel and use it for that . now let 's talk about the two kinds of gels that are most commonly used . the first is agarose , and the second is sds-page . so agarose is a gel that 's usually used for separating big pieces of dna . so if you think about the pore size in the agarose , it has pretty big pores , so imagine it looking kind of like this . the gel is pretty big . there 's big holes here , so that you 'll be able to separate out the big pieces of dna that come through . however , if you 're trying to separate out little pieces , it wo n't be that obvious , because they 'll all just race through these giant holes . so remember that this is for big dna fragments . usually , this is for dna that 's bigger than 50 base pairs . sds-page , on the other hand , can be used for very small things . so imagine that being a much finer weaving with smaller pores . although this can be used for small pieces of dna , it can also be used for proteins . you might be wondering , what does sds-page even stand for . the sds part is sodium dodecyl sulfate . this is a chemical agent that denatures proteins , disrupting any non-covalent interactions they may have . this makes it so that the charge of the proteins is n't a factor when they 're separating out onto the gel , and they 're only being separated strictly by size . the page part is polyacrylamide gel electrophoresis , or we 'll just leave it at ge . so polyacrylamide is the substance that gel 's made out . so how can we remember the difference between these two types of gels ? remember that sds-page is for small dna or protein cells . s for small , and s for sds . and agarose is for bigger fragments of dna . so today we 've talked about how you would setup a gel electrophoresis , why it works , and how you would want to pick the substance but your gel 's made of . if you were really doing this in the lab , now that you have your fragments of known size of dna or protein , you could either sequence them or use them in other molecular techniques .
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although this can be used for small pieces of dna , it can also be used for proteins . you might be wondering , what does sds-page even stand for . the sds part is sodium dodecyl sulfate .
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what 's the difference between native page , sds-page under reducing conditions & under non-reducing conditions ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . ''
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please can you explain what a verb is explicitly ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . ''
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so forms of the word `` to be '' are always linking verbs but other words that link ideas together can also be linking verbs under some circumstances ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely .
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what type of grammar is the word 'like ' in the sentence : the bear smells like cinnamon ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . ''
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is a state verb the same as a state of being verb ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is .
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in the sentence `` the bear is hungry '' why is n't `` hungry '' a noun ( an idea ) ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely .
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i am super confuse about past participle and also with 'been ' and 'being ' could some1 show me what it is or recommend a video on ka ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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what is a bear-like thing to do ? the bear eats a fish . that 's an action .
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could `` the bear is eating fish '' be both a linking and action verb ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . ''
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is `` a bear looked lonely '' belonged to past tense ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer .
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if the condition is under the present tense , the sentence should be changed to `` a bear looks lonely , '' right ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean .
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does a linking verb connect 2 nouns ( ideas ) or it can connect a noun to an adjective ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear .
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a follow up question : why is n't `` feels '' a noun ( an idea ) ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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hello , grammarians . today , we 're talking about verbs and bears .
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what is transitive and intransitive vebs ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . ''
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how many different linking verbs are there ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb .
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is n't the bear is sitting a state of being or an action ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer .
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so like `` am '' is different from `` is '' ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . ''
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how are the verbs different ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is .
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so can i say the bear were hungry ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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hello , grammarians . today , we 're talking about verbs and bears .
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why do cows wear bells ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that .
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and the actions and the state of being verbs are different ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . ''
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for example , one that describes what a `` predicate '' is ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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what is a bear-like thing to do ? the bear eats a fish . that 's an action .
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do bears actually eat fish ?
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hello , grammarians . today , we 're talking about verbs and bears . we had previously established at least one thing about the verb , and that was that it can show actions . but today i 'd like to introduce the idea that the verb can link ideas to one another . in fact , we have this whole class of verbs that we call `` linking verbs . '' or , if you want to call it something fancier , we call that `` state of being verbs . '' these linking verbs include all forms of the verb `` to be , '' which i have handily written out for you . so , i am , he is we are , be nice , they were being , they have been , he was , we were . now we use a linking verb when we want to connect one idea to another . so what i 'm gon na do is i 'm gon na divide the screen in half between the action side and the state of being side , just to show you what i mean . so we use action verbs to show what something does , whereas we use state of being verbs to show what something is . so let 's bring it back to this bear . let 's think of an action for this bear to do . what is a bear-like thing to do ? the bear eats a fish . that 's an action . that 's something the bear is doing . the bear is hungry , however , is not something that the bear is doing , it 's something that the bear is . so what `` is '' is doing here , is connecting the word `` hungry , '' to the word `` bear . '' it 's linking it . some verbs can be used both ways . they can be used both as actions , and as linking verbs . i 'll show you an example of that . you could say , `` the bear looked at me . '' which is to say the bear is doing a thing , looking , at something , namely me . but we could also say , using the same verb , `` the bear looked lonely . '' now in this case , this is describing how the bear looks . what the bear looks like . this looking is not something the bear is doing , it is how the bear appears to a viewer . so , `` looked '' is connecting `` lonely '' to `` bear . '' it is linking `` lonely '' to `` bear . '' it is a linking verb . by the same token , we could say , for an action , `` the bear smells a person . '' what is it smelling ? a person . but we could also say , `` the bear smells like cinnamon . '' which , i grant you , is pretty unlikely . do n't go smelling bears . but what `` smells '' is doing here is connecting the idea of cinnamon , to the bear . the bear is n't smelling the cinnamon , the bear smells like cinnamon . and that 's the difference between a linking very and an action verb . a linking verb shows what something is , an action verb shows what something does . so the bear is hungry , the bear looked lonely , the bear smells like cinnamon . these all reflect something about what the bear is . how it 's being . you can learn anything , david out .
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hello , grammarians . today , we 're talking about verbs and bears .
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which is right , '' i am he '' or `` i am him '' ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions .
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how is it that sin2c=2sinccosc ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ?
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why not numerator sin b ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ?
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could n't sal just have taken the three angles as x , ( x-c ) and ( x+c ) ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions .
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what is meant by cryptography ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ?
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to apply trignometry , should n't a or c be 90* ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is .
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in an arithmetic progression can the difference between the numbers be 0 ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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are we supposed to know the law of sines and/or cosines off the top of our head ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees .
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and why did we consider a constant n ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two .
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how to find equation of circle if diameters are known ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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can someone tell me how to find trignometric ratios of values like sin37 or cos37 without using calculator.. ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem .
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as it is related to trignometry , it should have a 90degrees angle , so we have found the 60 degrees , thus by using sum of angles of a triangle , we see that the angles are 30,60,90.. is this possible.. ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ?
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if we muliply ( sinb/b ) and ( b ) the dinominator b will be cancelled , right ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that .
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what is that thing in trigonometric equation sin and tan ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b .
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of sides in the polygons ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ?
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in the video , when sal said `` is to go from angle a to angle b- however much that is , is the same amount to go from angle b to angle c '' , is it the non-mathematical way of saying `` the difference between a and b is the same as the difference between b and c '' ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees .
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. since we are using trig , ca n't we assume that either a or c is 90 ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem .
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can the triangle abc be a 30 60 90 triangle , since 30 +30 = 60 and 60 + 30 = 90 ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b .
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how can we replace 2ab cosc = a^2 + b^2 - c^2 with 2bc cosa = c^2 + b^2 - a^2 ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two .
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what 's up with the equation sal is writing ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ?
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at about how does sal know to add a constant to angle a ( and then angle b ) instead of multiply ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is .
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do you only add constants in arithmetic sequences ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ?
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if so , what do you call a sequence where every term is multiplied by a constant ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared .
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how was sal able to swap the letters `` a '' and `` a '' with `` c '' and `` c '' and have the equation still equal the original ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ?
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how to find the solution if the angle were in a harmonic progression ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ?
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sir why we tell tan as tan only why not any other symbol ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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what does iit jee stand for ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem .
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how there is 60 in 2 ?
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if the angles a , b , and c of a triangle are in an arithmetic progression and if a , b and c - lower case a , b and c denote the length of the sides opposite to the capital , the angles a - capital a , capital b and capital c respectively then what is the value of this expression right over here so let 's if we can work our way through this , so let 's just draw the triangle so we have visualization of what all the letters represent so we have the angles a , b and c so let me just draw like this so we have the angles a b and c and then the sides opposites them are the lower case versions , so the side opposite the capital a is lower case a the side opposite to capital b is lower case b and the side opposite to the angle capital c is lower case c. now the first piece of information they tell us is that they the angles capital a , capital b and capital c of the triangle are in an arithmetic progression very fancy word but all an arithmetic progression is , is a series of numbers that are separated by the same amount , so and let me give you some examples , so 1,2,3 - that 's an arithmetic progression 2 , 4 , 6 arithmetic progression were separated by 2 every time i could do 10 , 20 , 30 also an arithmetic progression . these are all arithmetic progressions . so all they 're saying is is to go from angle a to angle b - however much that is , is this the same amount to go from angle b to angle c. so let 's see what that tells us what that tells us about or maybe tells us , maybe does n't tell us anything about those angles so if we could say we could say we have angle a and then we have the notion angle b so we could say that b is equal to a plus some constant we do n't know what that is . it could go by 1 , it could go by 2 , it could go by 10 we do n't know what it is , so a plus n and then c would be equal to b plus n which is the same thing which is the same thing as b is a plus n , so this is a plus n it plus plus n which is equal to a plus 2n so what does that us ? well , the other thing we know about three angles in a triangles that they have to add up to a 180 degrees . so : this , this and this have to add up to a 180 degrees let 's try it out so we have a plus a plus n plus a plus 2n , plus a plus 2n is going to be equal to 180 degrees we have one , two , three `` a '' s here , so we get 3a plus with one n and another two n : three a plus 3 n is equal to 180 degrees or you could divide both sides by 3 plus n is equal to and you get a a plus n is equal to 60 degrees so what does that what does that tell usłż well a could still be anything cause if n is 1 then a is 59 if n is 10 then a is going to be 50 , so it does n't give us much information about the angle a but if you look up here , do you see a+n anywhere ? you see it right over here . b is equal to a plus n and we just figured out that a plus n has to be equal to sixty degrees so using this first piece of information we are able to come up with something pretty tangible b must be equal to sixty degrees and you could try it out with a bunch of numbers that 's an arithmetic progression and once again b is the middle one right over here . but no matter what the arithmetic progression is in order for these three angles to add up to 180 the middle one has to be equal to has to be equal to 60 degrees . so that was a pretty that's.. we 're doing pretty well so far so let 's see what we can do it the next part when the next part of problem . i 'm trying to save some screen real estate right over here okay so they want us to figure out the value of the expression a over c sine of two c - capital c , plus c over a sine of two a . so let me just write it down so we have i 'll do it.. i 'll do it in blue a over c a over c sine of two times capital c plus c over a sine of two times capital a what 's that going to be equal to ? so , whenever you see stuff like this you got a 2 here a 2 here frankly the best things you do is just experiment with your trigonometric identities and see if anything pops out of you that might be useful and a little bit of a clue here the first part of the problem helped us figure out what b is it helped us figure out what b is but right now the expression has no b in it . so right now this information seems kind of useless but if we could put this somehow in terms of b then we 'll have will will be making progress as we know information about angle b so let 's see what we can do so the first thing i would use is with sine of 2a - let me just rewrite each of these so sine of i just say sine of two times anything that 's just the same thing as and this is called the double angle formula so this is i might be wrong i was forgot the actual names of them but sine of two time something is two sine of that something times the cosine of that times the cosine of that something and you 'll see that in any trigonometric book on the inside cover even a lot of calculus books let 's do that for this the same thing right over here so sine of 2a over here is going to be 2 sine of a cosine of a that 's just a standard trigonometric identity and we 've - in the trigonometric that we prove that identity i think we do it multiple times than on front we have our coefficients still we have a over c times this plus c over a times this now is there anything we can do and remembered in the back of our mind we should be thinking of how can we use this information that b is equal sixty so if we can somehow put this in the form get b here when i think about how do you get a b here i think well you know we have a triangle here so the things that relate the sides of the triangle when especially it 's not a right triangle we 're gon na deal with the law of sines or law of cosines and law of sines let me just rewrite it over here just for our reference so law sines would say sine of a over a is equal to sine of b over b which is equal to sine of c over c at looks like we might be able to use that let me just write the law of cosines here just in case it 's useful in the future so law of cosines c squared it 's really the pythagoras theorem with little adjustment for the fact that it 's not a right triangle so c squared is equal to a squared plus b squared minus 2ab cosine cosine of c - of capital c so it 's law of sines , law of cosines . if we can somehow use both of these to put these in terms of b which we have information about well the first thing is i could rewrite this . so this is sine of c over c and this is sine of a over a so let me do that so i have let me do this . so i have the two a i have two a cosine of c. let me write that separately so i have two a cosine cosine of capital c and then times sine of c over c times i 'll do it white . \4c & amp ; h000000 & amp ; sine of c that 's a capital c sine of capital c over lower case c that 's that term and that term right over there and then to that i 'm adding to that i 'm adding - i 'm doing the same thing over here i have two times i 'm going to separate these guys out actually now i want to do the sines . so let me separate this guy and this guy out and so i get plus two c cosine of a times sine of capital over lower case a times sine of capital a over lower case a and i want this do for me look that the sine right over there i have sine over c that 's that over there and then i have sine of a over a . that 's that over there , capital a over lower case a they 're both equal to sine of b over b so we were making progress . we have - we started introducing b into the equation and that 's the expression that we actually have information about so this to be rewritten as sine of b over b so this is the same thing as sine of capital b over lower case b and this is the same thing sine of capital b over lower case b and they are both being multiplied , or both of these terms are multiplying - be multiplied by that two a cosine of capital c times that and then plus two c it 's a lower case c cosine of capital a times that so we can factor out the sine of b over b let 's do that let 's factor it out so this is the same thing as it 's the same thing as two a and i really have a sense of what the next step is , so i leave a little space here two a times cosine of c plus , this - and these are being multiplied , i left some space there plus two c , two lower case c times a cosine of a and all of this all of this times the sine of b over b and we know - we already know the b is sixty degree so we can evaluate this uh ... pretty pretty easily but let 's just continue see if we can somehow somehow put this right over here in terms of b well if you look over here we have two a cosine of c , two c cosine of a it looks - it 's starting to look pretty darn close pretty darn close , each of these terms are pretty darn close to this part to this part of the law of cosines over there and actually let 's solve for that part of the law of cosines that 's what we can do . so if you add two ab cosine of c on both sides , you get two ab cosine of capital c plus c squared is equal to a squared plus b squared or if you subtract c squared from both sides you get two ab cosine of capital c is equal to a squared plus b squared minus c squared and this is interesting and we can , you know - switch around the letters later on but this looks pretty darn close to this so what if - and this looks pretty darn close of this except we 're here we 're dealing with an a instead of a c. we just switch the letters around and we could rewrite this actually let me rewrite it just for fun i could rewrite this over here as two two c b not rewrite it . i can swap the letters times the cosine of a here i 'm swaping the as and cs , is equal to c squared plus b squared minus a squared there 's nothing unique about sides c i can do this with all of the sides so here it 's a big c here you have an a and a b out front and then you have the a square plus b squared minus the small c squared . if you have a big a , then you have the cb in the front and then you are subtracting the a squared right over here and this is useful because this term right over here this term right over here looks almost like this term over here if we could just - if we could just multiply this by b so let 's do that we can multiply that by b but let 's multiply this whole numerator this whole term by b so if we multiply this whole term by b . what do we get ? we get a b there . we get a b there . and of course you can just arbitrarily multiply an expression by b that 'll change its value so what we could do is multiply the expression by b which we just did we distributed a b across here but they 'll also divide by b where which - so i 'll divide by b . that 's equivalent of multiplying the denominator the denominator there , not b squared that 's equivalent of multiplying the denominator by b that 's the same thing as dividing by b we multiplied by b , divided by b where that 's the same thing as just turning this to b squared now what does this give us well we have this term right over here this term right over here is now the exact same thing as that over there so it is now a square plus b squared minus c squared and then this term right over here is now the exact same thing as this thing over here which is the same thing as that we are using the law of cosines so this is plus c squared plus b squared minus a squared and then all of that times this sine of b sine of capital b over b squared now what does this give us ? we have an a square and a negative a square things are starting to simplify a squared , negative a squared we have a negative c squared and a positive c squared so what are we left is - so we just left the two b squared so our whole expression has simplified to two b squared sine of b sine of capital b over lower case b squared these cancel out so our whole expression simplifies to two sine of b and from the get go we knew what b was we know it 's sixty degrees so this is equal to two times the sine of sixty degrees and if you do n't have the sine of sixty degrees memorized you can always just break out a thirty sixty ninety triangle so let me draw - this is a right triangle right over here this is sixty degrees hypotenuse has length one we 're dealing with the unit circle this side is thirty degrees the side opposite the thirty degrees is one-half the side opposite the sixty degrees is square root of three times that . so it 's square root of three over two . you can even use the pythagoras theorem to figure out once you know one of them you could figure out the other one the sine of - sine is opposite over hypotenuse so square root of three over two over one or just square root of three over two . it 's equal to two times - it 's the home stretch , we are exciting square root of three over two these cancel out . so we are left with the square root of three that 's a pretty neat problem . and just in case you are curious where it came from , the two thousand and ten iit , iit are these - hard to get into uh ... engineering and science universities in india and gives you the example like you know hundreds of thousands of kids and you know the top the top i do n't know like two thousand actually get into one of iits that i just thought it was a pretty neat problem
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what do we get ? we get a b there . we get a b there .
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if we apply projection rule i.e acosc+ccosa=b then we can get it easily , right ?
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- andrey , andrey ? andrey starts working on his science project at 4 o'clock pm , and he finishes 50 minutes later . which number line below has a red dot marking when andrey finishes working on his science project ? so they tell us that he starts his science project at 4 o'clock pm . this first choice over here , puts the dot at 4 o'clock , but they did n't say when did he start working on his science project , they say put a red dot marking when andrey finishes working on his science project . so he started at 4:00 , and it says he finishes 50 minutes later . so if you start at four , if you add 50 minutes to four , that 's gon na put us at 4:50 . so where is 4:50 on this number line ? so , let 's see , or on this time line , i guess i could say . so , let 's see , this is 4:15 , so each of these marks must be five minutes . this is 4:05 , 4:10 , 4:15 , 4:20 , 4:25 , 4:30 , 4:35 , 4:40 , 4:45 , and then 4:50 would be right over here . so which of these choices has a dot at 4:50 ? this last choice has a dot , has the red dot , at 4:50 . notice this is 50 minutes after he started at 4 o'clock . let 's do a few more of these . amira starts raking leaves at 2:30 pm , and she finishes 25 minutes later . which number line below has a red dot marking when amira finishes raking the leaves ? so she starts at 2:30 , and she finishes 25 minutes later . so it looks like on this one , each of these spaces is five minutes . so this is 2:30 , so this five , 10 , 15 , 20 , 25 minutes later . and this space right over her is 2:55 . it looks like this first choice is actually the choice we 're looking for . they put the red dot 25 minutes after 2:30 . notice this is 2:30 , five minutes after 2:30 , 10 minutes after 2:30 , 15 minutes after 2:30 , 20 minutes after 2:30 , 25 minutes after 2:30 . let 's keep going , this is actually a lot of fun . the number line shows when scarlet starts and finishes art club . that 's when she starts . that 's when she finishes . what time did scarlet start art club ? well , let 's see , this is half way between 3 o'clock and 3:30 , so that 's going to be 3:15 . each of these marks go up by 15 minutes . so this is at 3:15 . what time did scarlet finish art club ? so this is right in between 4:30 and 5 o'clock , or you could say it 's 15 minutes more than 4:30 , so that 's going to be 4:45 , 4:45 . how long was scarlet in art club ? so let 's think about it . there 's a couple of ways you could do it . you could look at the line . let 's see , to go from this point to this point , that 's 15 minutes , then 30 , then 45 , then one hour , one hour 15 minutes , one hour 30 minutes . so you could say one hour 30 minutes , or actually they want it all in minutes . one hour is 60 minutes plus 30 minutes is 90 minutes . so actually let me just do it all in minutes again . so this is , as i move my mouse , i 'm gon na talk about how many minutes i 'm adding , relative to this , or from the start . so 15 minutes gets us to 3:30 , 30 minutes , 45 minutes , 60 minutes , 75 minutes , 90 minutes . so that 's 90 minutes right over there . another way to think about it is it would take 45 minutes to go from 3:15 to 4 o'clock , 45 minutes to go from 3:15 to 4 o'clock , that makes sense , then another 45 minutes to go from 4 o'clock to 4:45 . so 45 plus 45 is going to be 90 minutes . let 's do a couple of these . so the number line shows when anna starts and finishes reading a chapter in her book . what time did anna start reading ? it says right over there she started reading at 8:10 pm . 8:10 pm . what time did anna finish reading ? let 's see , let me make sure i 'm reading this , so this is 8:20 , this is 8:25 , and they have one , two , three , four , five marks , so each of these go up a minute . so , you could say it 's a minute before 8:25 , or four minutes after 8:24 , 8:21 , 8:22 , 8:23 , 8:24 . 8:24 . how long did anna read ? well , she starts reading at 8:10 and finishes reading at 8:24 . to go from 10 to 24 that means she read for 14 minutes . and you could even count it over here . one , two , three , four , five , six , seven , eight , nine , 10 , 11 , 12 , 13 , 14 . let 's do one more . mohamed speaks on the phone for 15 minutes and finishes his phone call at 9:20 am . which number line below has a green dot marking when mohamed starts his phone call ? alright , it finishes at 9:20 , so it finishes , let 's see , where 's 9:20 on this number line . so it looks like each of these marks go up by five minutes . so this is 9:15 , and i was able to figure that out because we have three marks that go to 9:15 , from 9:00 to 9:15 , so this is 9:05 , 9:10 , 9:15 . so this would be 9:20 . so this is when he finishes , 9:20 , and he was on the phone for 15 minutes . remember , we do n't want to mark when he finished , they tell us that it 's 9:20 . we want to mark when he started . so if he 's on the phone call for 15 minutes , and he finishes at 9:20 , he 's gon na finish 15 minutes , or sorry , if he 's on the phone for 15 minutes , and he finishes at 9:20 , he 's going to start 15 minutes before 9:20 . so five , 10 , 15 . he 's gon na start at 9:05 . so that 's this choice right over here . another way you could think about it is if i 'm at 9:20 , and i go 15 minutes before 9:20 , 20 minus 15 is five , so that 's gon na get me to 9:05 . and we can just make sure we feel good about it . if he starts his phone call at 9:05 and he speaks for 15 minutes , five , 10 , 15 , he 's gon na finish his phone call right over here at 9:20 , which is exactly what they described .
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it says right over there she started reading at 8:10 pm . 8:10 pm . what time did anna finish reading ?
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so if an old arcade game is scheduled to be unplugged at 9 am tomorrow , and it is pm , how long ( in hours and minutes ) will it be until the game is unplugged ?
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okay , now we 're going to work on our first op-amp circuit . here 's what the circuit 's going to look like . watch where it puts the plus sign is on the top on this one . and we 're going to have a voltage source over here . this will be plus or minus v in , that 's our input signal . and over on the output , we 'll have v out , and it 's hooked up this way . the resistor , another resistor , to ground , and this goes back to the inverting input . now we 're going to look at this circuit and see what it does . now we know that connected up here the power supply 's hooked up to these points here , and the ground symbol is zero volts . and we want to analyze this circuit . and what do we know about this ? we know that v out equals some gain , i 'll write the gain right there . a big , big number times v minus , sorry v plus minus v minus , and let 's label that . v plus is this point right here , and v minus is this point right here . and we also know that the currents , let 's call them i plus and i minus , equals zero , and that 's the currents going in here . this is i minus here , and that 's i plus , and we know those are both zero . so now what i want to do it describe what 's going on inside this triangle symbol in more detail by building a circuit model . alright , and a circuit model for an amplifier looks like this . we have v minus here , v plus here , so this is v in , and over on this side we have an , here 's a new symbol that you have n't seen before . it 's usually drawn as a diamond shape , and this is a voltage source , but it 's a special kind of voltage source . it 's called a voltage-dependent voltage source . and it 's the same as a regular ideal voltage source except for one thing , it says that the v , in this case v out , equals gain times v plus minus v minus . so the voltage here depends on the voltage somewhere else , and that 's what makes it a voltage-dependent , that 's what that means . so , we 've just taken our gain expression here , added , drawn circuit diagram that represents our voltage expression for our circuit . now , specifically over here we 've drawn an open circuit on v plus , and v minus so we know that those currents are zero . so this model , this circuit sketch represents our two properties of our op-amp . so i 'm going to take a second here and i 'm going to draw the rest of our circuit surrounding this model , but i need a little bit more space . so let 's put in the rest of our circuit here . we had our voltage source , connected to v plus , and that 's v in , and over here we had v out . let 's check , v out was connected to two resistors , and the bottom is connected to ground , and this was connected there . so what our goal is right now , we want to find v out as a function of v in . that 's what we 're shooting for . so let 's see if we can do that . let 's give our resistors some names . let 's call this r1 , and r2 , our favorite names always , and now everything is labeled . now and we can label this point here , and this point we can call v minus , v minus . so that 's our two unknowns . our unknowns are v not , v out , and v minus , so let 's see if we can find them . so what i 'm going to do is just start writing some expressions for things that i know are true . for example , i know that v out equals a times v plus minus v minus . alright , that 's what this op-amp is telling us is true . now what else do i know ? let 's look at this resistor chain here . this resistor chain actually looks a lot like a voltage divider , and it 's actually a very good voltage divider . remember we said this current here , what is this current here ? it 's zero . i can use the voltage divider expression that i know . in that case , i know that v minus , this is the voltage divider equation , equals v out times what ? times the bottom resistor remember this ? r2 over r1 plus r2 , so the voltage divider expression says that when you have a stack of resistors like this , with the voltage on the top and ground on the bottom , this is the expression for the voltage at the midpoint . kay , so what i 'm going to do next is i 'm going to take this expression and stuff it right in there . let 's do that . see if we got enough room , okay now let 's go over here . now i can say that v out equals a times v plus minus v out times r2 over r1 plus r2 , alright so far so good . let 's keep going , let 's keep working on this . v not equals a times v plus minus a v not , r2 over r1 plus r2 . alright , so now i 'm going to gather all the v not terms over on the left hand side . let 's try that . so that gives me , v not plus a v not , times r2 over r1 plus r2 and that equals a times v plus , and actually i can change that now v plus is what ? v plus is v in . okay let 's keep going i can factor out the v not . v not is one plus a r2 over r1 plus r2 and that equals av in . alright so we 're getting close , and our original goal , we want to find v out in terms of v in . so i 'm going to take this whole expression here and divide it over to the other side , so then i have just v not on this side , and v in on the other side . make some more room . i can do that , i can say v not equals a v in divided by this big old expression , one plus a r2 over r1 plus r2 . alright so that 's our answer . that 's the answer . that 's v out equals some function of v in . now i want to make a really important observation here . this is going to be a real cool simplification . okay , so this is the point where op-amp theory gets really cool . watch what happens here . we know that a is a giant number . a is something like 10 to the fifth , or 10 to the sixth , and it 's whatever we have here , if our resistors are sort of normal-sized resistors we know that a giant number times a normal number is still going to be a very big number compared to one . so this one is almost insignificant in this expression down here , so what i 'm going to do , bear with me , i 'm going to cross it out . i 'm going to say no , i do n't need that anymore . so if this , if this number here , if a is a million , 10 to the sixth , and this expression here is something like one half then this total thing is one half 10 to the sixth or a half a million , and that 's huge compared to one . so i can pretty safely ignore the one , it 's very , very small . now when i do that , well look what happens next , now i have a top and bottom in the expression , and i can cancel that too . so the a goes away , now this is pretty astonishing . we have this amplifier circuit and all of a sudden i have an expression here where a does n't appear , the gain does not appear , and what does this turn into ? this is called v not equals v in , times what ? times r1 plus r2 , divided by r2 . so our amplifier , our feedback circuit came down to v out is v in multiplied by the ratio of the resistors that we added to the circuit . this is one of the really cool properties of using op-amps in circuits , really high-gain amplifiers . what we 've done is we have chosen the gain of our circuit based on the components that we picked to add to the amplifier . it 's not determined by the gain of the amplifier as long as the amplifier gain is really , really big . and for op-amps , that 's a good assumption , it is really big . so this expression came out with a positive sign , right ? all the r 's are positive values , so this is referred to as a non-inverting op-amp circuit amplifier . so just to do a quick example , if r1 and r2 are the same , then we end up with an expression that looks like this v out equals r1 plus r2 , r plus r over r is equal to two so the gain is two times v in . so just to do a quick sketch just to remind ourselves what this looks like , this was v in , and we had what out here ? we had a resistor , we had a resistor to the ground , and this is v out . so this is the configuration of a non-inverting amplifier built with an op-amp , the two resistors in this voltage divider string connected to the negative input . so that 's what non-inverting op-amp circuit looks like , and it 's going to be one of the familiar patterns that you see over and over again as you read schematics and you design your own circuits .
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it 's usually drawn as a diamond shape , and this is a voltage source , but it 's a special kind of voltage source . it 's called a voltage-dependent voltage source . and it 's the same as a regular ideal voltage source except for one thing , it says that the v , in this case v out , equals gain times v plus minus v minus .
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and what does a -negative voltage even mean ?
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okay , now we 're going to work on our first op-amp circuit . here 's what the circuit 's going to look like . watch where it puts the plus sign is on the top on this one . and we 're going to have a voltage source over here . this will be plus or minus v in , that 's our input signal . and over on the output , we 'll have v out , and it 's hooked up this way . the resistor , another resistor , to ground , and this goes back to the inverting input . now we 're going to look at this circuit and see what it does . now we know that connected up here the power supply 's hooked up to these points here , and the ground symbol is zero volts . and we want to analyze this circuit . and what do we know about this ? we know that v out equals some gain , i 'll write the gain right there . a big , big number times v minus , sorry v plus minus v minus , and let 's label that . v plus is this point right here , and v minus is this point right here . and we also know that the currents , let 's call them i plus and i minus , equals zero , and that 's the currents going in here . this is i minus here , and that 's i plus , and we know those are both zero . so now what i want to do it describe what 's going on inside this triangle symbol in more detail by building a circuit model . alright , and a circuit model for an amplifier looks like this . we have v minus here , v plus here , so this is v in , and over on this side we have an , here 's a new symbol that you have n't seen before . it 's usually drawn as a diamond shape , and this is a voltage source , but it 's a special kind of voltage source . it 's called a voltage-dependent voltage source . and it 's the same as a regular ideal voltage source except for one thing , it says that the v , in this case v out , equals gain times v plus minus v minus . so the voltage here depends on the voltage somewhere else , and that 's what makes it a voltage-dependent , that 's what that means . so , we 've just taken our gain expression here , added , drawn circuit diagram that represents our voltage expression for our circuit . now , specifically over here we 've drawn an open circuit on v plus , and v minus so we know that those currents are zero . so this model , this circuit sketch represents our two properties of our op-amp . so i 'm going to take a second here and i 'm going to draw the rest of our circuit surrounding this model , but i need a little bit more space . so let 's put in the rest of our circuit here . we had our voltage source , connected to v plus , and that 's v in , and over here we had v out . let 's check , v out was connected to two resistors , and the bottom is connected to ground , and this was connected there . so what our goal is right now , we want to find v out as a function of v in . that 's what we 're shooting for . so let 's see if we can do that . let 's give our resistors some names . let 's call this r1 , and r2 , our favorite names always , and now everything is labeled . now and we can label this point here , and this point we can call v minus , v minus . so that 's our two unknowns . our unknowns are v not , v out , and v minus , so let 's see if we can find them . so what i 'm going to do is just start writing some expressions for things that i know are true . for example , i know that v out equals a times v plus minus v minus . alright , that 's what this op-amp is telling us is true . now what else do i know ? let 's look at this resistor chain here . this resistor chain actually looks a lot like a voltage divider , and it 's actually a very good voltage divider . remember we said this current here , what is this current here ? it 's zero . i can use the voltage divider expression that i know . in that case , i know that v minus , this is the voltage divider equation , equals v out times what ? times the bottom resistor remember this ? r2 over r1 plus r2 , so the voltage divider expression says that when you have a stack of resistors like this , with the voltage on the top and ground on the bottom , this is the expression for the voltage at the midpoint . kay , so what i 'm going to do next is i 'm going to take this expression and stuff it right in there . let 's do that . see if we got enough room , okay now let 's go over here . now i can say that v out equals a times v plus minus v out times r2 over r1 plus r2 , alright so far so good . let 's keep going , let 's keep working on this . v not equals a times v plus minus a v not , r2 over r1 plus r2 . alright , so now i 'm going to gather all the v not terms over on the left hand side . let 's try that . so that gives me , v not plus a v not , times r2 over r1 plus r2 and that equals a times v plus , and actually i can change that now v plus is what ? v plus is v in . okay let 's keep going i can factor out the v not . v not is one plus a r2 over r1 plus r2 and that equals av in . alright so we 're getting close , and our original goal , we want to find v out in terms of v in . so i 'm going to take this whole expression here and divide it over to the other side , so then i have just v not on this side , and v in on the other side . make some more room . i can do that , i can say v not equals a v in divided by this big old expression , one plus a r2 over r1 plus r2 . alright so that 's our answer . that 's the answer . that 's v out equals some function of v in . now i want to make a really important observation here . this is going to be a real cool simplification . okay , so this is the point where op-amp theory gets really cool . watch what happens here . we know that a is a giant number . a is something like 10 to the fifth , or 10 to the sixth , and it 's whatever we have here , if our resistors are sort of normal-sized resistors we know that a giant number times a normal number is still going to be a very big number compared to one . so this one is almost insignificant in this expression down here , so what i 'm going to do , bear with me , i 'm going to cross it out . i 'm going to say no , i do n't need that anymore . so if this , if this number here , if a is a million , 10 to the sixth , and this expression here is something like one half then this total thing is one half 10 to the sixth or a half a million , and that 's huge compared to one . so i can pretty safely ignore the one , it 's very , very small . now when i do that , well look what happens next , now i have a top and bottom in the expression , and i can cancel that too . so the a goes away , now this is pretty astonishing . we have this amplifier circuit and all of a sudden i have an expression here where a does n't appear , the gain does not appear , and what does this turn into ? this is called v not equals v in , times what ? times r1 plus r2 , divided by r2 . so our amplifier , our feedback circuit came down to v out is v in multiplied by the ratio of the resistors that we added to the circuit . this is one of the really cool properties of using op-amps in circuits , really high-gain amplifiers . what we 've done is we have chosen the gain of our circuit based on the components that we picked to add to the amplifier . it 's not determined by the gain of the amplifier as long as the amplifier gain is really , really big . and for op-amps , that 's a good assumption , it is really big . so this expression came out with a positive sign , right ? all the r 's are positive values , so this is referred to as a non-inverting op-amp circuit amplifier . so just to do a quick example , if r1 and r2 are the same , then we end up with an expression that looks like this v out equals r1 plus r2 , r plus r over r is equal to two so the gain is two times v in . so just to do a quick sketch just to remind ourselves what this looks like , this was v in , and we had what out here ? we had a resistor , we had a resistor to the ground , and this is v out . so this is the configuration of a non-inverting amplifier built with an op-amp , the two resistors in this voltage divider string connected to the negative input . so that 's what non-inverting op-amp circuit looks like , and it 's going to be one of the familiar patterns that you see over and over again as you read schematics and you design your own circuits .
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so that gives me , v not plus a v not , times r2 over r1 plus r2 and that equals a times v plus , and actually i can change that now v plus is what ? v plus is v in . okay let 's keep going i can factor out the v not .
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how can you say v+ is vin ?
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okay , now we 're going to work on our first op-amp circuit . here 's what the circuit 's going to look like . watch where it puts the plus sign is on the top on this one . and we 're going to have a voltage source over here . this will be plus or minus v in , that 's our input signal . and over on the output , we 'll have v out , and it 's hooked up this way . the resistor , another resistor , to ground , and this goes back to the inverting input . now we 're going to look at this circuit and see what it does . now we know that connected up here the power supply 's hooked up to these points here , and the ground symbol is zero volts . and we want to analyze this circuit . and what do we know about this ? we know that v out equals some gain , i 'll write the gain right there . a big , big number times v minus , sorry v plus minus v minus , and let 's label that . v plus is this point right here , and v minus is this point right here . and we also know that the currents , let 's call them i plus and i minus , equals zero , and that 's the currents going in here . this is i minus here , and that 's i plus , and we know those are both zero . so now what i want to do it describe what 's going on inside this triangle symbol in more detail by building a circuit model . alright , and a circuit model for an amplifier looks like this . we have v minus here , v plus here , so this is v in , and over on this side we have an , here 's a new symbol that you have n't seen before . it 's usually drawn as a diamond shape , and this is a voltage source , but it 's a special kind of voltage source . it 's called a voltage-dependent voltage source . and it 's the same as a regular ideal voltage source except for one thing , it says that the v , in this case v out , equals gain times v plus minus v minus . so the voltage here depends on the voltage somewhere else , and that 's what makes it a voltage-dependent , that 's what that means . so , we 've just taken our gain expression here , added , drawn circuit diagram that represents our voltage expression for our circuit . now , specifically over here we 've drawn an open circuit on v plus , and v minus so we know that those currents are zero . so this model , this circuit sketch represents our two properties of our op-amp . so i 'm going to take a second here and i 'm going to draw the rest of our circuit surrounding this model , but i need a little bit more space . so let 's put in the rest of our circuit here . we had our voltage source , connected to v plus , and that 's v in , and over here we had v out . let 's check , v out was connected to two resistors , and the bottom is connected to ground , and this was connected there . so what our goal is right now , we want to find v out as a function of v in . that 's what we 're shooting for . so let 's see if we can do that . let 's give our resistors some names . let 's call this r1 , and r2 , our favorite names always , and now everything is labeled . now and we can label this point here , and this point we can call v minus , v minus . so that 's our two unknowns . our unknowns are v not , v out , and v minus , so let 's see if we can find them . so what i 'm going to do is just start writing some expressions for things that i know are true . for example , i know that v out equals a times v plus minus v minus . alright , that 's what this op-amp is telling us is true . now what else do i know ? let 's look at this resistor chain here . this resistor chain actually looks a lot like a voltage divider , and it 's actually a very good voltage divider . remember we said this current here , what is this current here ? it 's zero . i can use the voltage divider expression that i know . in that case , i know that v minus , this is the voltage divider equation , equals v out times what ? times the bottom resistor remember this ? r2 over r1 plus r2 , so the voltage divider expression says that when you have a stack of resistors like this , with the voltage on the top and ground on the bottom , this is the expression for the voltage at the midpoint . kay , so what i 'm going to do next is i 'm going to take this expression and stuff it right in there . let 's do that . see if we got enough room , okay now let 's go over here . now i can say that v out equals a times v plus minus v out times r2 over r1 plus r2 , alright so far so good . let 's keep going , let 's keep working on this . v not equals a times v plus minus a v not , r2 over r1 plus r2 . alright , so now i 'm going to gather all the v not terms over on the left hand side . let 's try that . so that gives me , v not plus a v not , times r2 over r1 plus r2 and that equals a times v plus , and actually i can change that now v plus is what ? v plus is v in . okay let 's keep going i can factor out the v not . v not is one plus a r2 over r1 plus r2 and that equals av in . alright so we 're getting close , and our original goal , we want to find v out in terms of v in . so i 'm going to take this whole expression here and divide it over to the other side , so then i have just v not on this side , and v in on the other side . make some more room . i can do that , i can say v not equals a v in divided by this big old expression , one plus a r2 over r1 plus r2 . alright so that 's our answer . that 's the answer . that 's v out equals some function of v in . now i want to make a really important observation here . this is going to be a real cool simplification . okay , so this is the point where op-amp theory gets really cool . watch what happens here . we know that a is a giant number . a is something like 10 to the fifth , or 10 to the sixth , and it 's whatever we have here , if our resistors are sort of normal-sized resistors we know that a giant number times a normal number is still going to be a very big number compared to one . so this one is almost insignificant in this expression down here , so what i 'm going to do , bear with me , i 'm going to cross it out . i 'm going to say no , i do n't need that anymore . so if this , if this number here , if a is a million , 10 to the sixth , and this expression here is something like one half then this total thing is one half 10 to the sixth or a half a million , and that 's huge compared to one . so i can pretty safely ignore the one , it 's very , very small . now when i do that , well look what happens next , now i have a top and bottom in the expression , and i can cancel that too . so the a goes away , now this is pretty astonishing . we have this amplifier circuit and all of a sudden i have an expression here where a does n't appear , the gain does not appear , and what does this turn into ? this is called v not equals v in , times what ? times r1 plus r2 , divided by r2 . so our amplifier , our feedback circuit came down to v out is v in multiplied by the ratio of the resistors that we added to the circuit . this is one of the really cool properties of using op-amps in circuits , really high-gain amplifiers . what we 've done is we have chosen the gain of our circuit based on the components that we picked to add to the amplifier . it 's not determined by the gain of the amplifier as long as the amplifier gain is really , really big . and for op-amps , that 's a good assumption , it is really big . so this expression came out with a positive sign , right ? all the r 's are positive values , so this is referred to as a non-inverting op-amp circuit amplifier . so just to do a quick example , if r1 and r2 are the same , then we end up with an expression that looks like this v out equals r1 plus r2 , r plus r over r is equal to two so the gain is two times v in . so just to do a quick sketch just to remind ourselves what this looks like , this was v in , and we had what out here ? we had a resistor , we had a resistor to the ground , and this is v out . so this is the configuration of a non-inverting amplifier built with an op-amp , the two resistors in this voltage divider string connected to the negative input . so that 's what non-inverting op-amp circuit looks like , and it 's going to be one of the familiar patterns that you see over and over again as you read schematics and you design your own circuits .
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so that gives me , v not plus a v not , times r2 over r1 plus r2 and that equals a times v plus , and actually i can change that now v plus is what ? v plus is v in . okay let 's keep going i can factor out the v not .
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why cant the inverting input ( v- ) simply be connected to v_out ?
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okay , now we 're going to work on our first op-amp circuit . here 's what the circuit 's going to look like . watch where it puts the plus sign is on the top on this one . and we 're going to have a voltage source over here . this will be plus or minus v in , that 's our input signal . and over on the output , we 'll have v out , and it 's hooked up this way . the resistor , another resistor , to ground , and this goes back to the inverting input . now we 're going to look at this circuit and see what it does . now we know that connected up here the power supply 's hooked up to these points here , and the ground symbol is zero volts . and we want to analyze this circuit . and what do we know about this ? we know that v out equals some gain , i 'll write the gain right there . a big , big number times v minus , sorry v plus minus v minus , and let 's label that . v plus is this point right here , and v minus is this point right here . and we also know that the currents , let 's call them i plus and i minus , equals zero , and that 's the currents going in here . this is i minus here , and that 's i plus , and we know those are both zero . so now what i want to do it describe what 's going on inside this triangle symbol in more detail by building a circuit model . alright , and a circuit model for an amplifier looks like this . we have v minus here , v plus here , so this is v in , and over on this side we have an , here 's a new symbol that you have n't seen before . it 's usually drawn as a diamond shape , and this is a voltage source , but it 's a special kind of voltage source . it 's called a voltage-dependent voltage source . and it 's the same as a regular ideal voltage source except for one thing , it says that the v , in this case v out , equals gain times v plus minus v minus . so the voltage here depends on the voltage somewhere else , and that 's what makes it a voltage-dependent , that 's what that means . so , we 've just taken our gain expression here , added , drawn circuit diagram that represents our voltage expression for our circuit . now , specifically over here we 've drawn an open circuit on v plus , and v minus so we know that those currents are zero . so this model , this circuit sketch represents our two properties of our op-amp . so i 'm going to take a second here and i 'm going to draw the rest of our circuit surrounding this model , but i need a little bit more space . so let 's put in the rest of our circuit here . we had our voltage source , connected to v plus , and that 's v in , and over here we had v out . let 's check , v out was connected to two resistors , and the bottom is connected to ground , and this was connected there . so what our goal is right now , we want to find v out as a function of v in . that 's what we 're shooting for . so let 's see if we can do that . let 's give our resistors some names . let 's call this r1 , and r2 , our favorite names always , and now everything is labeled . now and we can label this point here , and this point we can call v minus , v minus . so that 's our two unknowns . our unknowns are v not , v out , and v minus , so let 's see if we can find them . so what i 'm going to do is just start writing some expressions for things that i know are true . for example , i know that v out equals a times v plus minus v minus . alright , that 's what this op-amp is telling us is true . now what else do i know ? let 's look at this resistor chain here . this resistor chain actually looks a lot like a voltage divider , and it 's actually a very good voltage divider . remember we said this current here , what is this current here ? it 's zero . i can use the voltage divider expression that i know . in that case , i know that v minus , this is the voltage divider equation , equals v out times what ? times the bottom resistor remember this ? r2 over r1 plus r2 , so the voltage divider expression says that when you have a stack of resistors like this , with the voltage on the top and ground on the bottom , this is the expression for the voltage at the midpoint . kay , so what i 'm going to do next is i 'm going to take this expression and stuff it right in there . let 's do that . see if we got enough room , okay now let 's go over here . now i can say that v out equals a times v plus minus v out times r2 over r1 plus r2 , alright so far so good . let 's keep going , let 's keep working on this . v not equals a times v plus minus a v not , r2 over r1 plus r2 . alright , so now i 'm going to gather all the v not terms over on the left hand side . let 's try that . so that gives me , v not plus a v not , times r2 over r1 plus r2 and that equals a times v plus , and actually i can change that now v plus is what ? v plus is v in . okay let 's keep going i can factor out the v not . v not is one plus a r2 over r1 plus r2 and that equals av in . alright so we 're getting close , and our original goal , we want to find v out in terms of v in . so i 'm going to take this whole expression here and divide it over to the other side , so then i have just v not on this side , and v in on the other side . make some more room . i can do that , i can say v not equals a v in divided by this big old expression , one plus a r2 over r1 plus r2 . alright so that 's our answer . that 's the answer . that 's v out equals some function of v in . now i want to make a really important observation here . this is going to be a real cool simplification . okay , so this is the point where op-amp theory gets really cool . watch what happens here . we know that a is a giant number . a is something like 10 to the fifth , or 10 to the sixth , and it 's whatever we have here , if our resistors are sort of normal-sized resistors we know that a giant number times a normal number is still going to be a very big number compared to one . so this one is almost insignificant in this expression down here , so what i 'm going to do , bear with me , i 'm going to cross it out . i 'm going to say no , i do n't need that anymore . so if this , if this number here , if a is a million , 10 to the sixth , and this expression here is something like one half then this total thing is one half 10 to the sixth or a half a million , and that 's huge compared to one . so i can pretty safely ignore the one , it 's very , very small . now when i do that , well look what happens next , now i have a top and bottom in the expression , and i can cancel that too . so the a goes away , now this is pretty astonishing . we have this amplifier circuit and all of a sudden i have an expression here where a does n't appear , the gain does not appear , and what does this turn into ? this is called v not equals v in , times what ? times r1 plus r2 , divided by r2 . so our amplifier , our feedback circuit came down to v out is v in multiplied by the ratio of the resistors that we added to the circuit . this is one of the really cool properties of using op-amps in circuits , really high-gain amplifiers . what we 've done is we have chosen the gain of our circuit based on the components that we picked to add to the amplifier . it 's not determined by the gain of the amplifier as long as the amplifier gain is really , really big . and for op-amps , that 's a good assumption , it is really big . so this expression came out with a positive sign , right ? all the r 's are positive values , so this is referred to as a non-inverting op-amp circuit amplifier . so just to do a quick example , if r1 and r2 are the same , then we end up with an expression that looks like this v out equals r1 plus r2 , r plus r over r is equal to two so the gain is two times v in . so just to do a quick sketch just to remind ourselves what this looks like , this was v in , and we had what out here ? we had a resistor , we had a resistor to the ground , and this is v out . so this is the configuration of a non-inverting amplifier built with an op-amp , the two resistors in this voltage divider string connected to the negative input . so that 's what non-inverting op-amp circuit looks like , and it 's going to be one of the familiar patterns that you see over and over again as you read schematics and you design your own circuits .
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so let 's see if we can do that . let 's give our resistors some names . let 's call this r1 , and r2 , our favorite names always , and now everything is labeled .
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why do we need tgose extra resistors ?
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okay , now we 're going to work on our first op-amp circuit . here 's what the circuit 's going to look like . watch where it puts the plus sign is on the top on this one . and we 're going to have a voltage source over here . this will be plus or minus v in , that 's our input signal . and over on the output , we 'll have v out , and it 's hooked up this way . the resistor , another resistor , to ground , and this goes back to the inverting input . now we 're going to look at this circuit and see what it does . now we know that connected up here the power supply 's hooked up to these points here , and the ground symbol is zero volts . and we want to analyze this circuit . and what do we know about this ? we know that v out equals some gain , i 'll write the gain right there . a big , big number times v minus , sorry v plus minus v minus , and let 's label that . v plus is this point right here , and v minus is this point right here . and we also know that the currents , let 's call them i plus and i minus , equals zero , and that 's the currents going in here . this is i minus here , and that 's i plus , and we know those are both zero . so now what i want to do it describe what 's going on inside this triangle symbol in more detail by building a circuit model . alright , and a circuit model for an amplifier looks like this . we have v minus here , v plus here , so this is v in , and over on this side we have an , here 's a new symbol that you have n't seen before . it 's usually drawn as a diamond shape , and this is a voltage source , but it 's a special kind of voltage source . it 's called a voltage-dependent voltage source . and it 's the same as a regular ideal voltage source except for one thing , it says that the v , in this case v out , equals gain times v plus minus v minus . so the voltage here depends on the voltage somewhere else , and that 's what makes it a voltage-dependent , that 's what that means . so , we 've just taken our gain expression here , added , drawn circuit diagram that represents our voltage expression for our circuit . now , specifically over here we 've drawn an open circuit on v plus , and v minus so we know that those currents are zero . so this model , this circuit sketch represents our two properties of our op-amp . so i 'm going to take a second here and i 'm going to draw the rest of our circuit surrounding this model , but i need a little bit more space . so let 's put in the rest of our circuit here . we had our voltage source , connected to v plus , and that 's v in , and over here we had v out . let 's check , v out was connected to two resistors , and the bottom is connected to ground , and this was connected there . so what our goal is right now , we want to find v out as a function of v in . that 's what we 're shooting for . so let 's see if we can do that . let 's give our resistors some names . let 's call this r1 , and r2 , our favorite names always , and now everything is labeled . now and we can label this point here , and this point we can call v minus , v minus . so that 's our two unknowns . our unknowns are v not , v out , and v minus , so let 's see if we can find them . so what i 'm going to do is just start writing some expressions for things that i know are true . for example , i know that v out equals a times v plus minus v minus . alright , that 's what this op-amp is telling us is true . now what else do i know ? let 's look at this resistor chain here . this resistor chain actually looks a lot like a voltage divider , and it 's actually a very good voltage divider . remember we said this current here , what is this current here ? it 's zero . i can use the voltage divider expression that i know . in that case , i know that v minus , this is the voltage divider equation , equals v out times what ? times the bottom resistor remember this ? r2 over r1 plus r2 , so the voltage divider expression says that when you have a stack of resistors like this , with the voltage on the top and ground on the bottom , this is the expression for the voltage at the midpoint . kay , so what i 'm going to do next is i 'm going to take this expression and stuff it right in there . let 's do that . see if we got enough room , okay now let 's go over here . now i can say that v out equals a times v plus minus v out times r2 over r1 plus r2 , alright so far so good . let 's keep going , let 's keep working on this . v not equals a times v plus minus a v not , r2 over r1 plus r2 . alright , so now i 'm going to gather all the v not terms over on the left hand side . let 's try that . so that gives me , v not plus a v not , times r2 over r1 plus r2 and that equals a times v plus , and actually i can change that now v plus is what ? v plus is v in . okay let 's keep going i can factor out the v not . v not is one plus a r2 over r1 plus r2 and that equals av in . alright so we 're getting close , and our original goal , we want to find v out in terms of v in . so i 'm going to take this whole expression here and divide it over to the other side , so then i have just v not on this side , and v in on the other side . make some more room . i can do that , i can say v not equals a v in divided by this big old expression , one plus a r2 over r1 plus r2 . alright so that 's our answer . that 's the answer . that 's v out equals some function of v in . now i want to make a really important observation here . this is going to be a real cool simplification . okay , so this is the point where op-amp theory gets really cool . watch what happens here . we know that a is a giant number . a is something like 10 to the fifth , or 10 to the sixth , and it 's whatever we have here , if our resistors are sort of normal-sized resistors we know that a giant number times a normal number is still going to be a very big number compared to one . so this one is almost insignificant in this expression down here , so what i 'm going to do , bear with me , i 'm going to cross it out . i 'm going to say no , i do n't need that anymore . so if this , if this number here , if a is a million , 10 to the sixth , and this expression here is something like one half then this total thing is one half 10 to the sixth or a half a million , and that 's huge compared to one . so i can pretty safely ignore the one , it 's very , very small . now when i do that , well look what happens next , now i have a top and bottom in the expression , and i can cancel that too . so the a goes away , now this is pretty astonishing . we have this amplifier circuit and all of a sudden i have an expression here where a does n't appear , the gain does not appear , and what does this turn into ? this is called v not equals v in , times what ? times r1 plus r2 , divided by r2 . so our amplifier , our feedback circuit came down to v out is v in multiplied by the ratio of the resistors that we added to the circuit . this is one of the really cool properties of using op-amps in circuits , really high-gain amplifiers . what we 've done is we have chosen the gain of our circuit based on the components that we picked to add to the amplifier . it 's not determined by the gain of the amplifier as long as the amplifier gain is really , really big . and for op-amps , that 's a good assumption , it is really big . so this expression came out with a positive sign , right ? all the r 's are positive values , so this is referred to as a non-inverting op-amp circuit amplifier . so just to do a quick example , if r1 and r2 are the same , then we end up with an expression that looks like this v out equals r1 plus r2 , r plus r over r is equal to two so the gain is two times v in . so just to do a quick sketch just to remind ourselves what this looks like , this was v in , and we had what out here ? we had a resistor , we had a resistor to the ground , and this is v out . so this is the configuration of a non-inverting amplifier built with an op-amp , the two resistors in this voltage divider string connected to the negative input . so that 's what non-inverting op-amp circuit looks like , and it 's going to be one of the familiar patterns that you see over and over again as you read schematics and you design your own circuits .
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this is called v not equals v in , times what ? times r1 plus r2 , divided by r2 . so our amplifier , our feedback circuit came down to v out is v in multiplied by the ratio of the resistors that we added to the circuit .
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why ca n't the voltage divider equation be vo = v in* ( r2 ) /r1+r2 ?
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okay , now we 're going to work on our first op-amp circuit . here 's what the circuit 's going to look like . watch where it puts the plus sign is on the top on this one . and we 're going to have a voltage source over here . this will be plus or minus v in , that 's our input signal . and over on the output , we 'll have v out , and it 's hooked up this way . the resistor , another resistor , to ground , and this goes back to the inverting input . now we 're going to look at this circuit and see what it does . now we know that connected up here the power supply 's hooked up to these points here , and the ground symbol is zero volts . and we want to analyze this circuit . and what do we know about this ? we know that v out equals some gain , i 'll write the gain right there . a big , big number times v minus , sorry v plus minus v minus , and let 's label that . v plus is this point right here , and v minus is this point right here . and we also know that the currents , let 's call them i plus and i minus , equals zero , and that 's the currents going in here . this is i minus here , and that 's i plus , and we know those are both zero . so now what i want to do it describe what 's going on inside this triangle symbol in more detail by building a circuit model . alright , and a circuit model for an amplifier looks like this . we have v minus here , v plus here , so this is v in , and over on this side we have an , here 's a new symbol that you have n't seen before . it 's usually drawn as a diamond shape , and this is a voltage source , but it 's a special kind of voltage source . it 's called a voltage-dependent voltage source . and it 's the same as a regular ideal voltage source except for one thing , it says that the v , in this case v out , equals gain times v plus minus v minus . so the voltage here depends on the voltage somewhere else , and that 's what makes it a voltage-dependent , that 's what that means . so , we 've just taken our gain expression here , added , drawn circuit diagram that represents our voltage expression for our circuit . now , specifically over here we 've drawn an open circuit on v plus , and v minus so we know that those currents are zero . so this model , this circuit sketch represents our two properties of our op-amp . so i 'm going to take a second here and i 'm going to draw the rest of our circuit surrounding this model , but i need a little bit more space . so let 's put in the rest of our circuit here . we had our voltage source , connected to v plus , and that 's v in , and over here we had v out . let 's check , v out was connected to two resistors , and the bottom is connected to ground , and this was connected there . so what our goal is right now , we want to find v out as a function of v in . that 's what we 're shooting for . so let 's see if we can do that . let 's give our resistors some names . let 's call this r1 , and r2 , our favorite names always , and now everything is labeled . now and we can label this point here , and this point we can call v minus , v minus . so that 's our two unknowns . our unknowns are v not , v out , and v minus , so let 's see if we can find them . so what i 'm going to do is just start writing some expressions for things that i know are true . for example , i know that v out equals a times v plus minus v minus . alright , that 's what this op-amp is telling us is true . now what else do i know ? let 's look at this resistor chain here . this resistor chain actually looks a lot like a voltage divider , and it 's actually a very good voltage divider . remember we said this current here , what is this current here ? it 's zero . i can use the voltage divider expression that i know . in that case , i know that v minus , this is the voltage divider equation , equals v out times what ? times the bottom resistor remember this ? r2 over r1 plus r2 , so the voltage divider expression says that when you have a stack of resistors like this , with the voltage on the top and ground on the bottom , this is the expression for the voltage at the midpoint . kay , so what i 'm going to do next is i 'm going to take this expression and stuff it right in there . let 's do that . see if we got enough room , okay now let 's go over here . now i can say that v out equals a times v plus minus v out times r2 over r1 plus r2 , alright so far so good . let 's keep going , let 's keep working on this . v not equals a times v plus minus a v not , r2 over r1 plus r2 . alright , so now i 'm going to gather all the v not terms over on the left hand side . let 's try that . so that gives me , v not plus a v not , times r2 over r1 plus r2 and that equals a times v plus , and actually i can change that now v plus is what ? v plus is v in . okay let 's keep going i can factor out the v not . v not is one plus a r2 over r1 plus r2 and that equals av in . alright so we 're getting close , and our original goal , we want to find v out in terms of v in . so i 'm going to take this whole expression here and divide it over to the other side , so then i have just v not on this side , and v in on the other side . make some more room . i can do that , i can say v not equals a v in divided by this big old expression , one plus a r2 over r1 plus r2 . alright so that 's our answer . that 's the answer . that 's v out equals some function of v in . now i want to make a really important observation here . this is going to be a real cool simplification . okay , so this is the point where op-amp theory gets really cool . watch what happens here . we know that a is a giant number . a is something like 10 to the fifth , or 10 to the sixth , and it 's whatever we have here , if our resistors are sort of normal-sized resistors we know that a giant number times a normal number is still going to be a very big number compared to one . so this one is almost insignificant in this expression down here , so what i 'm going to do , bear with me , i 'm going to cross it out . i 'm going to say no , i do n't need that anymore . so if this , if this number here , if a is a million , 10 to the sixth , and this expression here is something like one half then this total thing is one half 10 to the sixth or a half a million , and that 's huge compared to one . so i can pretty safely ignore the one , it 's very , very small . now when i do that , well look what happens next , now i have a top and bottom in the expression , and i can cancel that too . so the a goes away , now this is pretty astonishing . we have this amplifier circuit and all of a sudden i have an expression here where a does n't appear , the gain does not appear , and what does this turn into ? this is called v not equals v in , times what ? times r1 plus r2 , divided by r2 . so our amplifier , our feedback circuit came down to v out is v in multiplied by the ratio of the resistors that we added to the circuit . this is one of the really cool properties of using op-amps in circuits , really high-gain amplifiers . what we 've done is we have chosen the gain of our circuit based on the components that we picked to add to the amplifier . it 's not determined by the gain of the amplifier as long as the amplifier gain is really , really big . and for op-amps , that 's a good assumption , it is really big . so this expression came out with a positive sign , right ? all the r 's are positive values , so this is referred to as a non-inverting op-amp circuit amplifier . so just to do a quick example , if r1 and r2 are the same , then we end up with an expression that looks like this v out equals r1 plus r2 , r plus r over r is equal to two so the gain is two times v in . so just to do a quick sketch just to remind ourselves what this looks like , this was v in , and we had what out here ? we had a resistor , we had a resistor to the ground , and this is v out . so this is the configuration of a non-inverting amplifier built with an op-amp , the two resistors in this voltage divider string connected to the negative input . so that 's what non-inverting op-amp circuit looks like , and it 's going to be one of the familiar patterns that you see over and over again as you read schematics and you design your own circuits .
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this is called v not equals v in , times what ? times r1 plus r2 , divided by r2 . so our amplifier , our feedback circuit came down to v out is v in multiplied by the ratio of the resistors that we added to the circuit .
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why does r2 go on the numerator and not r1 ?
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okay , now we 're going to work on our first op-amp circuit . here 's what the circuit 's going to look like . watch where it puts the plus sign is on the top on this one . and we 're going to have a voltage source over here . this will be plus or minus v in , that 's our input signal . and over on the output , we 'll have v out , and it 's hooked up this way . the resistor , another resistor , to ground , and this goes back to the inverting input . now we 're going to look at this circuit and see what it does . now we know that connected up here the power supply 's hooked up to these points here , and the ground symbol is zero volts . and we want to analyze this circuit . and what do we know about this ? we know that v out equals some gain , i 'll write the gain right there . a big , big number times v minus , sorry v plus minus v minus , and let 's label that . v plus is this point right here , and v minus is this point right here . and we also know that the currents , let 's call them i plus and i minus , equals zero , and that 's the currents going in here . this is i minus here , and that 's i plus , and we know those are both zero . so now what i want to do it describe what 's going on inside this triangle symbol in more detail by building a circuit model . alright , and a circuit model for an amplifier looks like this . we have v minus here , v plus here , so this is v in , and over on this side we have an , here 's a new symbol that you have n't seen before . it 's usually drawn as a diamond shape , and this is a voltage source , but it 's a special kind of voltage source . it 's called a voltage-dependent voltage source . and it 's the same as a regular ideal voltage source except for one thing , it says that the v , in this case v out , equals gain times v plus minus v minus . so the voltage here depends on the voltage somewhere else , and that 's what makes it a voltage-dependent , that 's what that means . so , we 've just taken our gain expression here , added , drawn circuit diagram that represents our voltage expression for our circuit . now , specifically over here we 've drawn an open circuit on v plus , and v minus so we know that those currents are zero . so this model , this circuit sketch represents our two properties of our op-amp . so i 'm going to take a second here and i 'm going to draw the rest of our circuit surrounding this model , but i need a little bit more space . so let 's put in the rest of our circuit here . we had our voltage source , connected to v plus , and that 's v in , and over here we had v out . let 's check , v out was connected to two resistors , and the bottom is connected to ground , and this was connected there . so what our goal is right now , we want to find v out as a function of v in . that 's what we 're shooting for . so let 's see if we can do that . let 's give our resistors some names . let 's call this r1 , and r2 , our favorite names always , and now everything is labeled . now and we can label this point here , and this point we can call v minus , v minus . so that 's our two unknowns . our unknowns are v not , v out , and v minus , so let 's see if we can find them . so what i 'm going to do is just start writing some expressions for things that i know are true . for example , i know that v out equals a times v plus minus v minus . alright , that 's what this op-amp is telling us is true . now what else do i know ? let 's look at this resistor chain here . this resistor chain actually looks a lot like a voltage divider , and it 's actually a very good voltage divider . remember we said this current here , what is this current here ? it 's zero . i can use the voltage divider expression that i know . in that case , i know that v minus , this is the voltage divider equation , equals v out times what ? times the bottom resistor remember this ? r2 over r1 plus r2 , so the voltage divider expression says that when you have a stack of resistors like this , with the voltage on the top and ground on the bottom , this is the expression for the voltage at the midpoint . kay , so what i 'm going to do next is i 'm going to take this expression and stuff it right in there . let 's do that . see if we got enough room , okay now let 's go over here . now i can say that v out equals a times v plus minus v out times r2 over r1 plus r2 , alright so far so good . let 's keep going , let 's keep working on this . v not equals a times v plus minus a v not , r2 over r1 plus r2 . alright , so now i 'm going to gather all the v not terms over on the left hand side . let 's try that . so that gives me , v not plus a v not , times r2 over r1 plus r2 and that equals a times v plus , and actually i can change that now v plus is what ? v plus is v in . okay let 's keep going i can factor out the v not . v not is one plus a r2 over r1 plus r2 and that equals av in . alright so we 're getting close , and our original goal , we want to find v out in terms of v in . so i 'm going to take this whole expression here and divide it over to the other side , so then i have just v not on this side , and v in on the other side . make some more room . i can do that , i can say v not equals a v in divided by this big old expression , one plus a r2 over r1 plus r2 . alright so that 's our answer . that 's the answer . that 's v out equals some function of v in . now i want to make a really important observation here . this is going to be a real cool simplification . okay , so this is the point where op-amp theory gets really cool . watch what happens here . we know that a is a giant number . a is something like 10 to the fifth , or 10 to the sixth , and it 's whatever we have here , if our resistors are sort of normal-sized resistors we know that a giant number times a normal number is still going to be a very big number compared to one . so this one is almost insignificant in this expression down here , so what i 'm going to do , bear with me , i 'm going to cross it out . i 'm going to say no , i do n't need that anymore . so if this , if this number here , if a is a million , 10 to the sixth , and this expression here is something like one half then this total thing is one half 10 to the sixth or a half a million , and that 's huge compared to one . so i can pretty safely ignore the one , it 's very , very small . now when i do that , well look what happens next , now i have a top and bottom in the expression , and i can cancel that too . so the a goes away , now this is pretty astonishing . we have this amplifier circuit and all of a sudden i have an expression here where a does n't appear , the gain does not appear , and what does this turn into ? this is called v not equals v in , times what ? times r1 plus r2 , divided by r2 . so our amplifier , our feedback circuit came down to v out is v in multiplied by the ratio of the resistors that we added to the circuit . this is one of the really cool properties of using op-amps in circuits , really high-gain amplifiers . what we 've done is we have chosen the gain of our circuit based on the components that we picked to add to the amplifier . it 's not determined by the gain of the amplifier as long as the amplifier gain is really , really big . and for op-amps , that 's a good assumption , it is really big . so this expression came out with a positive sign , right ? all the r 's are positive values , so this is referred to as a non-inverting op-amp circuit amplifier . so just to do a quick example , if r1 and r2 are the same , then we end up with an expression that looks like this v out equals r1 plus r2 , r plus r over r is equal to two so the gain is two times v in . so just to do a quick sketch just to remind ourselves what this looks like , this was v in , and we had what out here ? we had a resistor , we had a resistor to the ground , and this is v out . so this is the configuration of a non-inverting amplifier built with an op-amp , the two resistors in this voltage divider string connected to the negative input . so that 's what non-inverting op-amp circuit looks like , and it 's going to be one of the familiar patterns that you see over and over again as you read schematics and you design your own circuits .
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alright so we 're getting close , and our original goal , we want to find v out in terms of v in . so i 'm going to take this whole expression here and divide it over to the other side , so then i have just v not on this side , and v in on the other side . make some more room .
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if the + side of a power supply is hooked to a circuit and the - side is simply grounded , how can we expect it to work ?
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okay , now we 're going to work on our first op-amp circuit . here 's what the circuit 's going to look like . watch where it puts the plus sign is on the top on this one . and we 're going to have a voltage source over here . this will be plus or minus v in , that 's our input signal . and over on the output , we 'll have v out , and it 's hooked up this way . the resistor , another resistor , to ground , and this goes back to the inverting input . now we 're going to look at this circuit and see what it does . now we know that connected up here the power supply 's hooked up to these points here , and the ground symbol is zero volts . and we want to analyze this circuit . and what do we know about this ? we know that v out equals some gain , i 'll write the gain right there . a big , big number times v minus , sorry v plus minus v minus , and let 's label that . v plus is this point right here , and v minus is this point right here . and we also know that the currents , let 's call them i plus and i minus , equals zero , and that 's the currents going in here . this is i minus here , and that 's i plus , and we know those are both zero . so now what i want to do it describe what 's going on inside this triangle symbol in more detail by building a circuit model . alright , and a circuit model for an amplifier looks like this . we have v minus here , v plus here , so this is v in , and over on this side we have an , here 's a new symbol that you have n't seen before . it 's usually drawn as a diamond shape , and this is a voltage source , but it 's a special kind of voltage source . it 's called a voltage-dependent voltage source . and it 's the same as a regular ideal voltage source except for one thing , it says that the v , in this case v out , equals gain times v plus minus v minus . so the voltage here depends on the voltage somewhere else , and that 's what makes it a voltage-dependent , that 's what that means . so , we 've just taken our gain expression here , added , drawn circuit diagram that represents our voltage expression for our circuit . now , specifically over here we 've drawn an open circuit on v plus , and v minus so we know that those currents are zero . so this model , this circuit sketch represents our two properties of our op-amp . so i 'm going to take a second here and i 'm going to draw the rest of our circuit surrounding this model , but i need a little bit more space . so let 's put in the rest of our circuit here . we had our voltage source , connected to v plus , and that 's v in , and over here we had v out . let 's check , v out was connected to two resistors , and the bottom is connected to ground , and this was connected there . so what our goal is right now , we want to find v out as a function of v in . that 's what we 're shooting for . so let 's see if we can do that . let 's give our resistors some names . let 's call this r1 , and r2 , our favorite names always , and now everything is labeled . now and we can label this point here , and this point we can call v minus , v minus . so that 's our two unknowns . our unknowns are v not , v out , and v minus , so let 's see if we can find them . so what i 'm going to do is just start writing some expressions for things that i know are true . for example , i know that v out equals a times v plus minus v minus . alright , that 's what this op-amp is telling us is true . now what else do i know ? let 's look at this resistor chain here . this resistor chain actually looks a lot like a voltage divider , and it 's actually a very good voltage divider . remember we said this current here , what is this current here ? it 's zero . i can use the voltage divider expression that i know . in that case , i know that v minus , this is the voltage divider equation , equals v out times what ? times the bottom resistor remember this ? r2 over r1 plus r2 , so the voltage divider expression says that when you have a stack of resistors like this , with the voltage on the top and ground on the bottom , this is the expression for the voltage at the midpoint . kay , so what i 'm going to do next is i 'm going to take this expression and stuff it right in there . let 's do that . see if we got enough room , okay now let 's go over here . now i can say that v out equals a times v plus minus v out times r2 over r1 plus r2 , alright so far so good . let 's keep going , let 's keep working on this . v not equals a times v plus minus a v not , r2 over r1 plus r2 . alright , so now i 'm going to gather all the v not terms over on the left hand side . let 's try that . so that gives me , v not plus a v not , times r2 over r1 plus r2 and that equals a times v plus , and actually i can change that now v plus is what ? v plus is v in . okay let 's keep going i can factor out the v not . v not is one plus a r2 over r1 plus r2 and that equals av in . alright so we 're getting close , and our original goal , we want to find v out in terms of v in . so i 'm going to take this whole expression here and divide it over to the other side , so then i have just v not on this side , and v in on the other side . make some more room . i can do that , i can say v not equals a v in divided by this big old expression , one plus a r2 over r1 plus r2 . alright so that 's our answer . that 's the answer . that 's v out equals some function of v in . now i want to make a really important observation here . this is going to be a real cool simplification . okay , so this is the point where op-amp theory gets really cool . watch what happens here . we know that a is a giant number . a is something like 10 to the fifth , or 10 to the sixth , and it 's whatever we have here , if our resistors are sort of normal-sized resistors we know that a giant number times a normal number is still going to be a very big number compared to one . so this one is almost insignificant in this expression down here , so what i 'm going to do , bear with me , i 'm going to cross it out . i 'm going to say no , i do n't need that anymore . so if this , if this number here , if a is a million , 10 to the sixth , and this expression here is something like one half then this total thing is one half 10 to the sixth or a half a million , and that 's huge compared to one . so i can pretty safely ignore the one , it 's very , very small . now when i do that , well look what happens next , now i have a top and bottom in the expression , and i can cancel that too . so the a goes away , now this is pretty astonishing . we have this amplifier circuit and all of a sudden i have an expression here where a does n't appear , the gain does not appear , and what does this turn into ? this is called v not equals v in , times what ? times r1 plus r2 , divided by r2 . so our amplifier , our feedback circuit came down to v out is v in multiplied by the ratio of the resistors that we added to the circuit . this is one of the really cool properties of using op-amps in circuits , really high-gain amplifiers . what we 've done is we have chosen the gain of our circuit based on the components that we picked to add to the amplifier . it 's not determined by the gain of the amplifier as long as the amplifier gain is really , really big . and for op-amps , that 's a good assumption , it is really big . so this expression came out with a positive sign , right ? all the r 's are positive values , so this is referred to as a non-inverting op-amp circuit amplifier . so just to do a quick example , if r1 and r2 are the same , then we end up with an expression that looks like this v out equals r1 plus r2 , r plus r over r is equal to two so the gain is two times v in . so just to do a quick sketch just to remind ourselves what this looks like , this was v in , and we had what out here ? we had a resistor , we had a resistor to the ground , and this is v out . so this is the configuration of a non-inverting amplifier built with an op-amp , the two resistors in this voltage divider string connected to the negative input . so that 's what non-inverting op-amp circuit looks like , and it 's going to be one of the familiar patterns that you see over and over again as you read schematics and you design your own circuits .
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so this expression came out with a positive sign , right ? all the r 's are positive values , so this is referred to as a non-inverting op-amp circuit amplifier . so just to do a quick example , if r1 and r2 are the same , then we end up with an expression that looks like this v out equals r1 plus r2 , r plus r over r is equal to two so the gain is two times v in .
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without those extra resistors will the gain of the non-inverting op amp be 1 ?
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okay , now we 're going to work on our first op-amp circuit . here 's what the circuit 's going to look like . watch where it puts the plus sign is on the top on this one . and we 're going to have a voltage source over here . this will be plus or minus v in , that 's our input signal . and over on the output , we 'll have v out , and it 's hooked up this way . the resistor , another resistor , to ground , and this goes back to the inverting input . now we 're going to look at this circuit and see what it does . now we know that connected up here the power supply 's hooked up to these points here , and the ground symbol is zero volts . and we want to analyze this circuit . and what do we know about this ? we know that v out equals some gain , i 'll write the gain right there . a big , big number times v minus , sorry v plus minus v minus , and let 's label that . v plus is this point right here , and v minus is this point right here . and we also know that the currents , let 's call them i plus and i minus , equals zero , and that 's the currents going in here . this is i minus here , and that 's i plus , and we know those are both zero . so now what i want to do it describe what 's going on inside this triangle symbol in more detail by building a circuit model . alright , and a circuit model for an amplifier looks like this . we have v minus here , v plus here , so this is v in , and over on this side we have an , here 's a new symbol that you have n't seen before . it 's usually drawn as a diamond shape , and this is a voltage source , but it 's a special kind of voltage source . it 's called a voltage-dependent voltage source . and it 's the same as a regular ideal voltage source except for one thing , it says that the v , in this case v out , equals gain times v plus minus v minus . so the voltage here depends on the voltage somewhere else , and that 's what makes it a voltage-dependent , that 's what that means . so , we 've just taken our gain expression here , added , drawn circuit diagram that represents our voltage expression for our circuit . now , specifically over here we 've drawn an open circuit on v plus , and v minus so we know that those currents are zero . so this model , this circuit sketch represents our two properties of our op-amp . so i 'm going to take a second here and i 'm going to draw the rest of our circuit surrounding this model , but i need a little bit more space . so let 's put in the rest of our circuit here . we had our voltage source , connected to v plus , and that 's v in , and over here we had v out . let 's check , v out was connected to two resistors , and the bottom is connected to ground , and this was connected there . so what our goal is right now , we want to find v out as a function of v in . that 's what we 're shooting for . so let 's see if we can do that . let 's give our resistors some names . let 's call this r1 , and r2 , our favorite names always , and now everything is labeled . now and we can label this point here , and this point we can call v minus , v minus . so that 's our two unknowns . our unknowns are v not , v out , and v minus , so let 's see if we can find them . so what i 'm going to do is just start writing some expressions for things that i know are true . for example , i know that v out equals a times v plus minus v minus . alright , that 's what this op-amp is telling us is true . now what else do i know ? let 's look at this resistor chain here . this resistor chain actually looks a lot like a voltage divider , and it 's actually a very good voltage divider . remember we said this current here , what is this current here ? it 's zero . i can use the voltage divider expression that i know . in that case , i know that v minus , this is the voltage divider equation , equals v out times what ? times the bottom resistor remember this ? r2 over r1 plus r2 , so the voltage divider expression says that when you have a stack of resistors like this , with the voltage on the top and ground on the bottom , this is the expression for the voltage at the midpoint . kay , so what i 'm going to do next is i 'm going to take this expression and stuff it right in there . let 's do that . see if we got enough room , okay now let 's go over here . now i can say that v out equals a times v plus minus v out times r2 over r1 plus r2 , alright so far so good . let 's keep going , let 's keep working on this . v not equals a times v plus minus a v not , r2 over r1 plus r2 . alright , so now i 'm going to gather all the v not terms over on the left hand side . let 's try that . so that gives me , v not plus a v not , times r2 over r1 plus r2 and that equals a times v plus , and actually i can change that now v plus is what ? v plus is v in . okay let 's keep going i can factor out the v not . v not is one plus a r2 over r1 plus r2 and that equals av in . alright so we 're getting close , and our original goal , we want to find v out in terms of v in . so i 'm going to take this whole expression here and divide it over to the other side , so then i have just v not on this side , and v in on the other side . make some more room . i can do that , i can say v not equals a v in divided by this big old expression , one plus a r2 over r1 plus r2 . alright so that 's our answer . that 's the answer . that 's v out equals some function of v in . now i want to make a really important observation here . this is going to be a real cool simplification . okay , so this is the point where op-amp theory gets really cool . watch what happens here . we know that a is a giant number . a is something like 10 to the fifth , or 10 to the sixth , and it 's whatever we have here , if our resistors are sort of normal-sized resistors we know that a giant number times a normal number is still going to be a very big number compared to one . so this one is almost insignificant in this expression down here , so what i 'm going to do , bear with me , i 'm going to cross it out . i 'm going to say no , i do n't need that anymore . so if this , if this number here , if a is a million , 10 to the sixth , and this expression here is something like one half then this total thing is one half 10 to the sixth or a half a million , and that 's huge compared to one . so i can pretty safely ignore the one , it 's very , very small . now when i do that , well look what happens next , now i have a top and bottom in the expression , and i can cancel that too . so the a goes away , now this is pretty astonishing . we have this amplifier circuit and all of a sudden i have an expression here where a does n't appear , the gain does not appear , and what does this turn into ? this is called v not equals v in , times what ? times r1 plus r2 , divided by r2 . so our amplifier , our feedback circuit came down to v out is v in multiplied by the ratio of the resistors that we added to the circuit . this is one of the really cool properties of using op-amps in circuits , really high-gain amplifiers . what we 've done is we have chosen the gain of our circuit based on the components that we picked to add to the amplifier . it 's not determined by the gain of the amplifier as long as the amplifier gain is really , really big . and for op-amps , that 's a good assumption , it is really big . so this expression came out with a positive sign , right ? all the r 's are positive values , so this is referred to as a non-inverting op-amp circuit amplifier . so just to do a quick example , if r1 and r2 are the same , then we end up with an expression that looks like this v out equals r1 plus r2 , r plus r over r is equal to two so the gain is two times v in . so just to do a quick sketch just to remind ourselves what this looks like , this was v in , and we had what out here ? we had a resistor , we had a resistor to the ground , and this is v out . so this is the configuration of a non-inverting amplifier built with an op-amp , the two resistors in this voltage divider string connected to the negative input . so that 's what non-inverting op-amp circuit looks like , and it 's going to be one of the familiar patterns that you see over and over again as you read schematics and you design your own circuits .
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it 's usually drawn as a diamond shape , and this is a voltage source , but it 's a special kind of voltage source . it 's called a voltage-dependent voltage source . and it 's the same as a regular ideal voltage source except for one thing , it says that the v , in this case v out , equals gain times v plus minus v minus .
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so the resisters decide the 'gain ' in the voltage ?
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okay , now we 're going to work on our first op-amp circuit . here 's what the circuit 's going to look like . watch where it puts the plus sign is on the top on this one . and we 're going to have a voltage source over here . this will be plus or minus v in , that 's our input signal . and over on the output , we 'll have v out , and it 's hooked up this way . the resistor , another resistor , to ground , and this goes back to the inverting input . now we 're going to look at this circuit and see what it does . now we know that connected up here the power supply 's hooked up to these points here , and the ground symbol is zero volts . and we want to analyze this circuit . and what do we know about this ? we know that v out equals some gain , i 'll write the gain right there . a big , big number times v minus , sorry v plus minus v minus , and let 's label that . v plus is this point right here , and v minus is this point right here . and we also know that the currents , let 's call them i plus and i minus , equals zero , and that 's the currents going in here . this is i minus here , and that 's i plus , and we know those are both zero . so now what i want to do it describe what 's going on inside this triangle symbol in more detail by building a circuit model . alright , and a circuit model for an amplifier looks like this . we have v minus here , v plus here , so this is v in , and over on this side we have an , here 's a new symbol that you have n't seen before . it 's usually drawn as a diamond shape , and this is a voltage source , but it 's a special kind of voltage source . it 's called a voltage-dependent voltage source . and it 's the same as a regular ideal voltage source except for one thing , it says that the v , in this case v out , equals gain times v plus minus v minus . so the voltage here depends on the voltage somewhere else , and that 's what makes it a voltage-dependent , that 's what that means . so , we 've just taken our gain expression here , added , drawn circuit diagram that represents our voltage expression for our circuit . now , specifically over here we 've drawn an open circuit on v plus , and v minus so we know that those currents are zero . so this model , this circuit sketch represents our two properties of our op-amp . so i 'm going to take a second here and i 'm going to draw the rest of our circuit surrounding this model , but i need a little bit more space . so let 's put in the rest of our circuit here . we had our voltage source , connected to v plus , and that 's v in , and over here we had v out . let 's check , v out was connected to two resistors , and the bottom is connected to ground , and this was connected there . so what our goal is right now , we want to find v out as a function of v in . that 's what we 're shooting for . so let 's see if we can do that . let 's give our resistors some names . let 's call this r1 , and r2 , our favorite names always , and now everything is labeled . now and we can label this point here , and this point we can call v minus , v minus . so that 's our two unknowns . our unknowns are v not , v out , and v minus , so let 's see if we can find them . so what i 'm going to do is just start writing some expressions for things that i know are true . for example , i know that v out equals a times v plus minus v minus . alright , that 's what this op-amp is telling us is true . now what else do i know ? let 's look at this resistor chain here . this resistor chain actually looks a lot like a voltage divider , and it 's actually a very good voltage divider . remember we said this current here , what is this current here ? it 's zero . i can use the voltage divider expression that i know . in that case , i know that v minus , this is the voltage divider equation , equals v out times what ? times the bottom resistor remember this ? r2 over r1 plus r2 , so the voltage divider expression says that when you have a stack of resistors like this , with the voltage on the top and ground on the bottom , this is the expression for the voltage at the midpoint . kay , so what i 'm going to do next is i 'm going to take this expression and stuff it right in there . let 's do that . see if we got enough room , okay now let 's go over here . now i can say that v out equals a times v plus minus v out times r2 over r1 plus r2 , alright so far so good . let 's keep going , let 's keep working on this . v not equals a times v plus minus a v not , r2 over r1 plus r2 . alright , so now i 'm going to gather all the v not terms over on the left hand side . let 's try that . so that gives me , v not plus a v not , times r2 over r1 plus r2 and that equals a times v plus , and actually i can change that now v plus is what ? v plus is v in . okay let 's keep going i can factor out the v not . v not is one plus a r2 over r1 plus r2 and that equals av in . alright so we 're getting close , and our original goal , we want to find v out in terms of v in . so i 'm going to take this whole expression here and divide it over to the other side , so then i have just v not on this side , and v in on the other side . make some more room . i can do that , i can say v not equals a v in divided by this big old expression , one plus a r2 over r1 plus r2 . alright so that 's our answer . that 's the answer . that 's v out equals some function of v in . now i want to make a really important observation here . this is going to be a real cool simplification . okay , so this is the point where op-amp theory gets really cool . watch what happens here . we know that a is a giant number . a is something like 10 to the fifth , or 10 to the sixth , and it 's whatever we have here , if our resistors are sort of normal-sized resistors we know that a giant number times a normal number is still going to be a very big number compared to one . so this one is almost insignificant in this expression down here , so what i 'm going to do , bear with me , i 'm going to cross it out . i 'm going to say no , i do n't need that anymore . so if this , if this number here , if a is a million , 10 to the sixth , and this expression here is something like one half then this total thing is one half 10 to the sixth or a half a million , and that 's huge compared to one . so i can pretty safely ignore the one , it 's very , very small . now when i do that , well look what happens next , now i have a top and bottom in the expression , and i can cancel that too . so the a goes away , now this is pretty astonishing . we have this amplifier circuit and all of a sudden i have an expression here where a does n't appear , the gain does not appear , and what does this turn into ? this is called v not equals v in , times what ? times r1 plus r2 , divided by r2 . so our amplifier , our feedback circuit came down to v out is v in multiplied by the ratio of the resistors that we added to the circuit . this is one of the really cool properties of using op-amps in circuits , really high-gain amplifiers . what we 've done is we have chosen the gain of our circuit based on the components that we picked to add to the amplifier . it 's not determined by the gain of the amplifier as long as the amplifier gain is really , really big . and for op-amps , that 's a good assumption , it is really big . so this expression came out with a positive sign , right ? all the r 's are positive values , so this is referred to as a non-inverting op-amp circuit amplifier . so just to do a quick example , if r1 and r2 are the same , then we end up with an expression that looks like this v out equals r1 plus r2 , r plus r over r is equal to two so the gain is two times v in . so just to do a quick sketch just to remind ourselves what this looks like , this was v in , and we had what out here ? we had a resistor , we had a resistor to the ground , and this is v out . so this is the configuration of a non-inverting amplifier built with an op-amp , the two resistors in this voltage divider string connected to the negative input . so that 's what non-inverting op-amp circuit looks like , and it 's going to be one of the familiar patterns that you see over and over again as you read schematics and you design your own circuits .
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we had a resistor , we had a resistor to the ground , and this is v out . so this is the configuration of a non-inverting amplifier built with an op-amp , the two resistors in this voltage divider string connected to the negative input . so that 's what non-inverting op-amp circuit looks like , and it 's going to be one of the familiar patterns that you see over and over again as you read schematics and you design your own circuits .
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is the non-inverting amplifier always designed to have two resistors stacked on one another while connected to vout ?
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okay , now we 're going to work on our first op-amp circuit . here 's what the circuit 's going to look like . watch where it puts the plus sign is on the top on this one . and we 're going to have a voltage source over here . this will be plus or minus v in , that 's our input signal . and over on the output , we 'll have v out , and it 's hooked up this way . the resistor , another resistor , to ground , and this goes back to the inverting input . now we 're going to look at this circuit and see what it does . now we know that connected up here the power supply 's hooked up to these points here , and the ground symbol is zero volts . and we want to analyze this circuit . and what do we know about this ? we know that v out equals some gain , i 'll write the gain right there . a big , big number times v minus , sorry v plus minus v minus , and let 's label that . v plus is this point right here , and v minus is this point right here . and we also know that the currents , let 's call them i plus and i minus , equals zero , and that 's the currents going in here . this is i minus here , and that 's i plus , and we know those are both zero . so now what i want to do it describe what 's going on inside this triangle symbol in more detail by building a circuit model . alright , and a circuit model for an amplifier looks like this . we have v minus here , v plus here , so this is v in , and over on this side we have an , here 's a new symbol that you have n't seen before . it 's usually drawn as a diamond shape , and this is a voltage source , but it 's a special kind of voltage source . it 's called a voltage-dependent voltage source . and it 's the same as a regular ideal voltage source except for one thing , it says that the v , in this case v out , equals gain times v plus minus v minus . so the voltage here depends on the voltage somewhere else , and that 's what makes it a voltage-dependent , that 's what that means . so , we 've just taken our gain expression here , added , drawn circuit diagram that represents our voltage expression for our circuit . now , specifically over here we 've drawn an open circuit on v plus , and v minus so we know that those currents are zero . so this model , this circuit sketch represents our two properties of our op-amp . so i 'm going to take a second here and i 'm going to draw the rest of our circuit surrounding this model , but i need a little bit more space . so let 's put in the rest of our circuit here . we had our voltage source , connected to v plus , and that 's v in , and over here we had v out . let 's check , v out was connected to two resistors , and the bottom is connected to ground , and this was connected there . so what our goal is right now , we want to find v out as a function of v in . that 's what we 're shooting for . so let 's see if we can do that . let 's give our resistors some names . let 's call this r1 , and r2 , our favorite names always , and now everything is labeled . now and we can label this point here , and this point we can call v minus , v minus . so that 's our two unknowns . our unknowns are v not , v out , and v minus , so let 's see if we can find them . so what i 'm going to do is just start writing some expressions for things that i know are true . for example , i know that v out equals a times v plus minus v minus . alright , that 's what this op-amp is telling us is true . now what else do i know ? let 's look at this resistor chain here . this resistor chain actually looks a lot like a voltage divider , and it 's actually a very good voltage divider . remember we said this current here , what is this current here ? it 's zero . i can use the voltage divider expression that i know . in that case , i know that v minus , this is the voltage divider equation , equals v out times what ? times the bottom resistor remember this ? r2 over r1 plus r2 , so the voltage divider expression says that when you have a stack of resistors like this , with the voltage on the top and ground on the bottom , this is the expression for the voltage at the midpoint . kay , so what i 'm going to do next is i 'm going to take this expression and stuff it right in there . let 's do that . see if we got enough room , okay now let 's go over here . now i can say that v out equals a times v plus minus v out times r2 over r1 plus r2 , alright so far so good . let 's keep going , let 's keep working on this . v not equals a times v plus minus a v not , r2 over r1 plus r2 . alright , so now i 'm going to gather all the v not terms over on the left hand side . let 's try that . so that gives me , v not plus a v not , times r2 over r1 plus r2 and that equals a times v plus , and actually i can change that now v plus is what ? v plus is v in . okay let 's keep going i can factor out the v not . v not is one plus a r2 over r1 plus r2 and that equals av in . alright so we 're getting close , and our original goal , we want to find v out in terms of v in . so i 'm going to take this whole expression here and divide it over to the other side , so then i have just v not on this side , and v in on the other side . make some more room . i can do that , i can say v not equals a v in divided by this big old expression , one plus a r2 over r1 plus r2 . alright so that 's our answer . that 's the answer . that 's v out equals some function of v in . now i want to make a really important observation here . this is going to be a real cool simplification . okay , so this is the point where op-amp theory gets really cool . watch what happens here . we know that a is a giant number . a is something like 10 to the fifth , or 10 to the sixth , and it 's whatever we have here , if our resistors are sort of normal-sized resistors we know that a giant number times a normal number is still going to be a very big number compared to one . so this one is almost insignificant in this expression down here , so what i 'm going to do , bear with me , i 'm going to cross it out . i 'm going to say no , i do n't need that anymore . so if this , if this number here , if a is a million , 10 to the sixth , and this expression here is something like one half then this total thing is one half 10 to the sixth or a half a million , and that 's huge compared to one . so i can pretty safely ignore the one , it 's very , very small . now when i do that , well look what happens next , now i have a top and bottom in the expression , and i can cancel that too . so the a goes away , now this is pretty astonishing . we have this amplifier circuit and all of a sudden i have an expression here where a does n't appear , the gain does not appear , and what does this turn into ? this is called v not equals v in , times what ? times r1 plus r2 , divided by r2 . so our amplifier , our feedback circuit came down to v out is v in multiplied by the ratio of the resistors that we added to the circuit . this is one of the really cool properties of using op-amps in circuits , really high-gain amplifiers . what we 've done is we have chosen the gain of our circuit based on the components that we picked to add to the amplifier . it 's not determined by the gain of the amplifier as long as the amplifier gain is really , really big . and for op-amps , that 's a good assumption , it is really big . so this expression came out with a positive sign , right ? all the r 's are positive values , so this is referred to as a non-inverting op-amp circuit amplifier . so just to do a quick example , if r1 and r2 are the same , then we end up with an expression that looks like this v out equals r1 plus r2 , r plus r over r is equal to two so the gain is two times v in . so just to do a quick sketch just to remind ourselves what this looks like , this was v in , and we had what out here ? we had a resistor , we had a resistor to the ground , and this is v out . so this is the configuration of a non-inverting amplifier built with an op-amp , the two resistors in this voltage divider string connected to the negative input . so that 's what non-inverting op-amp circuit looks like , and it 's going to be one of the familiar patterns that you see over and over again as you read schematics and you design your own circuits .
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and what do we know about this ? we know that v out equals some gain , i 'll write the gain right there . a big , big number times v minus , sorry v plus minus v minus , and let 's label that .
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is a is a symbol for gain ?
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okay , now we 're going to work on our first op-amp circuit . here 's what the circuit 's going to look like . watch where it puts the plus sign is on the top on this one . and we 're going to have a voltage source over here . this will be plus or minus v in , that 's our input signal . and over on the output , we 'll have v out , and it 's hooked up this way . the resistor , another resistor , to ground , and this goes back to the inverting input . now we 're going to look at this circuit and see what it does . now we know that connected up here the power supply 's hooked up to these points here , and the ground symbol is zero volts . and we want to analyze this circuit . and what do we know about this ? we know that v out equals some gain , i 'll write the gain right there . a big , big number times v minus , sorry v plus minus v minus , and let 's label that . v plus is this point right here , and v minus is this point right here . and we also know that the currents , let 's call them i plus and i minus , equals zero , and that 's the currents going in here . this is i minus here , and that 's i plus , and we know those are both zero . so now what i want to do it describe what 's going on inside this triangle symbol in more detail by building a circuit model . alright , and a circuit model for an amplifier looks like this . we have v minus here , v plus here , so this is v in , and over on this side we have an , here 's a new symbol that you have n't seen before . it 's usually drawn as a diamond shape , and this is a voltage source , but it 's a special kind of voltage source . it 's called a voltage-dependent voltage source . and it 's the same as a regular ideal voltage source except for one thing , it says that the v , in this case v out , equals gain times v plus minus v minus . so the voltage here depends on the voltage somewhere else , and that 's what makes it a voltage-dependent , that 's what that means . so , we 've just taken our gain expression here , added , drawn circuit diagram that represents our voltage expression for our circuit . now , specifically over here we 've drawn an open circuit on v plus , and v minus so we know that those currents are zero . so this model , this circuit sketch represents our two properties of our op-amp . so i 'm going to take a second here and i 'm going to draw the rest of our circuit surrounding this model , but i need a little bit more space . so let 's put in the rest of our circuit here . we had our voltage source , connected to v plus , and that 's v in , and over here we had v out . let 's check , v out was connected to two resistors , and the bottom is connected to ground , and this was connected there . so what our goal is right now , we want to find v out as a function of v in . that 's what we 're shooting for . so let 's see if we can do that . let 's give our resistors some names . let 's call this r1 , and r2 , our favorite names always , and now everything is labeled . now and we can label this point here , and this point we can call v minus , v minus . so that 's our two unknowns . our unknowns are v not , v out , and v minus , so let 's see if we can find them . so what i 'm going to do is just start writing some expressions for things that i know are true . for example , i know that v out equals a times v plus minus v minus . alright , that 's what this op-amp is telling us is true . now what else do i know ? let 's look at this resistor chain here . this resistor chain actually looks a lot like a voltage divider , and it 's actually a very good voltage divider . remember we said this current here , what is this current here ? it 's zero . i can use the voltage divider expression that i know . in that case , i know that v minus , this is the voltage divider equation , equals v out times what ? times the bottom resistor remember this ? r2 over r1 plus r2 , so the voltage divider expression says that when you have a stack of resistors like this , with the voltage on the top and ground on the bottom , this is the expression for the voltage at the midpoint . kay , so what i 'm going to do next is i 'm going to take this expression and stuff it right in there . let 's do that . see if we got enough room , okay now let 's go over here . now i can say that v out equals a times v plus minus v out times r2 over r1 plus r2 , alright so far so good . let 's keep going , let 's keep working on this . v not equals a times v plus minus a v not , r2 over r1 plus r2 . alright , so now i 'm going to gather all the v not terms over on the left hand side . let 's try that . so that gives me , v not plus a v not , times r2 over r1 plus r2 and that equals a times v plus , and actually i can change that now v plus is what ? v plus is v in . okay let 's keep going i can factor out the v not . v not is one plus a r2 over r1 plus r2 and that equals av in . alright so we 're getting close , and our original goal , we want to find v out in terms of v in . so i 'm going to take this whole expression here and divide it over to the other side , so then i have just v not on this side , and v in on the other side . make some more room . i can do that , i can say v not equals a v in divided by this big old expression , one plus a r2 over r1 plus r2 . alright so that 's our answer . that 's the answer . that 's v out equals some function of v in . now i want to make a really important observation here . this is going to be a real cool simplification . okay , so this is the point where op-amp theory gets really cool . watch what happens here . we know that a is a giant number . a is something like 10 to the fifth , or 10 to the sixth , and it 's whatever we have here , if our resistors are sort of normal-sized resistors we know that a giant number times a normal number is still going to be a very big number compared to one . so this one is almost insignificant in this expression down here , so what i 'm going to do , bear with me , i 'm going to cross it out . i 'm going to say no , i do n't need that anymore . so if this , if this number here , if a is a million , 10 to the sixth , and this expression here is something like one half then this total thing is one half 10 to the sixth or a half a million , and that 's huge compared to one . so i can pretty safely ignore the one , it 's very , very small . now when i do that , well look what happens next , now i have a top and bottom in the expression , and i can cancel that too . so the a goes away , now this is pretty astonishing . we have this amplifier circuit and all of a sudden i have an expression here where a does n't appear , the gain does not appear , and what does this turn into ? this is called v not equals v in , times what ? times r1 plus r2 , divided by r2 . so our amplifier , our feedback circuit came down to v out is v in multiplied by the ratio of the resistors that we added to the circuit . this is one of the really cool properties of using op-amps in circuits , really high-gain amplifiers . what we 've done is we have chosen the gain of our circuit based on the components that we picked to add to the amplifier . it 's not determined by the gain of the amplifier as long as the amplifier gain is really , really big . and for op-amps , that 's a good assumption , it is really big . so this expression came out with a positive sign , right ? all the r 's are positive values , so this is referred to as a non-inverting op-amp circuit amplifier . so just to do a quick example , if r1 and r2 are the same , then we end up with an expression that looks like this v out equals r1 plus r2 , r plus r over r is equal to two so the gain is two times v in . so just to do a quick sketch just to remind ourselves what this looks like , this was v in , and we had what out here ? we had a resistor , we had a resistor to the ground , and this is v out . so this is the configuration of a non-inverting amplifier built with an op-amp , the two resistors in this voltage divider string connected to the negative input . so that 's what non-inverting op-amp circuit looks like , and it 's going to be one of the familiar patterns that you see over and over again as you read schematics and you design your own circuits .
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it 's usually drawn as a diamond shape , and this is a voltage source , but it 's a special kind of voltage source . it 's called a voltage-dependent voltage source . and it 's the same as a regular ideal voltage source except for one thing , it says that the v , in this case v out , equals gain times v plus minus v minus .
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why is the circle used to denote a voltage source ?
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okay , now we 're going to work on our first op-amp circuit . here 's what the circuit 's going to look like . watch where it puts the plus sign is on the top on this one . and we 're going to have a voltage source over here . this will be plus or minus v in , that 's our input signal . and over on the output , we 'll have v out , and it 's hooked up this way . the resistor , another resistor , to ground , and this goes back to the inverting input . now we 're going to look at this circuit and see what it does . now we know that connected up here the power supply 's hooked up to these points here , and the ground symbol is zero volts . and we want to analyze this circuit . and what do we know about this ? we know that v out equals some gain , i 'll write the gain right there . a big , big number times v minus , sorry v plus minus v minus , and let 's label that . v plus is this point right here , and v minus is this point right here . and we also know that the currents , let 's call them i plus and i minus , equals zero , and that 's the currents going in here . this is i minus here , and that 's i plus , and we know those are both zero . so now what i want to do it describe what 's going on inside this triangle symbol in more detail by building a circuit model . alright , and a circuit model for an amplifier looks like this . we have v minus here , v plus here , so this is v in , and over on this side we have an , here 's a new symbol that you have n't seen before . it 's usually drawn as a diamond shape , and this is a voltage source , but it 's a special kind of voltage source . it 's called a voltage-dependent voltage source . and it 's the same as a regular ideal voltage source except for one thing , it says that the v , in this case v out , equals gain times v plus minus v minus . so the voltage here depends on the voltage somewhere else , and that 's what makes it a voltage-dependent , that 's what that means . so , we 've just taken our gain expression here , added , drawn circuit diagram that represents our voltage expression for our circuit . now , specifically over here we 've drawn an open circuit on v plus , and v minus so we know that those currents are zero . so this model , this circuit sketch represents our two properties of our op-amp . so i 'm going to take a second here and i 'm going to draw the rest of our circuit surrounding this model , but i need a little bit more space . so let 's put in the rest of our circuit here . we had our voltage source , connected to v plus , and that 's v in , and over here we had v out . let 's check , v out was connected to two resistors , and the bottom is connected to ground , and this was connected there . so what our goal is right now , we want to find v out as a function of v in . that 's what we 're shooting for . so let 's see if we can do that . let 's give our resistors some names . let 's call this r1 , and r2 , our favorite names always , and now everything is labeled . now and we can label this point here , and this point we can call v minus , v minus . so that 's our two unknowns . our unknowns are v not , v out , and v minus , so let 's see if we can find them . so what i 'm going to do is just start writing some expressions for things that i know are true . for example , i know that v out equals a times v plus minus v minus . alright , that 's what this op-amp is telling us is true . now what else do i know ? let 's look at this resistor chain here . this resistor chain actually looks a lot like a voltage divider , and it 's actually a very good voltage divider . remember we said this current here , what is this current here ? it 's zero . i can use the voltage divider expression that i know . in that case , i know that v minus , this is the voltage divider equation , equals v out times what ? times the bottom resistor remember this ? r2 over r1 plus r2 , so the voltage divider expression says that when you have a stack of resistors like this , with the voltage on the top and ground on the bottom , this is the expression for the voltage at the midpoint . kay , so what i 'm going to do next is i 'm going to take this expression and stuff it right in there . let 's do that . see if we got enough room , okay now let 's go over here . now i can say that v out equals a times v plus minus v out times r2 over r1 plus r2 , alright so far so good . let 's keep going , let 's keep working on this . v not equals a times v plus minus a v not , r2 over r1 plus r2 . alright , so now i 'm going to gather all the v not terms over on the left hand side . let 's try that . so that gives me , v not plus a v not , times r2 over r1 plus r2 and that equals a times v plus , and actually i can change that now v plus is what ? v plus is v in . okay let 's keep going i can factor out the v not . v not is one plus a r2 over r1 plus r2 and that equals av in . alright so we 're getting close , and our original goal , we want to find v out in terms of v in . so i 'm going to take this whole expression here and divide it over to the other side , so then i have just v not on this side , and v in on the other side . make some more room . i can do that , i can say v not equals a v in divided by this big old expression , one plus a r2 over r1 plus r2 . alright so that 's our answer . that 's the answer . that 's v out equals some function of v in . now i want to make a really important observation here . this is going to be a real cool simplification . okay , so this is the point where op-amp theory gets really cool . watch what happens here . we know that a is a giant number . a is something like 10 to the fifth , or 10 to the sixth , and it 's whatever we have here , if our resistors are sort of normal-sized resistors we know that a giant number times a normal number is still going to be a very big number compared to one . so this one is almost insignificant in this expression down here , so what i 'm going to do , bear with me , i 'm going to cross it out . i 'm going to say no , i do n't need that anymore . so if this , if this number here , if a is a million , 10 to the sixth , and this expression here is something like one half then this total thing is one half 10 to the sixth or a half a million , and that 's huge compared to one . so i can pretty safely ignore the one , it 's very , very small . now when i do that , well look what happens next , now i have a top and bottom in the expression , and i can cancel that too . so the a goes away , now this is pretty astonishing . we have this amplifier circuit and all of a sudden i have an expression here where a does n't appear , the gain does not appear , and what does this turn into ? this is called v not equals v in , times what ? times r1 plus r2 , divided by r2 . so our amplifier , our feedback circuit came down to v out is v in multiplied by the ratio of the resistors that we added to the circuit . this is one of the really cool properties of using op-amps in circuits , really high-gain amplifiers . what we 've done is we have chosen the gain of our circuit based on the components that we picked to add to the amplifier . it 's not determined by the gain of the amplifier as long as the amplifier gain is really , really big . and for op-amps , that 's a good assumption , it is really big . so this expression came out with a positive sign , right ? all the r 's are positive values , so this is referred to as a non-inverting op-amp circuit amplifier . so just to do a quick example , if r1 and r2 are the same , then we end up with an expression that looks like this v out equals r1 plus r2 , r plus r over r is equal to two so the gain is two times v in . so just to do a quick sketch just to remind ourselves what this looks like , this was v in , and we had what out here ? we had a resistor , we had a resistor to the ground , and this is v out . so this is the configuration of a non-inverting amplifier built with an op-amp , the two resistors in this voltage divider string connected to the negative input . so that 's what non-inverting op-amp circuit looks like , and it 's going to be one of the familiar patterns that you see over and over again as you read schematics and you design your own circuits .
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so that gives me , v not plus a v not , times r2 over r1 plus r2 and that equals a times v plus , and actually i can change that now v plus is what ? v plus is v in . okay let 's keep going i can factor out the v not .
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here v- is ground and one end of the resistance is ground what does it mean ?
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okay , now we 're going to work on our first op-amp circuit . here 's what the circuit 's going to look like . watch where it puts the plus sign is on the top on this one . and we 're going to have a voltage source over here . this will be plus or minus v in , that 's our input signal . and over on the output , we 'll have v out , and it 's hooked up this way . the resistor , another resistor , to ground , and this goes back to the inverting input . now we 're going to look at this circuit and see what it does . now we know that connected up here the power supply 's hooked up to these points here , and the ground symbol is zero volts . and we want to analyze this circuit . and what do we know about this ? we know that v out equals some gain , i 'll write the gain right there . a big , big number times v minus , sorry v plus minus v minus , and let 's label that . v plus is this point right here , and v minus is this point right here . and we also know that the currents , let 's call them i plus and i minus , equals zero , and that 's the currents going in here . this is i minus here , and that 's i plus , and we know those are both zero . so now what i want to do it describe what 's going on inside this triangle symbol in more detail by building a circuit model . alright , and a circuit model for an amplifier looks like this . we have v minus here , v plus here , so this is v in , and over on this side we have an , here 's a new symbol that you have n't seen before . it 's usually drawn as a diamond shape , and this is a voltage source , but it 's a special kind of voltage source . it 's called a voltage-dependent voltage source . and it 's the same as a regular ideal voltage source except for one thing , it says that the v , in this case v out , equals gain times v plus minus v minus . so the voltage here depends on the voltage somewhere else , and that 's what makes it a voltage-dependent , that 's what that means . so , we 've just taken our gain expression here , added , drawn circuit diagram that represents our voltage expression for our circuit . now , specifically over here we 've drawn an open circuit on v plus , and v minus so we know that those currents are zero . so this model , this circuit sketch represents our two properties of our op-amp . so i 'm going to take a second here and i 'm going to draw the rest of our circuit surrounding this model , but i need a little bit more space . so let 's put in the rest of our circuit here . we had our voltage source , connected to v plus , and that 's v in , and over here we had v out . let 's check , v out was connected to two resistors , and the bottom is connected to ground , and this was connected there . so what our goal is right now , we want to find v out as a function of v in . that 's what we 're shooting for . so let 's see if we can do that . let 's give our resistors some names . let 's call this r1 , and r2 , our favorite names always , and now everything is labeled . now and we can label this point here , and this point we can call v minus , v minus . so that 's our two unknowns . our unknowns are v not , v out , and v minus , so let 's see if we can find them . so what i 'm going to do is just start writing some expressions for things that i know are true . for example , i know that v out equals a times v plus minus v minus . alright , that 's what this op-amp is telling us is true . now what else do i know ? let 's look at this resistor chain here . this resistor chain actually looks a lot like a voltage divider , and it 's actually a very good voltage divider . remember we said this current here , what is this current here ? it 's zero . i can use the voltage divider expression that i know . in that case , i know that v minus , this is the voltage divider equation , equals v out times what ? times the bottom resistor remember this ? r2 over r1 plus r2 , so the voltage divider expression says that when you have a stack of resistors like this , with the voltage on the top and ground on the bottom , this is the expression for the voltage at the midpoint . kay , so what i 'm going to do next is i 'm going to take this expression and stuff it right in there . let 's do that . see if we got enough room , okay now let 's go over here . now i can say that v out equals a times v plus minus v out times r2 over r1 plus r2 , alright so far so good . let 's keep going , let 's keep working on this . v not equals a times v plus minus a v not , r2 over r1 plus r2 . alright , so now i 'm going to gather all the v not terms over on the left hand side . let 's try that . so that gives me , v not plus a v not , times r2 over r1 plus r2 and that equals a times v plus , and actually i can change that now v plus is what ? v plus is v in . okay let 's keep going i can factor out the v not . v not is one plus a r2 over r1 plus r2 and that equals av in . alright so we 're getting close , and our original goal , we want to find v out in terms of v in . so i 'm going to take this whole expression here and divide it over to the other side , so then i have just v not on this side , and v in on the other side . make some more room . i can do that , i can say v not equals a v in divided by this big old expression , one plus a r2 over r1 plus r2 . alright so that 's our answer . that 's the answer . that 's v out equals some function of v in . now i want to make a really important observation here . this is going to be a real cool simplification . okay , so this is the point where op-amp theory gets really cool . watch what happens here . we know that a is a giant number . a is something like 10 to the fifth , or 10 to the sixth , and it 's whatever we have here , if our resistors are sort of normal-sized resistors we know that a giant number times a normal number is still going to be a very big number compared to one . so this one is almost insignificant in this expression down here , so what i 'm going to do , bear with me , i 'm going to cross it out . i 'm going to say no , i do n't need that anymore . so if this , if this number here , if a is a million , 10 to the sixth , and this expression here is something like one half then this total thing is one half 10 to the sixth or a half a million , and that 's huge compared to one . so i can pretty safely ignore the one , it 's very , very small . now when i do that , well look what happens next , now i have a top and bottom in the expression , and i can cancel that too . so the a goes away , now this is pretty astonishing . we have this amplifier circuit and all of a sudden i have an expression here where a does n't appear , the gain does not appear , and what does this turn into ? this is called v not equals v in , times what ? times r1 plus r2 , divided by r2 . so our amplifier , our feedback circuit came down to v out is v in multiplied by the ratio of the resistors that we added to the circuit . this is one of the really cool properties of using op-amps in circuits , really high-gain amplifiers . what we 've done is we have chosen the gain of our circuit based on the components that we picked to add to the amplifier . it 's not determined by the gain of the amplifier as long as the amplifier gain is really , really big . and for op-amps , that 's a good assumption , it is really big . so this expression came out with a positive sign , right ? all the r 's are positive values , so this is referred to as a non-inverting op-amp circuit amplifier . so just to do a quick example , if r1 and r2 are the same , then we end up with an expression that looks like this v out equals r1 plus r2 , r plus r over r is equal to two so the gain is two times v in . so just to do a quick sketch just to remind ourselves what this looks like , this was v in , and we had what out here ? we had a resistor , we had a resistor to the ground , and this is v out . so this is the configuration of a non-inverting amplifier built with an op-amp , the two resistors in this voltage divider string connected to the negative input . so that 's what non-inverting op-amp circuit looks like , and it 's going to be one of the familiar patterns that you see over and over again as you read schematics and you design your own circuits .
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it 's usually drawn as a diamond shape , and this is a voltage source , but it 's a special kind of voltage source . it 's called a voltage-dependent voltage source . and it 's the same as a regular ideal voltage source except for one thing , it says that the v , in this case v out , equals gain times v plus minus v minus .
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does a ground voltage means a earthing current ?
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okay , now we 're going to work on our first op-amp circuit . here 's what the circuit 's going to look like . watch where it puts the plus sign is on the top on this one . and we 're going to have a voltage source over here . this will be plus or minus v in , that 's our input signal . and over on the output , we 'll have v out , and it 's hooked up this way . the resistor , another resistor , to ground , and this goes back to the inverting input . now we 're going to look at this circuit and see what it does . now we know that connected up here the power supply 's hooked up to these points here , and the ground symbol is zero volts . and we want to analyze this circuit . and what do we know about this ? we know that v out equals some gain , i 'll write the gain right there . a big , big number times v minus , sorry v plus minus v minus , and let 's label that . v plus is this point right here , and v minus is this point right here . and we also know that the currents , let 's call them i plus and i minus , equals zero , and that 's the currents going in here . this is i minus here , and that 's i plus , and we know those are both zero . so now what i want to do it describe what 's going on inside this triangle symbol in more detail by building a circuit model . alright , and a circuit model for an amplifier looks like this . we have v minus here , v plus here , so this is v in , and over on this side we have an , here 's a new symbol that you have n't seen before . it 's usually drawn as a diamond shape , and this is a voltage source , but it 's a special kind of voltage source . it 's called a voltage-dependent voltage source . and it 's the same as a regular ideal voltage source except for one thing , it says that the v , in this case v out , equals gain times v plus minus v minus . so the voltage here depends on the voltage somewhere else , and that 's what makes it a voltage-dependent , that 's what that means . so , we 've just taken our gain expression here , added , drawn circuit diagram that represents our voltage expression for our circuit . now , specifically over here we 've drawn an open circuit on v plus , and v minus so we know that those currents are zero . so this model , this circuit sketch represents our two properties of our op-amp . so i 'm going to take a second here and i 'm going to draw the rest of our circuit surrounding this model , but i need a little bit more space . so let 's put in the rest of our circuit here . we had our voltage source , connected to v plus , and that 's v in , and over here we had v out . let 's check , v out was connected to two resistors , and the bottom is connected to ground , and this was connected there . so what our goal is right now , we want to find v out as a function of v in . that 's what we 're shooting for . so let 's see if we can do that . let 's give our resistors some names . let 's call this r1 , and r2 , our favorite names always , and now everything is labeled . now and we can label this point here , and this point we can call v minus , v minus . so that 's our two unknowns . our unknowns are v not , v out , and v minus , so let 's see if we can find them . so what i 'm going to do is just start writing some expressions for things that i know are true . for example , i know that v out equals a times v plus minus v minus . alright , that 's what this op-amp is telling us is true . now what else do i know ? let 's look at this resistor chain here . this resistor chain actually looks a lot like a voltage divider , and it 's actually a very good voltage divider . remember we said this current here , what is this current here ? it 's zero . i can use the voltage divider expression that i know . in that case , i know that v minus , this is the voltage divider equation , equals v out times what ? times the bottom resistor remember this ? r2 over r1 plus r2 , so the voltage divider expression says that when you have a stack of resistors like this , with the voltage on the top and ground on the bottom , this is the expression for the voltage at the midpoint . kay , so what i 'm going to do next is i 'm going to take this expression and stuff it right in there . let 's do that . see if we got enough room , okay now let 's go over here . now i can say that v out equals a times v plus minus v out times r2 over r1 plus r2 , alright so far so good . let 's keep going , let 's keep working on this . v not equals a times v plus minus a v not , r2 over r1 plus r2 . alright , so now i 'm going to gather all the v not terms over on the left hand side . let 's try that . so that gives me , v not plus a v not , times r2 over r1 plus r2 and that equals a times v plus , and actually i can change that now v plus is what ? v plus is v in . okay let 's keep going i can factor out the v not . v not is one plus a r2 over r1 plus r2 and that equals av in . alright so we 're getting close , and our original goal , we want to find v out in terms of v in . so i 'm going to take this whole expression here and divide it over to the other side , so then i have just v not on this side , and v in on the other side . make some more room . i can do that , i can say v not equals a v in divided by this big old expression , one plus a r2 over r1 plus r2 . alright so that 's our answer . that 's the answer . that 's v out equals some function of v in . now i want to make a really important observation here . this is going to be a real cool simplification . okay , so this is the point where op-amp theory gets really cool . watch what happens here . we know that a is a giant number . a is something like 10 to the fifth , or 10 to the sixth , and it 's whatever we have here , if our resistors are sort of normal-sized resistors we know that a giant number times a normal number is still going to be a very big number compared to one . so this one is almost insignificant in this expression down here , so what i 'm going to do , bear with me , i 'm going to cross it out . i 'm going to say no , i do n't need that anymore . so if this , if this number here , if a is a million , 10 to the sixth , and this expression here is something like one half then this total thing is one half 10 to the sixth or a half a million , and that 's huge compared to one . so i can pretty safely ignore the one , it 's very , very small . now when i do that , well look what happens next , now i have a top and bottom in the expression , and i can cancel that too . so the a goes away , now this is pretty astonishing . we have this amplifier circuit and all of a sudden i have an expression here where a does n't appear , the gain does not appear , and what does this turn into ? this is called v not equals v in , times what ? times r1 plus r2 , divided by r2 . so our amplifier , our feedback circuit came down to v out is v in multiplied by the ratio of the resistors that we added to the circuit . this is one of the really cool properties of using op-amps in circuits , really high-gain amplifiers . what we 've done is we have chosen the gain of our circuit based on the components that we picked to add to the amplifier . it 's not determined by the gain of the amplifier as long as the amplifier gain is really , really big . and for op-amps , that 's a good assumption , it is really big . so this expression came out with a positive sign , right ? all the r 's are positive values , so this is referred to as a non-inverting op-amp circuit amplifier . so just to do a quick example , if r1 and r2 are the same , then we end up with an expression that looks like this v out equals r1 plus r2 , r plus r over r is equal to two so the gain is two times v in . so just to do a quick sketch just to remind ourselves what this looks like , this was v in , and we had what out here ? we had a resistor , we had a resistor to the ground , and this is v out . so this is the configuration of a non-inverting amplifier built with an op-amp , the two resistors in this voltage divider string connected to the negative input . so that 's what non-inverting op-amp circuit looks like , and it 's going to be one of the familiar patterns that you see over and over again as you read schematics and you design your own circuits .
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let 's keep going , let 's keep working on this . v not equals a times v plus minus a v not , r2 over r1 plus r2 . alright , so now i 'm going to gather all the v not terms over on the left hand side .
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also at 5.45 how does the expression for v- involve r2 rather than r1 in the numerator ?
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- [ voicover ] major depressive disorder , which is sometimes just referred to as depression , is characterized by prolonged helplessness and discouragement about the future . individuals with this disorder have low self-esteem and very powerful feelings of worthlessness . they also lack the energy to do the things that they used to enjoy , much less the things that they have to do , or the things that they do n't enjoy . they tend to feel socially isolated , and have trouble focusing on important tasks and have trouble making decisions . and this low mood tends to prevade all aspects of their life . there are a number of physical symptoms that also go along with depression . lethargy , so feeling fatigued . individuals with depression also tend to show fluctuations in weight . so either a lot of weight gain or a lot of weight loss . and they might also have trouble sleeping or they might sleep too much . and i think that these physical symptoms are often ignored , because in western cultures , like in the u.s. , we tend to think about depression in terms of moods or in terms of emotional states . but for individuals in some eastern cultures , especially cultures where it might be seen as inappropriate to delve into or talk about feelings and emotions , people in these cultures tend to think about depression and experience depression in terms of these bodily symptoms , and so it 's really important that these are n't discounted . depression or depressive symptoms are the number one reason people seek out mental health services . and , because of this , some people have taken to calling it the common cold of psychological disorders . and i like that term for some reasons and i dislike it for others . what i like about it is that it captures how pervasive this disorder is . it is estimated that 13 % of men and 22 % of women , worldwide , could meet the criteria for depression at least once in their lives . and one study has shown that as high as 31 % of college students might experience this disorder . and these are really high numbers . and so , in that way , i think that this term is appropriate . however , i think that the term common cold does n't really capture the seriousness of this disorder . because depression is n't just feeling down once in awhile . and it 's not feeling sadness or grief at appropriate times , which is just a normal part of life . and i think that this term kind of minimizes that part of the disorder . depression can be triggered by a life event , like a loss or a breakup , but it does n't have to be . it also does n't usually appear alone . it is actually really common for individuals with depression to have other disorders such as anxiety disorders . and i 've been writing in this blue color to kind of signify depression , but i 'm kind of getting bored with it now , so let me switch it up . so there are a number of factors that may be involved in depression and i 'm going to split them up into three categories : biological factors , psychological factors and sociocultural or environmental factors . and let me , let me take a minute to get all of that down . all right . first of all , we know from family and twin studies that there 's a genetic component to depression . and we also know from studies that use functional imaging that individuals with depression show a decreased activation in the prefrontal cortex . and this could be associated with the problems with decision making that people with depression tend to have . as well as their difficulties in generating actions . researchers have also found lower levels of activity in the rewards circuitry in the brain . and this could help to explain why individuals with depression might not find enjoyment in the actions that they once found pleasurable . depression has also been associated with certain neurotransmitters and neurotransmitter regulation . and i 'll abbreviate that here by writing nt for neurotransmitter . research has suggested that individuals with depression might have fewer receptors for serotonin and norepinephrine . and i think that all of this research is amazing . i think that it 's really important . and it 's also really compelling , in a way that research findings that include neuroscience typically are . but with that said , i really want to caution you against oversimplifying these biological factors . to give an example about why i 'm saying this , i wan na talk about the relationship between a certain serotonin transporter gene and depression . and this gene is known as 5-httlpr , and i 'll write that down . and a lot of findings have shown that this gene is involved in depression , but in actuality , it is only associated with depression if the individual with the gene is in a stressful environment . but the story does n't actually end there , it 's actually even more complicated . because it turns out that if an individual with this genetic feature is placed in a warm and positive environment , they actually show a decreased risk for depression . and this is something that we do n't totally understand yet , we 're still trying to figure out why this might be the case . but importantly , i think that this really shows us how complicated biological factors can be . let 's move on to some psychological factors that might influence depression . one theory is based on the concept of learned helplessness . and this theory supposes that if an individual is exposed to aversive situations over and over again , without any power to change or control them , they might begin to feel powerless in a way that might lead to depression . so if someone is exposed to prolonged stress due to family life or bullying or some other cause that they do n't have control over , their helplessness could spiral out of control and they might stop trying to change their situation because they perceive it to be completely helpless . and that 's a behavioral theory or way of thinking about depression , but there are cognitive theories about it as well . and these theories tend to focus on thoughts or beliefs that , with repetition , could trigger depression . and while it 's true that everyone has negative or self-destructive thoughts every once in awhile , generally we are able to step back from them , and we 're able to realize that what we 're thinking is n't completely logical . but sometimes people can get trapped in these thought patterns and they might put too much emphasis on negative thoughts and actions and experiences . and when they ruminate on these things , when they turn them over and over in their minds , it 's possible that these cognitive distortions might lead to depression . another cognitive theory about depression focuses on the concept of attribution or explanatory style . now as we go about our daily lives , we naturally try to understand and explain the events that go on around us . and when we do this , we can either attribute the things we see to internal or external causes . so is it something that i did ? or is it something that happened because of something that is completely out of my control ? did i get a bad grade on a test because i did n't study ? that would be an internal cause . or did i get a bad grade because the teacher made a really unfair test ? which would be an external cause . individuals with depression tend to attribute negative experiences to internal causes . so maybe they 'll think that a friend did n't call or text back because they are unlikeable or unlovable and not because they were at the movies with their family and maybe their phone was off . in addition to this , they tend to see negative experiences as being stable , so they think that they 'll continue to happen in the future . and they also tend to think that they 're global , so they might assume that one friend not calling them back somehow signifies that none of their friends like them and together these things , these internal attributions , these stable attributions and global attributions , these things form a pessimistic attributional style and it might make certain individuals particularly vulnerable to depression . and there are many other psychological theories about depression . things that have to do with coping style or self-esteem , but it can actually be hard to know whether these things cause depression or if they are the result of it . so does a pessimistic attributional style lead to depression or do people with depression tend to have a pessimistic attributional style ? it is n't always clear . environmental and sociocultural factors can also have a strong influence on depression . having a friend or partner or roommate with depression can actually increase the likelihood that individuals around them will also develop depression . and although we do n't know exactly why that is , some researchers suggest that it might have to do with co-rumination , where friends talk about problems and negative events . but instead of discussing how to solve them , they focus on the negative emotions and dwell on future problems and occurrences . and on some level , this is kind of normal . it is perfectly normal for close friends to take on some of the negative feelings of the other , like being sad when they lose someone who is close to them or being angry if they were dumped by their partner . this is just natural empathy . but the same empathy that allows us to comfort our friends when they 're in distress , might also be the reason that depression seems to spread . we also know that individuals with a low socioeconomic status , especially those living in poverty , are more likely to develop depression , as are those who are struggling to keep a job or have just lost a job . and there are other environmental factors as well . social isolation , child abuse , even prejudice have all been implicated in causing depression . and let 's think about this in terms of prejudice . if someone grows up in a household that has negative feelings about homosexuality and if they grow up and begin to have same sex attractions , they 've probably internalized the prejudice after years of hearing it and this could lead to depression . all right , so stepping back for a second , we said that we had biological factors , psychological factors and sociocultural and environmental factors . and when we put all of these things together , we get what is referred to as a biopsychosocial model of depression and this theory acknowledges that all of these factors play a role . so some people are genetically predisposed to the condition , but it only comes about if the situation is right or if we develop certain patterns of thinking .
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however , i think that the term common cold does n't really capture the seriousness of this disorder . because depression is n't just feeling down once in awhile . and it 's not feeling sadness or grief at appropriate times , which is just a normal part of life .
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is it possible a person with clinical depression does n't want to be helped ?
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- [ voicover ] major depressive disorder , which is sometimes just referred to as depression , is characterized by prolonged helplessness and discouragement about the future . individuals with this disorder have low self-esteem and very powerful feelings of worthlessness . they also lack the energy to do the things that they used to enjoy , much less the things that they have to do , or the things that they do n't enjoy . they tend to feel socially isolated , and have trouble focusing on important tasks and have trouble making decisions . and this low mood tends to prevade all aspects of their life . there are a number of physical symptoms that also go along with depression . lethargy , so feeling fatigued . individuals with depression also tend to show fluctuations in weight . so either a lot of weight gain or a lot of weight loss . and they might also have trouble sleeping or they might sleep too much . and i think that these physical symptoms are often ignored , because in western cultures , like in the u.s. , we tend to think about depression in terms of moods or in terms of emotional states . but for individuals in some eastern cultures , especially cultures where it might be seen as inappropriate to delve into or talk about feelings and emotions , people in these cultures tend to think about depression and experience depression in terms of these bodily symptoms , and so it 's really important that these are n't discounted . depression or depressive symptoms are the number one reason people seek out mental health services . and , because of this , some people have taken to calling it the common cold of psychological disorders . and i like that term for some reasons and i dislike it for others . what i like about it is that it captures how pervasive this disorder is . it is estimated that 13 % of men and 22 % of women , worldwide , could meet the criteria for depression at least once in their lives . and one study has shown that as high as 31 % of college students might experience this disorder . and these are really high numbers . and so , in that way , i think that this term is appropriate . however , i think that the term common cold does n't really capture the seriousness of this disorder . because depression is n't just feeling down once in awhile . and it 's not feeling sadness or grief at appropriate times , which is just a normal part of life . and i think that this term kind of minimizes that part of the disorder . depression can be triggered by a life event , like a loss or a breakup , but it does n't have to be . it also does n't usually appear alone . it is actually really common for individuals with depression to have other disorders such as anxiety disorders . and i 've been writing in this blue color to kind of signify depression , but i 'm kind of getting bored with it now , so let me switch it up . so there are a number of factors that may be involved in depression and i 'm going to split them up into three categories : biological factors , psychological factors and sociocultural or environmental factors . and let me , let me take a minute to get all of that down . all right . first of all , we know from family and twin studies that there 's a genetic component to depression . and we also know from studies that use functional imaging that individuals with depression show a decreased activation in the prefrontal cortex . and this could be associated with the problems with decision making that people with depression tend to have . as well as their difficulties in generating actions . researchers have also found lower levels of activity in the rewards circuitry in the brain . and this could help to explain why individuals with depression might not find enjoyment in the actions that they once found pleasurable . depression has also been associated with certain neurotransmitters and neurotransmitter regulation . and i 'll abbreviate that here by writing nt for neurotransmitter . research has suggested that individuals with depression might have fewer receptors for serotonin and norepinephrine . and i think that all of this research is amazing . i think that it 's really important . and it 's also really compelling , in a way that research findings that include neuroscience typically are . but with that said , i really want to caution you against oversimplifying these biological factors . to give an example about why i 'm saying this , i wan na talk about the relationship between a certain serotonin transporter gene and depression . and this gene is known as 5-httlpr , and i 'll write that down . and a lot of findings have shown that this gene is involved in depression , but in actuality , it is only associated with depression if the individual with the gene is in a stressful environment . but the story does n't actually end there , it 's actually even more complicated . because it turns out that if an individual with this genetic feature is placed in a warm and positive environment , they actually show a decreased risk for depression . and this is something that we do n't totally understand yet , we 're still trying to figure out why this might be the case . but importantly , i think that this really shows us how complicated biological factors can be . let 's move on to some psychological factors that might influence depression . one theory is based on the concept of learned helplessness . and this theory supposes that if an individual is exposed to aversive situations over and over again , without any power to change or control them , they might begin to feel powerless in a way that might lead to depression . so if someone is exposed to prolonged stress due to family life or bullying or some other cause that they do n't have control over , their helplessness could spiral out of control and they might stop trying to change their situation because they perceive it to be completely helpless . and that 's a behavioral theory or way of thinking about depression , but there are cognitive theories about it as well . and these theories tend to focus on thoughts or beliefs that , with repetition , could trigger depression . and while it 's true that everyone has negative or self-destructive thoughts every once in awhile , generally we are able to step back from them , and we 're able to realize that what we 're thinking is n't completely logical . but sometimes people can get trapped in these thought patterns and they might put too much emphasis on negative thoughts and actions and experiences . and when they ruminate on these things , when they turn them over and over in their minds , it 's possible that these cognitive distortions might lead to depression . another cognitive theory about depression focuses on the concept of attribution or explanatory style . now as we go about our daily lives , we naturally try to understand and explain the events that go on around us . and when we do this , we can either attribute the things we see to internal or external causes . so is it something that i did ? or is it something that happened because of something that is completely out of my control ? did i get a bad grade on a test because i did n't study ? that would be an internal cause . or did i get a bad grade because the teacher made a really unfair test ? which would be an external cause . individuals with depression tend to attribute negative experiences to internal causes . so maybe they 'll think that a friend did n't call or text back because they are unlikeable or unlovable and not because they were at the movies with their family and maybe their phone was off . in addition to this , they tend to see negative experiences as being stable , so they think that they 'll continue to happen in the future . and they also tend to think that they 're global , so they might assume that one friend not calling them back somehow signifies that none of their friends like them and together these things , these internal attributions , these stable attributions and global attributions , these things form a pessimistic attributional style and it might make certain individuals particularly vulnerable to depression . and there are many other psychological theories about depression . things that have to do with coping style or self-esteem , but it can actually be hard to know whether these things cause depression or if they are the result of it . so does a pessimistic attributional style lead to depression or do people with depression tend to have a pessimistic attributional style ? it is n't always clear . environmental and sociocultural factors can also have a strong influence on depression . having a friend or partner or roommate with depression can actually increase the likelihood that individuals around them will also develop depression . and although we do n't know exactly why that is , some researchers suggest that it might have to do with co-rumination , where friends talk about problems and negative events . but instead of discussing how to solve them , they focus on the negative emotions and dwell on future problems and occurrences . and on some level , this is kind of normal . it is perfectly normal for close friends to take on some of the negative feelings of the other , like being sad when they lose someone who is close to them or being angry if they were dumped by their partner . this is just natural empathy . but the same empathy that allows us to comfort our friends when they 're in distress , might also be the reason that depression seems to spread . we also know that individuals with a low socioeconomic status , especially those living in poverty , are more likely to develop depression , as are those who are struggling to keep a job or have just lost a job . and there are other environmental factors as well . social isolation , child abuse , even prejudice have all been implicated in causing depression . and let 's think about this in terms of prejudice . if someone grows up in a household that has negative feelings about homosexuality and if they grow up and begin to have same sex attractions , they 've probably internalized the prejudice after years of hearing it and this could lead to depression . all right , so stepping back for a second , we said that we had biological factors , psychological factors and sociocultural and environmental factors . and when we put all of these things together , we get what is referred to as a biopsychosocial model of depression and this theory acknowledges that all of these factors play a role . so some people are genetically predisposed to the condition , but it only comes about if the situation is right or if we develop certain patterns of thinking .
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so is it something that i did ? or is it something that happened because of something that is completely out of my control ? did i get a bad grade on a test because i did n't study ?
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is it completely ruled out that depression is caused by some kind of pathogen , like a virus or something ?
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- [ voicover ] major depressive disorder , which is sometimes just referred to as depression , is characterized by prolonged helplessness and discouragement about the future . individuals with this disorder have low self-esteem and very powerful feelings of worthlessness . they also lack the energy to do the things that they used to enjoy , much less the things that they have to do , or the things that they do n't enjoy . they tend to feel socially isolated , and have trouble focusing on important tasks and have trouble making decisions . and this low mood tends to prevade all aspects of their life . there are a number of physical symptoms that also go along with depression . lethargy , so feeling fatigued . individuals with depression also tend to show fluctuations in weight . so either a lot of weight gain or a lot of weight loss . and they might also have trouble sleeping or they might sleep too much . and i think that these physical symptoms are often ignored , because in western cultures , like in the u.s. , we tend to think about depression in terms of moods or in terms of emotional states . but for individuals in some eastern cultures , especially cultures where it might be seen as inappropriate to delve into or talk about feelings and emotions , people in these cultures tend to think about depression and experience depression in terms of these bodily symptoms , and so it 's really important that these are n't discounted . depression or depressive symptoms are the number one reason people seek out mental health services . and , because of this , some people have taken to calling it the common cold of psychological disorders . and i like that term for some reasons and i dislike it for others . what i like about it is that it captures how pervasive this disorder is . it is estimated that 13 % of men and 22 % of women , worldwide , could meet the criteria for depression at least once in their lives . and one study has shown that as high as 31 % of college students might experience this disorder . and these are really high numbers . and so , in that way , i think that this term is appropriate . however , i think that the term common cold does n't really capture the seriousness of this disorder . because depression is n't just feeling down once in awhile . and it 's not feeling sadness or grief at appropriate times , which is just a normal part of life . and i think that this term kind of minimizes that part of the disorder . depression can be triggered by a life event , like a loss or a breakup , but it does n't have to be . it also does n't usually appear alone . it is actually really common for individuals with depression to have other disorders such as anxiety disorders . and i 've been writing in this blue color to kind of signify depression , but i 'm kind of getting bored with it now , so let me switch it up . so there are a number of factors that may be involved in depression and i 'm going to split them up into three categories : biological factors , psychological factors and sociocultural or environmental factors . and let me , let me take a minute to get all of that down . all right . first of all , we know from family and twin studies that there 's a genetic component to depression . and we also know from studies that use functional imaging that individuals with depression show a decreased activation in the prefrontal cortex . and this could be associated with the problems with decision making that people with depression tend to have . as well as their difficulties in generating actions . researchers have also found lower levels of activity in the rewards circuitry in the brain . and this could help to explain why individuals with depression might not find enjoyment in the actions that they once found pleasurable . depression has also been associated with certain neurotransmitters and neurotransmitter regulation . and i 'll abbreviate that here by writing nt for neurotransmitter . research has suggested that individuals with depression might have fewer receptors for serotonin and norepinephrine . and i think that all of this research is amazing . i think that it 's really important . and it 's also really compelling , in a way that research findings that include neuroscience typically are . but with that said , i really want to caution you against oversimplifying these biological factors . to give an example about why i 'm saying this , i wan na talk about the relationship between a certain serotonin transporter gene and depression . and this gene is known as 5-httlpr , and i 'll write that down . and a lot of findings have shown that this gene is involved in depression , but in actuality , it is only associated with depression if the individual with the gene is in a stressful environment . but the story does n't actually end there , it 's actually even more complicated . because it turns out that if an individual with this genetic feature is placed in a warm and positive environment , they actually show a decreased risk for depression . and this is something that we do n't totally understand yet , we 're still trying to figure out why this might be the case . but importantly , i think that this really shows us how complicated biological factors can be . let 's move on to some psychological factors that might influence depression . one theory is based on the concept of learned helplessness . and this theory supposes that if an individual is exposed to aversive situations over and over again , without any power to change or control them , they might begin to feel powerless in a way that might lead to depression . so if someone is exposed to prolonged stress due to family life or bullying or some other cause that they do n't have control over , their helplessness could spiral out of control and they might stop trying to change their situation because they perceive it to be completely helpless . and that 's a behavioral theory or way of thinking about depression , but there are cognitive theories about it as well . and these theories tend to focus on thoughts or beliefs that , with repetition , could trigger depression . and while it 's true that everyone has negative or self-destructive thoughts every once in awhile , generally we are able to step back from them , and we 're able to realize that what we 're thinking is n't completely logical . but sometimes people can get trapped in these thought patterns and they might put too much emphasis on negative thoughts and actions and experiences . and when they ruminate on these things , when they turn them over and over in their minds , it 's possible that these cognitive distortions might lead to depression . another cognitive theory about depression focuses on the concept of attribution or explanatory style . now as we go about our daily lives , we naturally try to understand and explain the events that go on around us . and when we do this , we can either attribute the things we see to internal or external causes . so is it something that i did ? or is it something that happened because of something that is completely out of my control ? did i get a bad grade on a test because i did n't study ? that would be an internal cause . or did i get a bad grade because the teacher made a really unfair test ? which would be an external cause . individuals with depression tend to attribute negative experiences to internal causes . so maybe they 'll think that a friend did n't call or text back because they are unlikeable or unlovable and not because they were at the movies with their family and maybe their phone was off . in addition to this , they tend to see negative experiences as being stable , so they think that they 'll continue to happen in the future . and they also tend to think that they 're global , so they might assume that one friend not calling them back somehow signifies that none of their friends like them and together these things , these internal attributions , these stable attributions and global attributions , these things form a pessimistic attributional style and it might make certain individuals particularly vulnerable to depression . and there are many other psychological theories about depression . things that have to do with coping style or self-esteem , but it can actually be hard to know whether these things cause depression or if they are the result of it . so does a pessimistic attributional style lead to depression or do people with depression tend to have a pessimistic attributional style ? it is n't always clear . environmental and sociocultural factors can also have a strong influence on depression . having a friend or partner or roommate with depression can actually increase the likelihood that individuals around them will also develop depression . and although we do n't know exactly why that is , some researchers suggest that it might have to do with co-rumination , where friends talk about problems and negative events . but instead of discussing how to solve them , they focus on the negative emotions and dwell on future problems and occurrences . and on some level , this is kind of normal . it is perfectly normal for close friends to take on some of the negative feelings of the other , like being sad when they lose someone who is close to them or being angry if they were dumped by their partner . this is just natural empathy . but the same empathy that allows us to comfort our friends when they 're in distress , might also be the reason that depression seems to spread . we also know that individuals with a low socioeconomic status , especially those living in poverty , are more likely to develop depression , as are those who are struggling to keep a job or have just lost a job . and there are other environmental factors as well . social isolation , child abuse , even prejudice have all been implicated in causing depression . and let 's think about this in terms of prejudice . if someone grows up in a household that has negative feelings about homosexuality and if they grow up and begin to have same sex attractions , they 've probably internalized the prejudice after years of hearing it and this could lead to depression . all right , so stepping back for a second , we said that we had biological factors , psychological factors and sociocultural and environmental factors . and when we put all of these things together , we get what is referred to as a biopsychosocial model of depression and this theory acknowledges that all of these factors play a role . so some people are genetically predisposed to the condition , but it only comes about if the situation is right or if we develop certain patterns of thinking .
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and i 'll abbreviate that here by writing nt for neurotransmitter . research has suggested that individuals with depression might have fewer receptors for serotonin and norepinephrine . and i think that all of this research is amazing .
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..i wan na ask if the depression of a genetic cause like fewer receptors for serotonin why the symptoms of depression appear suddenly however , this patient has this number of receptors throughout life ?
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- [ voicover ] major depressive disorder , which is sometimes just referred to as depression , is characterized by prolonged helplessness and discouragement about the future . individuals with this disorder have low self-esteem and very powerful feelings of worthlessness . they also lack the energy to do the things that they used to enjoy , much less the things that they have to do , or the things that they do n't enjoy . they tend to feel socially isolated , and have trouble focusing on important tasks and have trouble making decisions . and this low mood tends to prevade all aspects of their life . there are a number of physical symptoms that also go along with depression . lethargy , so feeling fatigued . individuals with depression also tend to show fluctuations in weight . so either a lot of weight gain or a lot of weight loss . and they might also have trouble sleeping or they might sleep too much . and i think that these physical symptoms are often ignored , because in western cultures , like in the u.s. , we tend to think about depression in terms of moods or in terms of emotional states . but for individuals in some eastern cultures , especially cultures where it might be seen as inappropriate to delve into or talk about feelings and emotions , people in these cultures tend to think about depression and experience depression in terms of these bodily symptoms , and so it 's really important that these are n't discounted . depression or depressive symptoms are the number one reason people seek out mental health services . and , because of this , some people have taken to calling it the common cold of psychological disorders . and i like that term for some reasons and i dislike it for others . what i like about it is that it captures how pervasive this disorder is . it is estimated that 13 % of men and 22 % of women , worldwide , could meet the criteria for depression at least once in their lives . and one study has shown that as high as 31 % of college students might experience this disorder . and these are really high numbers . and so , in that way , i think that this term is appropriate . however , i think that the term common cold does n't really capture the seriousness of this disorder . because depression is n't just feeling down once in awhile . and it 's not feeling sadness or grief at appropriate times , which is just a normal part of life . and i think that this term kind of minimizes that part of the disorder . depression can be triggered by a life event , like a loss or a breakup , but it does n't have to be . it also does n't usually appear alone . it is actually really common for individuals with depression to have other disorders such as anxiety disorders . and i 've been writing in this blue color to kind of signify depression , but i 'm kind of getting bored with it now , so let me switch it up . so there are a number of factors that may be involved in depression and i 'm going to split them up into three categories : biological factors , psychological factors and sociocultural or environmental factors . and let me , let me take a minute to get all of that down . all right . first of all , we know from family and twin studies that there 's a genetic component to depression . and we also know from studies that use functional imaging that individuals with depression show a decreased activation in the prefrontal cortex . and this could be associated with the problems with decision making that people with depression tend to have . as well as their difficulties in generating actions . researchers have also found lower levels of activity in the rewards circuitry in the brain . and this could help to explain why individuals with depression might not find enjoyment in the actions that they once found pleasurable . depression has also been associated with certain neurotransmitters and neurotransmitter regulation . and i 'll abbreviate that here by writing nt for neurotransmitter . research has suggested that individuals with depression might have fewer receptors for serotonin and norepinephrine . and i think that all of this research is amazing . i think that it 's really important . and it 's also really compelling , in a way that research findings that include neuroscience typically are . but with that said , i really want to caution you against oversimplifying these biological factors . to give an example about why i 'm saying this , i wan na talk about the relationship between a certain serotonin transporter gene and depression . and this gene is known as 5-httlpr , and i 'll write that down . and a lot of findings have shown that this gene is involved in depression , but in actuality , it is only associated with depression if the individual with the gene is in a stressful environment . but the story does n't actually end there , it 's actually even more complicated . because it turns out that if an individual with this genetic feature is placed in a warm and positive environment , they actually show a decreased risk for depression . and this is something that we do n't totally understand yet , we 're still trying to figure out why this might be the case . but importantly , i think that this really shows us how complicated biological factors can be . let 's move on to some psychological factors that might influence depression . one theory is based on the concept of learned helplessness . and this theory supposes that if an individual is exposed to aversive situations over and over again , without any power to change or control them , they might begin to feel powerless in a way that might lead to depression . so if someone is exposed to prolonged stress due to family life or bullying or some other cause that they do n't have control over , their helplessness could spiral out of control and they might stop trying to change their situation because they perceive it to be completely helpless . and that 's a behavioral theory or way of thinking about depression , but there are cognitive theories about it as well . and these theories tend to focus on thoughts or beliefs that , with repetition , could trigger depression . and while it 's true that everyone has negative or self-destructive thoughts every once in awhile , generally we are able to step back from them , and we 're able to realize that what we 're thinking is n't completely logical . but sometimes people can get trapped in these thought patterns and they might put too much emphasis on negative thoughts and actions and experiences . and when they ruminate on these things , when they turn them over and over in their minds , it 's possible that these cognitive distortions might lead to depression . another cognitive theory about depression focuses on the concept of attribution or explanatory style . now as we go about our daily lives , we naturally try to understand and explain the events that go on around us . and when we do this , we can either attribute the things we see to internal or external causes . so is it something that i did ? or is it something that happened because of something that is completely out of my control ? did i get a bad grade on a test because i did n't study ? that would be an internal cause . or did i get a bad grade because the teacher made a really unfair test ? which would be an external cause . individuals with depression tend to attribute negative experiences to internal causes . so maybe they 'll think that a friend did n't call or text back because they are unlikeable or unlovable and not because they were at the movies with their family and maybe their phone was off . in addition to this , they tend to see negative experiences as being stable , so they think that they 'll continue to happen in the future . and they also tend to think that they 're global , so they might assume that one friend not calling them back somehow signifies that none of their friends like them and together these things , these internal attributions , these stable attributions and global attributions , these things form a pessimistic attributional style and it might make certain individuals particularly vulnerable to depression . and there are many other psychological theories about depression . things that have to do with coping style or self-esteem , but it can actually be hard to know whether these things cause depression or if they are the result of it . so does a pessimistic attributional style lead to depression or do people with depression tend to have a pessimistic attributional style ? it is n't always clear . environmental and sociocultural factors can also have a strong influence on depression . having a friend or partner or roommate with depression can actually increase the likelihood that individuals around them will also develop depression . and although we do n't know exactly why that is , some researchers suggest that it might have to do with co-rumination , where friends talk about problems and negative events . but instead of discussing how to solve them , they focus on the negative emotions and dwell on future problems and occurrences . and on some level , this is kind of normal . it is perfectly normal for close friends to take on some of the negative feelings of the other , like being sad when they lose someone who is close to them or being angry if they were dumped by their partner . this is just natural empathy . but the same empathy that allows us to comfort our friends when they 're in distress , might also be the reason that depression seems to spread . we also know that individuals with a low socioeconomic status , especially those living in poverty , are more likely to develop depression , as are those who are struggling to keep a job or have just lost a job . and there are other environmental factors as well . social isolation , child abuse , even prejudice have all been implicated in causing depression . and let 's think about this in terms of prejudice . if someone grows up in a household that has negative feelings about homosexuality and if they grow up and begin to have same sex attractions , they 've probably internalized the prejudice after years of hearing it and this could lead to depression . all right , so stepping back for a second , we said that we had biological factors , psychological factors and sociocultural and environmental factors . and when we put all of these things together , we get what is referred to as a biopsychosocial model of depression and this theory acknowledges that all of these factors play a role . so some people are genetically predisposed to the condition , but it only comes about if the situation is right or if we develop certain patterns of thinking .
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and i 've been writing in this blue color to kind of signify depression , but i 'm kind of getting bored with it now , so let me switch it up . so there are a number of factors that may be involved in depression and i 'm going to split them up into three categories : biological factors , psychological factors and sociocultural or environmental factors . and let me , let me take a minute to get all of that down .
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7 would it not be more accurate if you call it `` biochemical factors '' instead of `` biological factors '' ?
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- [ voicover ] major depressive disorder , which is sometimes just referred to as depression , is characterized by prolonged helplessness and discouragement about the future . individuals with this disorder have low self-esteem and very powerful feelings of worthlessness . they also lack the energy to do the things that they used to enjoy , much less the things that they have to do , or the things that they do n't enjoy . they tend to feel socially isolated , and have trouble focusing on important tasks and have trouble making decisions . and this low mood tends to prevade all aspects of their life . there are a number of physical symptoms that also go along with depression . lethargy , so feeling fatigued . individuals with depression also tend to show fluctuations in weight . so either a lot of weight gain or a lot of weight loss . and they might also have trouble sleeping or they might sleep too much . and i think that these physical symptoms are often ignored , because in western cultures , like in the u.s. , we tend to think about depression in terms of moods or in terms of emotional states . but for individuals in some eastern cultures , especially cultures where it might be seen as inappropriate to delve into or talk about feelings and emotions , people in these cultures tend to think about depression and experience depression in terms of these bodily symptoms , and so it 's really important that these are n't discounted . depression or depressive symptoms are the number one reason people seek out mental health services . and , because of this , some people have taken to calling it the common cold of psychological disorders . and i like that term for some reasons and i dislike it for others . what i like about it is that it captures how pervasive this disorder is . it is estimated that 13 % of men and 22 % of women , worldwide , could meet the criteria for depression at least once in their lives . and one study has shown that as high as 31 % of college students might experience this disorder . and these are really high numbers . and so , in that way , i think that this term is appropriate . however , i think that the term common cold does n't really capture the seriousness of this disorder . because depression is n't just feeling down once in awhile . and it 's not feeling sadness or grief at appropriate times , which is just a normal part of life . and i think that this term kind of minimizes that part of the disorder . depression can be triggered by a life event , like a loss or a breakup , but it does n't have to be . it also does n't usually appear alone . it is actually really common for individuals with depression to have other disorders such as anxiety disorders . and i 've been writing in this blue color to kind of signify depression , but i 'm kind of getting bored with it now , so let me switch it up . so there are a number of factors that may be involved in depression and i 'm going to split them up into three categories : biological factors , psychological factors and sociocultural or environmental factors . and let me , let me take a minute to get all of that down . all right . first of all , we know from family and twin studies that there 's a genetic component to depression . and we also know from studies that use functional imaging that individuals with depression show a decreased activation in the prefrontal cortex . and this could be associated with the problems with decision making that people with depression tend to have . as well as their difficulties in generating actions . researchers have also found lower levels of activity in the rewards circuitry in the brain . and this could help to explain why individuals with depression might not find enjoyment in the actions that they once found pleasurable . depression has also been associated with certain neurotransmitters and neurotransmitter regulation . and i 'll abbreviate that here by writing nt for neurotransmitter . research has suggested that individuals with depression might have fewer receptors for serotonin and norepinephrine . and i think that all of this research is amazing . i think that it 's really important . and it 's also really compelling , in a way that research findings that include neuroscience typically are . but with that said , i really want to caution you against oversimplifying these biological factors . to give an example about why i 'm saying this , i wan na talk about the relationship between a certain serotonin transporter gene and depression . and this gene is known as 5-httlpr , and i 'll write that down . and a lot of findings have shown that this gene is involved in depression , but in actuality , it is only associated with depression if the individual with the gene is in a stressful environment . but the story does n't actually end there , it 's actually even more complicated . because it turns out that if an individual with this genetic feature is placed in a warm and positive environment , they actually show a decreased risk for depression . and this is something that we do n't totally understand yet , we 're still trying to figure out why this might be the case . but importantly , i think that this really shows us how complicated biological factors can be . let 's move on to some psychological factors that might influence depression . one theory is based on the concept of learned helplessness . and this theory supposes that if an individual is exposed to aversive situations over and over again , without any power to change or control them , they might begin to feel powerless in a way that might lead to depression . so if someone is exposed to prolonged stress due to family life or bullying or some other cause that they do n't have control over , their helplessness could spiral out of control and they might stop trying to change their situation because they perceive it to be completely helpless . and that 's a behavioral theory or way of thinking about depression , but there are cognitive theories about it as well . and these theories tend to focus on thoughts or beliefs that , with repetition , could trigger depression . and while it 's true that everyone has negative or self-destructive thoughts every once in awhile , generally we are able to step back from them , and we 're able to realize that what we 're thinking is n't completely logical . but sometimes people can get trapped in these thought patterns and they might put too much emphasis on negative thoughts and actions and experiences . and when they ruminate on these things , when they turn them over and over in their minds , it 's possible that these cognitive distortions might lead to depression . another cognitive theory about depression focuses on the concept of attribution or explanatory style . now as we go about our daily lives , we naturally try to understand and explain the events that go on around us . and when we do this , we can either attribute the things we see to internal or external causes . so is it something that i did ? or is it something that happened because of something that is completely out of my control ? did i get a bad grade on a test because i did n't study ? that would be an internal cause . or did i get a bad grade because the teacher made a really unfair test ? which would be an external cause . individuals with depression tend to attribute negative experiences to internal causes . so maybe they 'll think that a friend did n't call or text back because they are unlikeable or unlovable and not because they were at the movies with their family and maybe their phone was off . in addition to this , they tend to see negative experiences as being stable , so they think that they 'll continue to happen in the future . and they also tend to think that they 're global , so they might assume that one friend not calling them back somehow signifies that none of their friends like them and together these things , these internal attributions , these stable attributions and global attributions , these things form a pessimistic attributional style and it might make certain individuals particularly vulnerable to depression . and there are many other psychological theories about depression . things that have to do with coping style or self-esteem , but it can actually be hard to know whether these things cause depression or if they are the result of it . so does a pessimistic attributional style lead to depression or do people with depression tend to have a pessimistic attributional style ? it is n't always clear . environmental and sociocultural factors can also have a strong influence on depression . having a friend or partner or roommate with depression can actually increase the likelihood that individuals around them will also develop depression . and although we do n't know exactly why that is , some researchers suggest that it might have to do with co-rumination , where friends talk about problems and negative events . but instead of discussing how to solve them , they focus on the negative emotions and dwell on future problems and occurrences . and on some level , this is kind of normal . it is perfectly normal for close friends to take on some of the negative feelings of the other , like being sad when they lose someone who is close to them or being angry if they were dumped by their partner . this is just natural empathy . but the same empathy that allows us to comfort our friends when they 're in distress , might also be the reason that depression seems to spread . we also know that individuals with a low socioeconomic status , especially those living in poverty , are more likely to develop depression , as are those who are struggling to keep a job or have just lost a job . and there are other environmental factors as well . social isolation , child abuse , even prejudice have all been implicated in causing depression . and let 's think about this in terms of prejudice . if someone grows up in a household that has negative feelings about homosexuality and if they grow up and begin to have same sex attractions , they 've probably internalized the prejudice after years of hearing it and this could lead to depression . all right , so stepping back for a second , we said that we had biological factors , psychological factors and sociocultural and environmental factors . and when we put all of these things together , we get what is referred to as a biopsychosocial model of depression and this theory acknowledges that all of these factors play a role . so some people are genetically predisposed to the condition , but it only comes about if the situation is right or if we develop certain patterns of thinking .
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it is n't always clear . environmental and sociocultural factors can also have a strong influence on depression . having a friend or partner or roommate with depression can actually increase the likelihood that individuals around them will also develop depression . and although we do n't know exactly why that is , some researchers suggest that it might have to do with co-rumination , where friends talk about problems and negative events .
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which age group mostly suffers from depression ?
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- [ voicover ] major depressive disorder , which is sometimes just referred to as depression , is characterized by prolonged helplessness and discouragement about the future . individuals with this disorder have low self-esteem and very powerful feelings of worthlessness . they also lack the energy to do the things that they used to enjoy , much less the things that they have to do , or the things that they do n't enjoy . they tend to feel socially isolated , and have trouble focusing on important tasks and have trouble making decisions . and this low mood tends to prevade all aspects of their life . there are a number of physical symptoms that also go along with depression . lethargy , so feeling fatigued . individuals with depression also tend to show fluctuations in weight . so either a lot of weight gain or a lot of weight loss . and they might also have trouble sleeping or they might sleep too much . and i think that these physical symptoms are often ignored , because in western cultures , like in the u.s. , we tend to think about depression in terms of moods or in terms of emotional states . but for individuals in some eastern cultures , especially cultures where it might be seen as inappropriate to delve into or talk about feelings and emotions , people in these cultures tend to think about depression and experience depression in terms of these bodily symptoms , and so it 's really important that these are n't discounted . depression or depressive symptoms are the number one reason people seek out mental health services . and , because of this , some people have taken to calling it the common cold of psychological disorders . and i like that term for some reasons and i dislike it for others . what i like about it is that it captures how pervasive this disorder is . it is estimated that 13 % of men and 22 % of women , worldwide , could meet the criteria for depression at least once in their lives . and one study has shown that as high as 31 % of college students might experience this disorder . and these are really high numbers . and so , in that way , i think that this term is appropriate . however , i think that the term common cold does n't really capture the seriousness of this disorder . because depression is n't just feeling down once in awhile . and it 's not feeling sadness or grief at appropriate times , which is just a normal part of life . and i think that this term kind of minimizes that part of the disorder . depression can be triggered by a life event , like a loss or a breakup , but it does n't have to be . it also does n't usually appear alone . it is actually really common for individuals with depression to have other disorders such as anxiety disorders . and i 've been writing in this blue color to kind of signify depression , but i 'm kind of getting bored with it now , so let me switch it up . so there are a number of factors that may be involved in depression and i 'm going to split them up into three categories : biological factors , psychological factors and sociocultural or environmental factors . and let me , let me take a minute to get all of that down . all right . first of all , we know from family and twin studies that there 's a genetic component to depression . and we also know from studies that use functional imaging that individuals with depression show a decreased activation in the prefrontal cortex . and this could be associated with the problems with decision making that people with depression tend to have . as well as their difficulties in generating actions . researchers have also found lower levels of activity in the rewards circuitry in the brain . and this could help to explain why individuals with depression might not find enjoyment in the actions that they once found pleasurable . depression has also been associated with certain neurotransmitters and neurotransmitter regulation . and i 'll abbreviate that here by writing nt for neurotransmitter . research has suggested that individuals with depression might have fewer receptors for serotonin and norepinephrine . and i think that all of this research is amazing . i think that it 's really important . and it 's also really compelling , in a way that research findings that include neuroscience typically are . but with that said , i really want to caution you against oversimplifying these biological factors . to give an example about why i 'm saying this , i wan na talk about the relationship between a certain serotonin transporter gene and depression . and this gene is known as 5-httlpr , and i 'll write that down . and a lot of findings have shown that this gene is involved in depression , but in actuality , it is only associated with depression if the individual with the gene is in a stressful environment . but the story does n't actually end there , it 's actually even more complicated . because it turns out that if an individual with this genetic feature is placed in a warm and positive environment , they actually show a decreased risk for depression . and this is something that we do n't totally understand yet , we 're still trying to figure out why this might be the case . but importantly , i think that this really shows us how complicated biological factors can be . let 's move on to some psychological factors that might influence depression . one theory is based on the concept of learned helplessness . and this theory supposes that if an individual is exposed to aversive situations over and over again , without any power to change or control them , they might begin to feel powerless in a way that might lead to depression . so if someone is exposed to prolonged stress due to family life or bullying or some other cause that they do n't have control over , their helplessness could spiral out of control and they might stop trying to change their situation because they perceive it to be completely helpless . and that 's a behavioral theory or way of thinking about depression , but there are cognitive theories about it as well . and these theories tend to focus on thoughts or beliefs that , with repetition , could trigger depression . and while it 's true that everyone has negative or self-destructive thoughts every once in awhile , generally we are able to step back from them , and we 're able to realize that what we 're thinking is n't completely logical . but sometimes people can get trapped in these thought patterns and they might put too much emphasis on negative thoughts and actions and experiences . and when they ruminate on these things , when they turn them over and over in their minds , it 's possible that these cognitive distortions might lead to depression . another cognitive theory about depression focuses on the concept of attribution or explanatory style . now as we go about our daily lives , we naturally try to understand and explain the events that go on around us . and when we do this , we can either attribute the things we see to internal or external causes . so is it something that i did ? or is it something that happened because of something that is completely out of my control ? did i get a bad grade on a test because i did n't study ? that would be an internal cause . or did i get a bad grade because the teacher made a really unfair test ? which would be an external cause . individuals with depression tend to attribute negative experiences to internal causes . so maybe they 'll think that a friend did n't call or text back because they are unlikeable or unlovable and not because they were at the movies with their family and maybe their phone was off . in addition to this , they tend to see negative experiences as being stable , so they think that they 'll continue to happen in the future . and they also tend to think that they 're global , so they might assume that one friend not calling them back somehow signifies that none of their friends like them and together these things , these internal attributions , these stable attributions and global attributions , these things form a pessimistic attributional style and it might make certain individuals particularly vulnerable to depression . and there are many other psychological theories about depression . things that have to do with coping style or self-esteem , but it can actually be hard to know whether these things cause depression or if they are the result of it . so does a pessimistic attributional style lead to depression or do people with depression tend to have a pessimistic attributional style ? it is n't always clear . environmental and sociocultural factors can also have a strong influence on depression . having a friend or partner or roommate with depression can actually increase the likelihood that individuals around them will also develop depression . and although we do n't know exactly why that is , some researchers suggest that it might have to do with co-rumination , where friends talk about problems and negative events . but instead of discussing how to solve them , they focus on the negative emotions and dwell on future problems and occurrences . and on some level , this is kind of normal . it is perfectly normal for close friends to take on some of the negative feelings of the other , like being sad when they lose someone who is close to them or being angry if they were dumped by their partner . this is just natural empathy . but the same empathy that allows us to comfort our friends when they 're in distress , might also be the reason that depression seems to spread . we also know that individuals with a low socioeconomic status , especially those living in poverty , are more likely to develop depression , as are those who are struggling to keep a job or have just lost a job . and there are other environmental factors as well . social isolation , child abuse , even prejudice have all been implicated in causing depression . and let 's think about this in terms of prejudice . if someone grows up in a household that has negative feelings about homosexuality and if they grow up and begin to have same sex attractions , they 've probably internalized the prejudice after years of hearing it and this could lead to depression . all right , so stepping back for a second , we said that we had biological factors , psychological factors and sociocultural and environmental factors . and when we put all of these things together , we get what is referred to as a biopsychosocial model of depression and this theory acknowledges that all of these factors play a role . so some people are genetically predisposed to the condition , but it only comes about if the situation is right or if we develop certain patterns of thinking .
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and on some level , this is kind of normal . it is perfectly normal for close friends to take on some of the negative feelings of the other , like being sad when they lose someone who is close to them or being angry if they were dumped by their partner . this is just natural empathy .
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can someone be depressed but not be suicidal ?
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- [ voicover ] major depressive disorder , which is sometimes just referred to as depression , is characterized by prolonged helplessness and discouragement about the future . individuals with this disorder have low self-esteem and very powerful feelings of worthlessness . they also lack the energy to do the things that they used to enjoy , much less the things that they have to do , or the things that they do n't enjoy . they tend to feel socially isolated , and have trouble focusing on important tasks and have trouble making decisions . and this low mood tends to prevade all aspects of their life . there are a number of physical symptoms that also go along with depression . lethargy , so feeling fatigued . individuals with depression also tend to show fluctuations in weight . so either a lot of weight gain or a lot of weight loss . and they might also have trouble sleeping or they might sleep too much . and i think that these physical symptoms are often ignored , because in western cultures , like in the u.s. , we tend to think about depression in terms of moods or in terms of emotional states . but for individuals in some eastern cultures , especially cultures where it might be seen as inappropriate to delve into or talk about feelings and emotions , people in these cultures tend to think about depression and experience depression in terms of these bodily symptoms , and so it 's really important that these are n't discounted . depression or depressive symptoms are the number one reason people seek out mental health services . and , because of this , some people have taken to calling it the common cold of psychological disorders . and i like that term for some reasons and i dislike it for others . what i like about it is that it captures how pervasive this disorder is . it is estimated that 13 % of men and 22 % of women , worldwide , could meet the criteria for depression at least once in their lives . and one study has shown that as high as 31 % of college students might experience this disorder . and these are really high numbers . and so , in that way , i think that this term is appropriate . however , i think that the term common cold does n't really capture the seriousness of this disorder . because depression is n't just feeling down once in awhile . and it 's not feeling sadness or grief at appropriate times , which is just a normal part of life . and i think that this term kind of minimizes that part of the disorder . depression can be triggered by a life event , like a loss or a breakup , but it does n't have to be . it also does n't usually appear alone . it is actually really common for individuals with depression to have other disorders such as anxiety disorders . and i 've been writing in this blue color to kind of signify depression , but i 'm kind of getting bored with it now , so let me switch it up . so there are a number of factors that may be involved in depression and i 'm going to split them up into three categories : biological factors , psychological factors and sociocultural or environmental factors . and let me , let me take a minute to get all of that down . all right . first of all , we know from family and twin studies that there 's a genetic component to depression . and we also know from studies that use functional imaging that individuals with depression show a decreased activation in the prefrontal cortex . and this could be associated with the problems with decision making that people with depression tend to have . as well as their difficulties in generating actions . researchers have also found lower levels of activity in the rewards circuitry in the brain . and this could help to explain why individuals with depression might not find enjoyment in the actions that they once found pleasurable . depression has also been associated with certain neurotransmitters and neurotransmitter regulation . and i 'll abbreviate that here by writing nt for neurotransmitter . research has suggested that individuals with depression might have fewer receptors for serotonin and norepinephrine . and i think that all of this research is amazing . i think that it 's really important . and it 's also really compelling , in a way that research findings that include neuroscience typically are . but with that said , i really want to caution you against oversimplifying these biological factors . to give an example about why i 'm saying this , i wan na talk about the relationship between a certain serotonin transporter gene and depression . and this gene is known as 5-httlpr , and i 'll write that down . and a lot of findings have shown that this gene is involved in depression , but in actuality , it is only associated with depression if the individual with the gene is in a stressful environment . but the story does n't actually end there , it 's actually even more complicated . because it turns out that if an individual with this genetic feature is placed in a warm and positive environment , they actually show a decreased risk for depression . and this is something that we do n't totally understand yet , we 're still trying to figure out why this might be the case . but importantly , i think that this really shows us how complicated biological factors can be . let 's move on to some psychological factors that might influence depression . one theory is based on the concept of learned helplessness . and this theory supposes that if an individual is exposed to aversive situations over and over again , without any power to change or control them , they might begin to feel powerless in a way that might lead to depression . so if someone is exposed to prolonged stress due to family life or bullying or some other cause that they do n't have control over , their helplessness could spiral out of control and they might stop trying to change their situation because they perceive it to be completely helpless . and that 's a behavioral theory or way of thinking about depression , but there are cognitive theories about it as well . and these theories tend to focus on thoughts or beliefs that , with repetition , could trigger depression . and while it 's true that everyone has negative or self-destructive thoughts every once in awhile , generally we are able to step back from them , and we 're able to realize that what we 're thinking is n't completely logical . but sometimes people can get trapped in these thought patterns and they might put too much emphasis on negative thoughts and actions and experiences . and when they ruminate on these things , when they turn them over and over in their minds , it 's possible that these cognitive distortions might lead to depression . another cognitive theory about depression focuses on the concept of attribution or explanatory style . now as we go about our daily lives , we naturally try to understand and explain the events that go on around us . and when we do this , we can either attribute the things we see to internal or external causes . so is it something that i did ? or is it something that happened because of something that is completely out of my control ? did i get a bad grade on a test because i did n't study ? that would be an internal cause . or did i get a bad grade because the teacher made a really unfair test ? which would be an external cause . individuals with depression tend to attribute negative experiences to internal causes . so maybe they 'll think that a friend did n't call or text back because they are unlikeable or unlovable and not because they were at the movies with their family and maybe their phone was off . in addition to this , they tend to see negative experiences as being stable , so they think that they 'll continue to happen in the future . and they also tend to think that they 're global , so they might assume that one friend not calling them back somehow signifies that none of their friends like them and together these things , these internal attributions , these stable attributions and global attributions , these things form a pessimistic attributional style and it might make certain individuals particularly vulnerable to depression . and there are many other psychological theories about depression . things that have to do with coping style or self-esteem , but it can actually be hard to know whether these things cause depression or if they are the result of it . so does a pessimistic attributional style lead to depression or do people with depression tend to have a pessimistic attributional style ? it is n't always clear . environmental and sociocultural factors can also have a strong influence on depression . having a friend or partner or roommate with depression can actually increase the likelihood that individuals around them will also develop depression . and although we do n't know exactly why that is , some researchers suggest that it might have to do with co-rumination , where friends talk about problems and negative events . but instead of discussing how to solve them , they focus on the negative emotions and dwell on future problems and occurrences . and on some level , this is kind of normal . it is perfectly normal for close friends to take on some of the negative feelings of the other , like being sad when they lose someone who is close to them or being angry if they were dumped by their partner . this is just natural empathy . but the same empathy that allows us to comfort our friends when they 're in distress , might also be the reason that depression seems to spread . we also know that individuals with a low socioeconomic status , especially those living in poverty , are more likely to develop depression , as are those who are struggling to keep a job or have just lost a job . and there are other environmental factors as well . social isolation , child abuse , even prejudice have all been implicated in causing depression . and let 's think about this in terms of prejudice . if someone grows up in a household that has negative feelings about homosexuality and if they grow up and begin to have same sex attractions , they 've probably internalized the prejudice after years of hearing it and this could lead to depression . all right , so stepping back for a second , we said that we had biological factors , psychological factors and sociocultural and environmental factors . and when we put all of these things together , we get what is referred to as a biopsychosocial model of depression and this theory acknowledges that all of these factors play a role . so some people are genetically predisposed to the condition , but it only comes about if the situation is right or if we develop certain patterns of thinking .
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however , i think that the term common cold does n't really capture the seriousness of this disorder . because depression is n't just feeling down once in awhile . and it 's not feeling sadness or grief at appropriate times , which is just a normal part of life .
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or is depression always accompanied with feeling suicidal ?
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- [ voicover ] major depressive disorder , which is sometimes just referred to as depression , is characterized by prolonged helplessness and discouragement about the future . individuals with this disorder have low self-esteem and very powerful feelings of worthlessness . they also lack the energy to do the things that they used to enjoy , much less the things that they have to do , or the things that they do n't enjoy . they tend to feel socially isolated , and have trouble focusing on important tasks and have trouble making decisions . and this low mood tends to prevade all aspects of their life . there are a number of physical symptoms that also go along with depression . lethargy , so feeling fatigued . individuals with depression also tend to show fluctuations in weight . so either a lot of weight gain or a lot of weight loss . and they might also have trouble sleeping or they might sleep too much . and i think that these physical symptoms are often ignored , because in western cultures , like in the u.s. , we tend to think about depression in terms of moods or in terms of emotional states . but for individuals in some eastern cultures , especially cultures where it might be seen as inappropriate to delve into or talk about feelings and emotions , people in these cultures tend to think about depression and experience depression in terms of these bodily symptoms , and so it 's really important that these are n't discounted . depression or depressive symptoms are the number one reason people seek out mental health services . and , because of this , some people have taken to calling it the common cold of psychological disorders . and i like that term for some reasons and i dislike it for others . what i like about it is that it captures how pervasive this disorder is . it is estimated that 13 % of men and 22 % of women , worldwide , could meet the criteria for depression at least once in their lives . and one study has shown that as high as 31 % of college students might experience this disorder . and these are really high numbers . and so , in that way , i think that this term is appropriate . however , i think that the term common cold does n't really capture the seriousness of this disorder . because depression is n't just feeling down once in awhile . and it 's not feeling sadness or grief at appropriate times , which is just a normal part of life . and i think that this term kind of minimizes that part of the disorder . depression can be triggered by a life event , like a loss or a breakup , but it does n't have to be . it also does n't usually appear alone . it is actually really common for individuals with depression to have other disorders such as anxiety disorders . and i 've been writing in this blue color to kind of signify depression , but i 'm kind of getting bored with it now , so let me switch it up . so there are a number of factors that may be involved in depression and i 'm going to split them up into three categories : biological factors , psychological factors and sociocultural or environmental factors . and let me , let me take a minute to get all of that down . all right . first of all , we know from family and twin studies that there 's a genetic component to depression . and we also know from studies that use functional imaging that individuals with depression show a decreased activation in the prefrontal cortex . and this could be associated with the problems with decision making that people with depression tend to have . as well as their difficulties in generating actions . researchers have also found lower levels of activity in the rewards circuitry in the brain . and this could help to explain why individuals with depression might not find enjoyment in the actions that they once found pleasurable . depression has also been associated with certain neurotransmitters and neurotransmitter regulation . and i 'll abbreviate that here by writing nt for neurotransmitter . research has suggested that individuals with depression might have fewer receptors for serotonin and norepinephrine . and i think that all of this research is amazing . i think that it 's really important . and it 's also really compelling , in a way that research findings that include neuroscience typically are . but with that said , i really want to caution you against oversimplifying these biological factors . to give an example about why i 'm saying this , i wan na talk about the relationship between a certain serotonin transporter gene and depression . and this gene is known as 5-httlpr , and i 'll write that down . and a lot of findings have shown that this gene is involved in depression , but in actuality , it is only associated with depression if the individual with the gene is in a stressful environment . but the story does n't actually end there , it 's actually even more complicated . because it turns out that if an individual with this genetic feature is placed in a warm and positive environment , they actually show a decreased risk for depression . and this is something that we do n't totally understand yet , we 're still trying to figure out why this might be the case . but importantly , i think that this really shows us how complicated biological factors can be . let 's move on to some psychological factors that might influence depression . one theory is based on the concept of learned helplessness . and this theory supposes that if an individual is exposed to aversive situations over and over again , without any power to change or control them , they might begin to feel powerless in a way that might lead to depression . so if someone is exposed to prolonged stress due to family life or bullying or some other cause that they do n't have control over , their helplessness could spiral out of control and they might stop trying to change their situation because they perceive it to be completely helpless . and that 's a behavioral theory or way of thinking about depression , but there are cognitive theories about it as well . and these theories tend to focus on thoughts or beliefs that , with repetition , could trigger depression . and while it 's true that everyone has negative or self-destructive thoughts every once in awhile , generally we are able to step back from them , and we 're able to realize that what we 're thinking is n't completely logical . but sometimes people can get trapped in these thought patterns and they might put too much emphasis on negative thoughts and actions and experiences . and when they ruminate on these things , when they turn them over and over in their minds , it 's possible that these cognitive distortions might lead to depression . another cognitive theory about depression focuses on the concept of attribution or explanatory style . now as we go about our daily lives , we naturally try to understand and explain the events that go on around us . and when we do this , we can either attribute the things we see to internal or external causes . so is it something that i did ? or is it something that happened because of something that is completely out of my control ? did i get a bad grade on a test because i did n't study ? that would be an internal cause . or did i get a bad grade because the teacher made a really unfair test ? which would be an external cause . individuals with depression tend to attribute negative experiences to internal causes . so maybe they 'll think that a friend did n't call or text back because they are unlikeable or unlovable and not because they were at the movies with their family and maybe their phone was off . in addition to this , they tend to see negative experiences as being stable , so they think that they 'll continue to happen in the future . and they also tend to think that they 're global , so they might assume that one friend not calling them back somehow signifies that none of their friends like them and together these things , these internal attributions , these stable attributions and global attributions , these things form a pessimistic attributional style and it might make certain individuals particularly vulnerable to depression . and there are many other psychological theories about depression . things that have to do with coping style or self-esteem , but it can actually be hard to know whether these things cause depression or if they are the result of it . so does a pessimistic attributional style lead to depression or do people with depression tend to have a pessimistic attributional style ? it is n't always clear . environmental and sociocultural factors can also have a strong influence on depression . having a friend or partner or roommate with depression can actually increase the likelihood that individuals around them will also develop depression . and although we do n't know exactly why that is , some researchers suggest that it might have to do with co-rumination , where friends talk about problems and negative events . but instead of discussing how to solve them , they focus on the negative emotions and dwell on future problems and occurrences . and on some level , this is kind of normal . it is perfectly normal for close friends to take on some of the negative feelings of the other , like being sad when they lose someone who is close to them or being angry if they were dumped by their partner . this is just natural empathy . but the same empathy that allows us to comfort our friends when they 're in distress , might also be the reason that depression seems to spread . we also know that individuals with a low socioeconomic status , especially those living in poverty , are more likely to develop depression , as are those who are struggling to keep a job or have just lost a job . and there are other environmental factors as well . social isolation , child abuse , even prejudice have all been implicated in causing depression . and let 's think about this in terms of prejudice . if someone grows up in a household that has negative feelings about homosexuality and if they grow up and begin to have same sex attractions , they 've probably internalized the prejudice after years of hearing it and this could lead to depression . all right , so stepping back for a second , we said that we had biological factors , psychological factors and sociocultural and environmental factors . and when we put all of these things together , we get what is referred to as a biopsychosocial model of depression and this theory acknowledges that all of these factors play a role . so some people are genetically predisposed to the condition , but it only comes about if the situation is right or if we develop certain patterns of thinking .
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individuals with depression also tend to show fluctuations in weight . so either a lot of weight gain or a lot of weight loss . and they might also have trouble sleeping or they might sleep too much .
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what would constitute as a lot of weight loss or gain ?
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- [ voicover ] major depressive disorder , which is sometimes just referred to as depression , is characterized by prolonged helplessness and discouragement about the future . individuals with this disorder have low self-esteem and very powerful feelings of worthlessness . they also lack the energy to do the things that they used to enjoy , much less the things that they have to do , or the things that they do n't enjoy . they tend to feel socially isolated , and have trouble focusing on important tasks and have trouble making decisions . and this low mood tends to prevade all aspects of their life . there are a number of physical symptoms that also go along with depression . lethargy , so feeling fatigued . individuals with depression also tend to show fluctuations in weight . so either a lot of weight gain or a lot of weight loss . and they might also have trouble sleeping or they might sleep too much . and i think that these physical symptoms are often ignored , because in western cultures , like in the u.s. , we tend to think about depression in terms of moods or in terms of emotional states . but for individuals in some eastern cultures , especially cultures where it might be seen as inappropriate to delve into or talk about feelings and emotions , people in these cultures tend to think about depression and experience depression in terms of these bodily symptoms , and so it 's really important that these are n't discounted . depression or depressive symptoms are the number one reason people seek out mental health services . and , because of this , some people have taken to calling it the common cold of psychological disorders . and i like that term for some reasons and i dislike it for others . what i like about it is that it captures how pervasive this disorder is . it is estimated that 13 % of men and 22 % of women , worldwide , could meet the criteria for depression at least once in their lives . and one study has shown that as high as 31 % of college students might experience this disorder . and these are really high numbers . and so , in that way , i think that this term is appropriate . however , i think that the term common cold does n't really capture the seriousness of this disorder . because depression is n't just feeling down once in awhile . and it 's not feeling sadness or grief at appropriate times , which is just a normal part of life . and i think that this term kind of minimizes that part of the disorder . depression can be triggered by a life event , like a loss or a breakup , but it does n't have to be . it also does n't usually appear alone . it is actually really common for individuals with depression to have other disorders such as anxiety disorders . and i 've been writing in this blue color to kind of signify depression , but i 'm kind of getting bored with it now , so let me switch it up . so there are a number of factors that may be involved in depression and i 'm going to split them up into three categories : biological factors , psychological factors and sociocultural or environmental factors . and let me , let me take a minute to get all of that down . all right . first of all , we know from family and twin studies that there 's a genetic component to depression . and we also know from studies that use functional imaging that individuals with depression show a decreased activation in the prefrontal cortex . and this could be associated with the problems with decision making that people with depression tend to have . as well as their difficulties in generating actions . researchers have also found lower levels of activity in the rewards circuitry in the brain . and this could help to explain why individuals with depression might not find enjoyment in the actions that they once found pleasurable . depression has also been associated with certain neurotransmitters and neurotransmitter regulation . and i 'll abbreviate that here by writing nt for neurotransmitter . research has suggested that individuals with depression might have fewer receptors for serotonin and norepinephrine . and i think that all of this research is amazing . i think that it 's really important . and it 's also really compelling , in a way that research findings that include neuroscience typically are . but with that said , i really want to caution you against oversimplifying these biological factors . to give an example about why i 'm saying this , i wan na talk about the relationship between a certain serotonin transporter gene and depression . and this gene is known as 5-httlpr , and i 'll write that down . and a lot of findings have shown that this gene is involved in depression , but in actuality , it is only associated with depression if the individual with the gene is in a stressful environment . but the story does n't actually end there , it 's actually even more complicated . because it turns out that if an individual with this genetic feature is placed in a warm and positive environment , they actually show a decreased risk for depression . and this is something that we do n't totally understand yet , we 're still trying to figure out why this might be the case . but importantly , i think that this really shows us how complicated biological factors can be . let 's move on to some psychological factors that might influence depression . one theory is based on the concept of learned helplessness . and this theory supposes that if an individual is exposed to aversive situations over and over again , without any power to change or control them , they might begin to feel powerless in a way that might lead to depression . so if someone is exposed to prolonged stress due to family life or bullying or some other cause that they do n't have control over , their helplessness could spiral out of control and they might stop trying to change their situation because they perceive it to be completely helpless . and that 's a behavioral theory or way of thinking about depression , but there are cognitive theories about it as well . and these theories tend to focus on thoughts or beliefs that , with repetition , could trigger depression . and while it 's true that everyone has negative or self-destructive thoughts every once in awhile , generally we are able to step back from them , and we 're able to realize that what we 're thinking is n't completely logical . but sometimes people can get trapped in these thought patterns and they might put too much emphasis on negative thoughts and actions and experiences . and when they ruminate on these things , when they turn them over and over in their minds , it 's possible that these cognitive distortions might lead to depression . another cognitive theory about depression focuses on the concept of attribution or explanatory style . now as we go about our daily lives , we naturally try to understand and explain the events that go on around us . and when we do this , we can either attribute the things we see to internal or external causes . so is it something that i did ? or is it something that happened because of something that is completely out of my control ? did i get a bad grade on a test because i did n't study ? that would be an internal cause . or did i get a bad grade because the teacher made a really unfair test ? which would be an external cause . individuals with depression tend to attribute negative experiences to internal causes . so maybe they 'll think that a friend did n't call or text back because they are unlikeable or unlovable and not because they were at the movies with their family and maybe their phone was off . in addition to this , they tend to see negative experiences as being stable , so they think that they 'll continue to happen in the future . and they also tend to think that they 're global , so they might assume that one friend not calling them back somehow signifies that none of their friends like them and together these things , these internal attributions , these stable attributions and global attributions , these things form a pessimistic attributional style and it might make certain individuals particularly vulnerable to depression . and there are many other psychological theories about depression . things that have to do with coping style or self-esteem , but it can actually be hard to know whether these things cause depression or if they are the result of it . so does a pessimistic attributional style lead to depression or do people with depression tend to have a pessimistic attributional style ? it is n't always clear . environmental and sociocultural factors can also have a strong influence on depression . having a friend or partner or roommate with depression can actually increase the likelihood that individuals around them will also develop depression . and although we do n't know exactly why that is , some researchers suggest that it might have to do with co-rumination , where friends talk about problems and negative events . but instead of discussing how to solve them , they focus on the negative emotions and dwell on future problems and occurrences . and on some level , this is kind of normal . it is perfectly normal for close friends to take on some of the negative feelings of the other , like being sad when they lose someone who is close to them or being angry if they were dumped by their partner . this is just natural empathy . but the same empathy that allows us to comfort our friends when they 're in distress , might also be the reason that depression seems to spread . we also know that individuals with a low socioeconomic status , especially those living in poverty , are more likely to develop depression , as are those who are struggling to keep a job or have just lost a job . and there are other environmental factors as well . social isolation , child abuse , even prejudice have all been implicated in causing depression . and let 's think about this in terms of prejudice . if someone grows up in a household that has negative feelings about homosexuality and if they grow up and begin to have same sex attractions , they 've probably internalized the prejudice after years of hearing it and this could lead to depression . all right , so stepping back for a second , we said that we had biological factors , psychological factors and sociocultural and environmental factors . and when we put all of these things together , we get what is referred to as a biopsychosocial model of depression and this theory acknowledges that all of these factors play a role . so some people are genetically predisposed to the condition , but it only comes about if the situation is right or if we develop certain patterns of thinking .
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but for individuals in some eastern cultures , especially cultures where it might be seen as inappropriate to delve into or talk about feelings and emotions , people in these cultures tend to think about depression and experience depression in terms of these bodily symptoms , and so it 's really important that these are n't discounted . depression or depressive symptoms are the number one reason people seek out mental health services . and , because of this , some people have taken to calling it the common cold of psychological disorders .
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why mental health care is not included in the curriculum of developing country education system like pakistan and india ?
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- [ voicover ] major depressive disorder , which is sometimes just referred to as depression , is characterized by prolonged helplessness and discouragement about the future . individuals with this disorder have low self-esteem and very powerful feelings of worthlessness . they also lack the energy to do the things that they used to enjoy , much less the things that they have to do , or the things that they do n't enjoy . they tend to feel socially isolated , and have trouble focusing on important tasks and have trouble making decisions . and this low mood tends to prevade all aspects of their life . there are a number of physical symptoms that also go along with depression . lethargy , so feeling fatigued . individuals with depression also tend to show fluctuations in weight . so either a lot of weight gain or a lot of weight loss . and they might also have trouble sleeping or they might sleep too much . and i think that these physical symptoms are often ignored , because in western cultures , like in the u.s. , we tend to think about depression in terms of moods or in terms of emotional states . but for individuals in some eastern cultures , especially cultures where it might be seen as inappropriate to delve into or talk about feelings and emotions , people in these cultures tend to think about depression and experience depression in terms of these bodily symptoms , and so it 's really important that these are n't discounted . depression or depressive symptoms are the number one reason people seek out mental health services . and , because of this , some people have taken to calling it the common cold of psychological disorders . and i like that term for some reasons and i dislike it for others . what i like about it is that it captures how pervasive this disorder is . it is estimated that 13 % of men and 22 % of women , worldwide , could meet the criteria for depression at least once in their lives . and one study has shown that as high as 31 % of college students might experience this disorder . and these are really high numbers . and so , in that way , i think that this term is appropriate . however , i think that the term common cold does n't really capture the seriousness of this disorder . because depression is n't just feeling down once in awhile . and it 's not feeling sadness or grief at appropriate times , which is just a normal part of life . and i think that this term kind of minimizes that part of the disorder . depression can be triggered by a life event , like a loss or a breakup , but it does n't have to be . it also does n't usually appear alone . it is actually really common for individuals with depression to have other disorders such as anxiety disorders . and i 've been writing in this blue color to kind of signify depression , but i 'm kind of getting bored with it now , so let me switch it up . so there are a number of factors that may be involved in depression and i 'm going to split them up into three categories : biological factors , psychological factors and sociocultural or environmental factors . and let me , let me take a minute to get all of that down . all right . first of all , we know from family and twin studies that there 's a genetic component to depression . and we also know from studies that use functional imaging that individuals with depression show a decreased activation in the prefrontal cortex . and this could be associated with the problems with decision making that people with depression tend to have . as well as their difficulties in generating actions . researchers have also found lower levels of activity in the rewards circuitry in the brain . and this could help to explain why individuals with depression might not find enjoyment in the actions that they once found pleasurable . depression has also been associated with certain neurotransmitters and neurotransmitter regulation . and i 'll abbreviate that here by writing nt for neurotransmitter . research has suggested that individuals with depression might have fewer receptors for serotonin and norepinephrine . and i think that all of this research is amazing . i think that it 's really important . and it 's also really compelling , in a way that research findings that include neuroscience typically are . but with that said , i really want to caution you against oversimplifying these biological factors . to give an example about why i 'm saying this , i wan na talk about the relationship between a certain serotonin transporter gene and depression . and this gene is known as 5-httlpr , and i 'll write that down . and a lot of findings have shown that this gene is involved in depression , but in actuality , it is only associated with depression if the individual with the gene is in a stressful environment . but the story does n't actually end there , it 's actually even more complicated . because it turns out that if an individual with this genetic feature is placed in a warm and positive environment , they actually show a decreased risk for depression . and this is something that we do n't totally understand yet , we 're still trying to figure out why this might be the case . but importantly , i think that this really shows us how complicated biological factors can be . let 's move on to some psychological factors that might influence depression . one theory is based on the concept of learned helplessness . and this theory supposes that if an individual is exposed to aversive situations over and over again , without any power to change or control them , they might begin to feel powerless in a way that might lead to depression . so if someone is exposed to prolonged stress due to family life or bullying or some other cause that they do n't have control over , their helplessness could spiral out of control and they might stop trying to change their situation because they perceive it to be completely helpless . and that 's a behavioral theory or way of thinking about depression , but there are cognitive theories about it as well . and these theories tend to focus on thoughts or beliefs that , with repetition , could trigger depression . and while it 's true that everyone has negative or self-destructive thoughts every once in awhile , generally we are able to step back from them , and we 're able to realize that what we 're thinking is n't completely logical . but sometimes people can get trapped in these thought patterns and they might put too much emphasis on negative thoughts and actions and experiences . and when they ruminate on these things , when they turn them over and over in their minds , it 's possible that these cognitive distortions might lead to depression . another cognitive theory about depression focuses on the concept of attribution or explanatory style . now as we go about our daily lives , we naturally try to understand and explain the events that go on around us . and when we do this , we can either attribute the things we see to internal or external causes . so is it something that i did ? or is it something that happened because of something that is completely out of my control ? did i get a bad grade on a test because i did n't study ? that would be an internal cause . or did i get a bad grade because the teacher made a really unfair test ? which would be an external cause . individuals with depression tend to attribute negative experiences to internal causes . so maybe they 'll think that a friend did n't call or text back because they are unlikeable or unlovable and not because they were at the movies with their family and maybe their phone was off . in addition to this , they tend to see negative experiences as being stable , so they think that they 'll continue to happen in the future . and they also tend to think that they 're global , so they might assume that one friend not calling them back somehow signifies that none of their friends like them and together these things , these internal attributions , these stable attributions and global attributions , these things form a pessimistic attributional style and it might make certain individuals particularly vulnerable to depression . and there are many other psychological theories about depression . things that have to do with coping style or self-esteem , but it can actually be hard to know whether these things cause depression or if they are the result of it . so does a pessimistic attributional style lead to depression or do people with depression tend to have a pessimistic attributional style ? it is n't always clear . environmental and sociocultural factors can also have a strong influence on depression . having a friend or partner or roommate with depression can actually increase the likelihood that individuals around them will also develop depression . and although we do n't know exactly why that is , some researchers suggest that it might have to do with co-rumination , where friends talk about problems and negative events . but instead of discussing how to solve them , they focus on the negative emotions and dwell on future problems and occurrences . and on some level , this is kind of normal . it is perfectly normal for close friends to take on some of the negative feelings of the other , like being sad when they lose someone who is close to them or being angry if they were dumped by their partner . this is just natural empathy . but the same empathy that allows us to comfort our friends when they 're in distress , might also be the reason that depression seems to spread . we also know that individuals with a low socioeconomic status , especially those living in poverty , are more likely to develop depression , as are those who are struggling to keep a job or have just lost a job . and there are other environmental factors as well . social isolation , child abuse , even prejudice have all been implicated in causing depression . and let 's think about this in terms of prejudice . if someone grows up in a household that has negative feelings about homosexuality and if they grow up and begin to have same sex attractions , they 've probably internalized the prejudice after years of hearing it and this could lead to depression . all right , so stepping back for a second , we said that we had biological factors , psychological factors and sociocultural and environmental factors . and when we put all of these things together , we get what is referred to as a biopsychosocial model of depression and this theory acknowledges that all of these factors play a role . so some people are genetically predisposed to the condition , but it only comes about if the situation is right or if we develop certain patterns of thinking .
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and this could help to explain why individuals with depression might not find enjoyment in the actions that they once found pleasurable . depression has also been associated with certain neurotransmitters and neurotransmitter regulation . and i 'll abbreviate that here by writing nt for neurotransmitter .
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is it possible that certain personality types are prone to depression ?
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- [ voicover ] major depressive disorder , which is sometimes just referred to as depression , is characterized by prolonged helplessness and discouragement about the future . individuals with this disorder have low self-esteem and very powerful feelings of worthlessness . they also lack the energy to do the things that they used to enjoy , much less the things that they have to do , or the things that they do n't enjoy . they tend to feel socially isolated , and have trouble focusing on important tasks and have trouble making decisions . and this low mood tends to prevade all aspects of their life . there are a number of physical symptoms that also go along with depression . lethargy , so feeling fatigued . individuals with depression also tend to show fluctuations in weight . so either a lot of weight gain or a lot of weight loss . and they might also have trouble sleeping or they might sleep too much . and i think that these physical symptoms are often ignored , because in western cultures , like in the u.s. , we tend to think about depression in terms of moods or in terms of emotional states . but for individuals in some eastern cultures , especially cultures where it might be seen as inappropriate to delve into or talk about feelings and emotions , people in these cultures tend to think about depression and experience depression in terms of these bodily symptoms , and so it 's really important that these are n't discounted . depression or depressive symptoms are the number one reason people seek out mental health services . and , because of this , some people have taken to calling it the common cold of psychological disorders . and i like that term for some reasons and i dislike it for others . what i like about it is that it captures how pervasive this disorder is . it is estimated that 13 % of men and 22 % of women , worldwide , could meet the criteria for depression at least once in their lives . and one study has shown that as high as 31 % of college students might experience this disorder . and these are really high numbers . and so , in that way , i think that this term is appropriate . however , i think that the term common cold does n't really capture the seriousness of this disorder . because depression is n't just feeling down once in awhile . and it 's not feeling sadness or grief at appropriate times , which is just a normal part of life . and i think that this term kind of minimizes that part of the disorder . depression can be triggered by a life event , like a loss or a breakup , but it does n't have to be . it also does n't usually appear alone . it is actually really common for individuals with depression to have other disorders such as anxiety disorders . and i 've been writing in this blue color to kind of signify depression , but i 'm kind of getting bored with it now , so let me switch it up . so there are a number of factors that may be involved in depression and i 'm going to split them up into three categories : biological factors , psychological factors and sociocultural or environmental factors . and let me , let me take a minute to get all of that down . all right . first of all , we know from family and twin studies that there 's a genetic component to depression . and we also know from studies that use functional imaging that individuals with depression show a decreased activation in the prefrontal cortex . and this could be associated with the problems with decision making that people with depression tend to have . as well as their difficulties in generating actions . researchers have also found lower levels of activity in the rewards circuitry in the brain . and this could help to explain why individuals with depression might not find enjoyment in the actions that they once found pleasurable . depression has also been associated with certain neurotransmitters and neurotransmitter regulation . and i 'll abbreviate that here by writing nt for neurotransmitter . research has suggested that individuals with depression might have fewer receptors for serotonin and norepinephrine . and i think that all of this research is amazing . i think that it 's really important . and it 's also really compelling , in a way that research findings that include neuroscience typically are . but with that said , i really want to caution you against oversimplifying these biological factors . to give an example about why i 'm saying this , i wan na talk about the relationship between a certain serotonin transporter gene and depression . and this gene is known as 5-httlpr , and i 'll write that down . and a lot of findings have shown that this gene is involved in depression , but in actuality , it is only associated with depression if the individual with the gene is in a stressful environment . but the story does n't actually end there , it 's actually even more complicated . because it turns out that if an individual with this genetic feature is placed in a warm and positive environment , they actually show a decreased risk for depression . and this is something that we do n't totally understand yet , we 're still trying to figure out why this might be the case . but importantly , i think that this really shows us how complicated biological factors can be . let 's move on to some psychological factors that might influence depression . one theory is based on the concept of learned helplessness . and this theory supposes that if an individual is exposed to aversive situations over and over again , without any power to change or control them , they might begin to feel powerless in a way that might lead to depression . so if someone is exposed to prolonged stress due to family life or bullying or some other cause that they do n't have control over , their helplessness could spiral out of control and they might stop trying to change their situation because they perceive it to be completely helpless . and that 's a behavioral theory or way of thinking about depression , but there are cognitive theories about it as well . and these theories tend to focus on thoughts or beliefs that , with repetition , could trigger depression . and while it 's true that everyone has negative or self-destructive thoughts every once in awhile , generally we are able to step back from them , and we 're able to realize that what we 're thinking is n't completely logical . but sometimes people can get trapped in these thought patterns and they might put too much emphasis on negative thoughts and actions and experiences . and when they ruminate on these things , when they turn them over and over in their minds , it 's possible that these cognitive distortions might lead to depression . another cognitive theory about depression focuses on the concept of attribution or explanatory style . now as we go about our daily lives , we naturally try to understand and explain the events that go on around us . and when we do this , we can either attribute the things we see to internal or external causes . so is it something that i did ? or is it something that happened because of something that is completely out of my control ? did i get a bad grade on a test because i did n't study ? that would be an internal cause . or did i get a bad grade because the teacher made a really unfair test ? which would be an external cause . individuals with depression tend to attribute negative experiences to internal causes . so maybe they 'll think that a friend did n't call or text back because they are unlikeable or unlovable and not because they were at the movies with their family and maybe their phone was off . in addition to this , they tend to see negative experiences as being stable , so they think that they 'll continue to happen in the future . and they also tend to think that they 're global , so they might assume that one friend not calling them back somehow signifies that none of their friends like them and together these things , these internal attributions , these stable attributions and global attributions , these things form a pessimistic attributional style and it might make certain individuals particularly vulnerable to depression . and there are many other psychological theories about depression . things that have to do with coping style or self-esteem , but it can actually be hard to know whether these things cause depression or if they are the result of it . so does a pessimistic attributional style lead to depression or do people with depression tend to have a pessimistic attributional style ? it is n't always clear . environmental and sociocultural factors can also have a strong influence on depression . having a friend or partner or roommate with depression can actually increase the likelihood that individuals around them will also develop depression . and although we do n't know exactly why that is , some researchers suggest that it might have to do with co-rumination , where friends talk about problems and negative events . but instead of discussing how to solve them , they focus on the negative emotions and dwell on future problems and occurrences . and on some level , this is kind of normal . it is perfectly normal for close friends to take on some of the negative feelings of the other , like being sad when they lose someone who is close to them or being angry if they were dumped by their partner . this is just natural empathy . but the same empathy that allows us to comfort our friends when they 're in distress , might also be the reason that depression seems to spread . we also know that individuals with a low socioeconomic status , especially those living in poverty , are more likely to develop depression , as are those who are struggling to keep a job or have just lost a job . and there are other environmental factors as well . social isolation , child abuse , even prejudice have all been implicated in causing depression . and let 's think about this in terms of prejudice . if someone grows up in a household that has negative feelings about homosexuality and if they grow up and begin to have same sex attractions , they 've probably internalized the prejudice after years of hearing it and this could lead to depression . all right , so stepping back for a second , we said that we had biological factors , psychological factors and sociocultural and environmental factors . and when we put all of these things together , we get what is referred to as a biopsychosocial model of depression and this theory acknowledges that all of these factors play a role . so some people are genetically predisposed to the condition , but it only comes about if the situation is right or if we develop certain patterns of thinking .
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it is n't always clear . environmental and sociocultural factors can also have a strong influence on depression . having a friend or partner or roommate with depression can actually increase the likelihood that individuals around them will also develop depression . and although we do n't know exactly why that is , some researchers suggest that it might have to do with co-rumination , where friends talk about problems and negative events .
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so is depression the only study of psychology ?
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- [ voicover ] major depressive disorder , which is sometimes just referred to as depression , is characterized by prolonged helplessness and discouragement about the future . individuals with this disorder have low self-esteem and very powerful feelings of worthlessness . they also lack the energy to do the things that they used to enjoy , much less the things that they have to do , or the things that they do n't enjoy . they tend to feel socially isolated , and have trouble focusing on important tasks and have trouble making decisions . and this low mood tends to prevade all aspects of their life . there are a number of physical symptoms that also go along with depression . lethargy , so feeling fatigued . individuals with depression also tend to show fluctuations in weight . so either a lot of weight gain or a lot of weight loss . and they might also have trouble sleeping or they might sleep too much . and i think that these physical symptoms are often ignored , because in western cultures , like in the u.s. , we tend to think about depression in terms of moods or in terms of emotional states . but for individuals in some eastern cultures , especially cultures where it might be seen as inappropriate to delve into or talk about feelings and emotions , people in these cultures tend to think about depression and experience depression in terms of these bodily symptoms , and so it 's really important that these are n't discounted . depression or depressive symptoms are the number one reason people seek out mental health services . and , because of this , some people have taken to calling it the common cold of psychological disorders . and i like that term for some reasons and i dislike it for others . what i like about it is that it captures how pervasive this disorder is . it is estimated that 13 % of men and 22 % of women , worldwide , could meet the criteria for depression at least once in their lives . and one study has shown that as high as 31 % of college students might experience this disorder . and these are really high numbers . and so , in that way , i think that this term is appropriate . however , i think that the term common cold does n't really capture the seriousness of this disorder . because depression is n't just feeling down once in awhile . and it 's not feeling sadness or grief at appropriate times , which is just a normal part of life . and i think that this term kind of minimizes that part of the disorder . depression can be triggered by a life event , like a loss or a breakup , but it does n't have to be . it also does n't usually appear alone . it is actually really common for individuals with depression to have other disorders such as anxiety disorders . and i 've been writing in this blue color to kind of signify depression , but i 'm kind of getting bored with it now , so let me switch it up . so there are a number of factors that may be involved in depression and i 'm going to split them up into three categories : biological factors , psychological factors and sociocultural or environmental factors . and let me , let me take a minute to get all of that down . all right . first of all , we know from family and twin studies that there 's a genetic component to depression . and we also know from studies that use functional imaging that individuals with depression show a decreased activation in the prefrontal cortex . and this could be associated with the problems with decision making that people with depression tend to have . as well as their difficulties in generating actions . researchers have also found lower levels of activity in the rewards circuitry in the brain . and this could help to explain why individuals with depression might not find enjoyment in the actions that they once found pleasurable . depression has also been associated with certain neurotransmitters and neurotransmitter regulation . and i 'll abbreviate that here by writing nt for neurotransmitter . research has suggested that individuals with depression might have fewer receptors for serotonin and norepinephrine . and i think that all of this research is amazing . i think that it 's really important . and it 's also really compelling , in a way that research findings that include neuroscience typically are . but with that said , i really want to caution you against oversimplifying these biological factors . to give an example about why i 'm saying this , i wan na talk about the relationship between a certain serotonin transporter gene and depression . and this gene is known as 5-httlpr , and i 'll write that down . and a lot of findings have shown that this gene is involved in depression , but in actuality , it is only associated with depression if the individual with the gene is in a stressful environment . but the story does n't actually end there , it 's actually even more complicated . because it turns out that if an individual with this genetic feature is placed in a warm and positive environment , they actually show a decreased risk for depression . and this is something that we do n't totally understand yet , we 're still trying to figure out why this might be the case . but importantly , i think that this really shows us how complicated biological factors can be . let 's move on to some psychological factors that might influence depression . one theory is based on the concept of learned helplessness . and this theory supposes that if an individual is exposed to aversive situations over and over again , without any power to change or control them , they might begin to feel powerless in a way that might lead to depression . so if someone is exposed to prolonged stress due to family life or bullying or some other cause that they do n't have control over , their helplessness could spiral out of control and they might stop trying to change their situation because they perceive it to be completely helpless . and that 's a behavioral theory or way of thinking about depression , but there are cognitive theories about it as well . and these theories tend to focus on thoughts or beliefs that , with repetition , could trigger depression . and while it 's true that everyone has negative or self-destructive thoughts every once in awhile , generally we are able to step back from them , and we 're able to realize that what we 're thinking is n't completely logical . but sometimes people can get trapped in these thought patterns and they might put too much emphasis on negative thoughts and actions and experiences . and when they ruminate on these things , when they turn them over and over in their minds , it 's possible that these cognitive distortions might lead to depression . another cognitive theory about depression focuses on the concept of attribution or explanatory style . now as we go about our daily lives , we naturally try to understand and explain the events that go on around us . and when we do this , we can either attribute the things we see to internal or external causes . so is it something that i did ? or is it something that happened because of something that is completely out of my control ? did i get a bad grade on a test because i did n't study ? that would be an internal cause . or did i get a bad grade because the teacher made a really unfair test ? which would be an external cause . individuals with depression tend to attribute negative experiences to internal causes . so maybe they 'll think that a friend did n't call or text back because they are unlikeable or unlovable and not because they were at the movies with their family and maybe their phone was off . in addition to this , they tend to see negative experiences as being stable , so they think that they 'll continue to happen in the future . and they also tend to think that they 're global , so they might assume that one friend not calling them back somehow signifies that none of their friends like them and together these things , these internal attributions , these stable attributions and global attributions , these things form a pessimistic attributional style and it might make certain individuals particularly vulnerable to depression . and there are many other psychological theories about depression . things that have to do with coping style or self-esteem , but it can actually be hard to know whether these things cause depression or if they are the result of it . so does a pessimistic attributional style lead to depression or do people with depression tend to have a pessimistic attributional style ? it is n't always clear . environmental and sociocultural factors can also have a strong influence on depression . having a friend or partner or roommate with depression can actually increase the likelihood that individuals around them will also develop depression . and although we do n't know exactly why that is , some researchers suggest that it might have to do with co-rumination , where friends talk about problems and negative events . but instead of discussing how to solve them , they focus on the negative emotions and dwell on future problems and occurrences . and on some level , this is kind of normal . it is perfectly normal for close friends to take on some of the negative feelings of the other , like being sad when they lose someone who is close to them or being angry if they were dumped by their partner . this is just natural empathy . but the same empathy that allows us to comfort our friends when they 're in distress , might also be the reason that depression seems to spread . we also know that individuals with a low socioeconomic status , especially those living in poverty , are more likely to develop depression , as are those who are struggling to keep a job or have just lost a job . and there are other environmental factors as well . social isolation , child abuse , even prejudice have all been implicated in causing depression . and let 's think about this in terms of prejudice . if someone grows up in a household that has negative feelings about homosexuality and if they grow up and begin to have same sex attractions , they 've probably internalized the prejudice after years of hearing it and this could lead to depression . all right , so stepping back for a second , we said that we had biological factors , psychological factors and sociocultural and environmental factors . and when we put all of these things together , we get what is referred to as a biopsychosocial model of depression and this theory acknowledges that all of these factors play a role . so some people are genetically predisposed to the condition , but it only comes about if the situation is right or if we develop certain patterns of thinking .
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- [ voicover ] major depressive disorder , which is sometimes just referred to as depression , is characterized by prolonged helplessness and discouragement about the future . individuals with this disorder have low self-esteem and very powerful feelings of worthlessness .
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what is an ecological variable ?
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- [ voicover ] major depressive disorder , which is sometimes just referred to as depression , is characterized by prolonged helplessness and discouragement about the future . individuals with this disorder have low self-esteem and very powerful feelings of worthlessness . they also lack the energy to do the things that they used to enjoy , much less the things that they have to do , or the things that they do n't enjoy . they tend to feel socially isolated , and have trouble focusing on important tasks and have trouble making decisions . and this low mood tends to prevade all aspects of their life . there are a number of physical symptoms that also go along with depression . lethargy , so feeling fatigued . individuals with depression also tend to show fluctuations in weight . so either a lot of weight gain or a lot of weight loss . and they might also have trouble sleeping or they might sleep too much . and i think that these physical symptoms are often ignored , because in western cultures , like in the u.s. , we tend to think about depression in terms of moods or in terms of emotional states . but for individuals in some eastern cultures , especially cultures where it might be seen as inappropriate to delve into or talk about feelings and emotions , people in these cultures tend to think about depression and experience depression in terms of these bodily symptoms , and so it 's really important that these are n't discounted . depression or depressive symptoms are the number one reason people seek out mental health services . and , because of this , some people have taken to calling it the common cold of psychological disorders . and i like that term for some reasons and i dislike it for others . what i like about it is that it captures how pervasive this disorder is . it is estimated that 13 % of men and 22 % of women , worldwide , could meet the criteria for depression at least once in their lives . and one study has shown that as high as 31 % of college students might experience this disorder . and these are really high numbers . and so , in that way , i think that this term is appropriate . however , i think that the term common cold does n't really capture the seriousness of this disorder . because depression is n't just feeling down once in awhile . and it 's not feeling sadness or grief at appropriate times , which is just a normal part of life . and i think that this term kind of minimizes that part of the disorder . depression can be triggered by a life event , like a loss or a breakup , but it does n't have to be . it also does n't usually appear alone . it is actually really common for individuals with depression to have other disorders such as anxiety disorders . and i 've been writing in this blue color to kind of signify depression , but i 'm kind of getting bored with it now , so let me switch it up . so there are a number of factors that may be involved in depression and i 'm going to split them up into three categories : biological factors , psychological factors and sociocultural or environmental factors . and let me , let me take a minute to get all of that down . all right . first of all , we know from family and twin studies that there 's a genetic component to depression . and we also know from studies that use functional imaging that individuals with depression show a decreased activation in the prefrontal cortex . and this could be associated with the problems with decision making that people with depression tend to have . as well as their difficulties in generating actions . researchers have also found lower levels of activity in the rewards circuitry in the brain . and this could help to explain why individuals with depression might not find enjoyment in the actions that they once found pleasurable . depression has also been associated with certain neurotransmitters and neurotransmitter regulation . and i 'll abbreviate that here by writing nt for neurotransmitter . research has suggested that individuals with depression might have fewer receptors for serotonin and norepinephrine . and i think that all of this research is amazing . i think that it 's really important . and it 's also really compelling , in a way that research findings that include neuroscience typically are . but with that said , i really want to caution you against oversimplifying these biological factors . to give an example about why i 'm saying this , i wan na talk about the relationship between a certain serotonin transporter gene and depression . and this gene is known as 5-httlpr , and i 'll write that down . and a lot of findings have shown that this gene is involved in depression , but in actuality , it is only associated with depression if the individual with the gene is in a stressful environment . but the story does n't actually end there , it 's actually even more complicated . because it turns out that if an individual with this genetic feature is placed in a warm and positive environment , they actually show a decreased risk for depression . and this is something that we do n't totally understand yet , we 're still trying to figure out why this might be the case . but importantly , i think that this really shows us how complicated biological factors can be . let 's move on to some psychological factors that might influence depression . one theory is based on the concept of learned helplessness . and this theory supposes that if an individual is exposed to aversive situations over and over again , without any power to change or control them , they might begin to feel powerless in a way that might lead to depression . so if someone is exposed to prolonged stress due to family life or bullying or some other cause that they do n't have control over , their helplessness could spiral out of control and they might stop trying to change their situation because they perceive it to be completely helpless . and that 's a behavioral theory or way of thinking about depression , but there are cognitive theories about it as well . and these theories tend to focus on thoughts or beliefs that , with repetition , could trigger depression . and while it 's true that everyone has negative or self-destructive thoughts every once in awhile , generally we are able to step back from them , and we 're able to realize that what we 're thinking is n't completely logical . but sometimes people can get trapped in these thought patterns and they might put too much emphasis on negative thoughts and actions and experiences . and when they ruminate on these things , when they turn them over and over in their minds , it 's possible that these cognitive distortions might lead to depression . another cognitive theory about depression focuses on the concept of attribution or explanatory style . now as we go about our daily lives , we naturally try to understand and explain the events that go on around us . and when we do this , we can either attribute the things we see to internal or external causes . so is it something that i did ? or is it something that happened because of something that is completely out of my control ? did i get a bad grade on a test because i did n't study ? that would be an internal cause . or did i get a bad grade because the teacher made a really unfair test ? which would be an external cause . individuals with depression tend to attribute negative experiences to internal causes . so maybe they 'll think that a friend did n't call or text back because they are unlikeable or unlovable and not because they were at the movies with their family and maybe their phone was off . in addition to this , they tend to see negative experiences as being stable , so they think that they 'll continue to happen in the future . and they also tend to think that they 're global , so they might assume that one friend not calling them back somehow signifies that none of their friends like them and together these things , these internal attributions , these stable attributions and global attributions , these things form a pessimistic attributional style and it might make certain individuals particularly vulnerable to depression . and there are many other psychological theories about depression . things that have to do with coping style or self-esteem , but it can actually be hard to know whether these things cause depression or if they are the result of it . so does a pessimistic attributional style lead to depression or do people with depression tend to have a pessimistic attributional style ? it is n't always clear . environmental and sociocultural factors can also have a strong influence on depression . having a friend or partner or roommate with depression can actually increase the likelihood that individuals around them will also develop depression . and although we do n't know exactly why that is , some researchers suggest that it might have to do with co-rumination , where friends talk about problems and negative events . but instead of discussing how to solve them , they focus on the negative emotions and dwell on future problems and occurrences . and on some level , this is kind of normal . it is perfectly normal for close friends to take on some of the negative feelings of the other , like being sad when they lose someone who is close to them or being angry if they were dumped by their partner . this is just natural empathy . but the same empathy that allows us to comfort our friends when they 're in distress , might also be the reason that depression seems to spread . we also know that individuals with a low socioeconomic status , especially those living in poverty , are more likely to develop depression , as are those who are struggling to keep a job or have just lost a job . and there are other environmental factors as well . social isolation , child abuse , even prejudice have all been implicated in causing depression . and let 's think about this in terms of prejudice . if someone grows up in a household that has negative feelings about homosexuality and if they grow up and begin to have same sex attractions , they 've probably internalized the prejudice after years of hearing it and this could lead to depression . all right , so stepping back for a second , we said that we had biological factors , psychological factors and sociocultural and environmental factors . and when we put all of these things together , we get what is referred to as a biopsychosocial model of depression and this theory acknowledges that all of these factors play a role . so some people are genetically predisposed to the condition , but it only comes about if the situation is right or if we develop certain patterns of thinking .
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- [ voicover ] major depressive disorder , which is sometimes just referred to as depression , is characterized by prolonged helplessness and discouragement about the future . individuals with this disorder have low self-esteem and very powerful feelings of worthlessness .
|
depression is a chemical imbalance of the brain , however , if something major happen , how does that chemical imbalance occur ?
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- [ voicover ] major depressive disorder , which is sometimes just referred to as depression , is characterized by prolonged helplessness and discouragement about the future . individuals with this disorder have low self-esteem and very powerful feelings of worthlessness . they also lack the energy to do the things that they used to enjoy , much less the things that they have to do , or the things that they do n't enjoy . they tend to feel socially isolated , and have trouble focusing on important tasks and have trouble making decisions . and this low mood tends to prevade all aspects of their life . there are a number of physical symptoms that also go along with depression . lethargy , so feeling fatigued . individuals with depression also tend to show fluctuations in weight . so either a lot of weight gain or a lot of weight loss . and they might also have trouble sleeping or they might sleep too much . and i think that these physical symptoms are often ignored , because in western cultures , like in the u.s. , we tend to think about depression in terms of moods or in terms of emotional states . but for individuals in some eastern cultures , especially cultures where it might be seen as inappropriate to delve into or talk about feelings and emotions , people in these cultures tend to think about depression and experience depression in terms of these bodily symptoms , and so it 's really important that these are n't discounted . depression or depressive symptoms are the number one reason people seek out mental health services . and , because of this , some people have taken to calling it the common cold of psychological disorders . and i like that term for some reasons and i dislike it for others . what i like about it is that it captures how pervasive this disorder is . it is estimated that 13 % of men and 22 % of women , worldwide , could meet the criteria for depression at least once in their lives . and one study has shown that as high as 31 % of college students might experience this disorder . and these are really high numbers . and so , in that way , i think that this term is appropriate . however , i think that the term common cold does n't really capture the seriousness of this disorder . because depression is n't just feeling down once in awhile . and it 's not feeling sadness or grief at appropriate times , which is just a normal part of life . and i think that this term kind of minimizes that part of the disorder . depression can be triggered by a life event , like a loss or a breakup , but it does n't have to be . it also does n't usually appear alone . it is actually really common for individuals with depression to have other disorders such as anxiety disorders . and i 've been writing in this blue color to kind of signify depression , but i 'm kind of getting bored with it now , so let me switch it up . so there are a number of factors that may be involved in depression and i 'm going to split them up into three categories : biological factors , psychological factors and sociocultural or environmental factors . and let me , let me take a minute to get all of that down . all right . first of all , we know from family and twin studies that there 's a genetic component to depression . and we also know from studies that use functional imaging that individuals with depression show a decreased activation in the prefrontal cortex . and this could be associated with the problems with decision making that people with depression tend to have . as well as their difficulties in generating actions . researchers have also found lower levels of activity in the rewards circuitry in the brain . and this could help to explain why individuals with depression might not find enjoyment in the actions that they once found pleasurable . depression has also been associated with certain neurotransmitters and neurotransmitter regulation . and i 'll abbreviate that here by writing nt for neurotransmitter . research has suggested that individuals with depression might have fewer receptors for serotonin and norepinephrine . and i think that all of this research is amazing . i think that it 's really important . and it 's also really compelling , in a way that research findings that include neuroscience typically are . but with that said , i really want to caution you against oversimplifying these biological factors . to give an example about why i 'm saying this , i wan na talk about the relationship between a certain serotonin transporter gene and depression . and this gene is known as 5-httlpr , and i 'll write that down . and a lot of findings have shown that this gene is involved in depression , but in actuality , it is only associated with depression if the individual with the gene is in a stressful environment . but the story does n't actually end there , it 's actually even more complicated . because it turns out that if an individual with this genetic feature is placed in a warm and positive environment , they actually show a decreased risk for depression . and this is something that we do n't totally understand yet , we 're still trying to figure out why this might be the case . but importantly , i think that this really shows us how complicated biological factors can be . let 's move on to some psychological factors that might influence depression . one theory is based on the concept of learned helplessness . and this theory supposes that if an individual is exposed to aversive situations over and over again , without any power to change or control them , they might begin to feel powerless in a way that might lead to depression . so if someone is exposed to prolonged stress due to family life or bullying or some other cause that they do n't have control over , their helplessness could spiral out of control and they might stop trying to change their situation because they perceive it to be completely helpless . and that 's a behavioral theory or way of thinking about depression , but there are cognitive theories about it as well . and these theories tend to focus on thoughts or beliefs that , with repetition , could trigger depression . and while it 's true that everyone has negative or self-destructive thoughts every once in awhile , generally we are able to step back from them , and we 're able to realize that what we 're thinking is n't completely logical . but sometimes people can get trapped in these thought patterns and they might put too much emphasis on negative thoughts and actions and experiences . and when they ruminate on these things , when they turn them over and over in their minds , it 's possible that these cognitive distortions might lead to depression . another cognitive theory about depression focuses on the concept of attribution or explanatory style . now as we go about our daily lives , we naturally try to understand and explain the events that go on around us . and when we do this , we can either attribute the things we see to internal or external causes . so is it something that i did ? or is it something that happened because of something that is completely out of my control ? did i get a bad grade on a test because i did n't study ? that would be an internal cause . or did i get a bad grade because the teacher made a really unfair test ? which would be an external cause . individuals with depression tend to attribute negative experiences to internal causes . so maybe they 'll think that a friend did n't call or text back because they are unlikeable or unlovable and not because they were at the movies with their family and maybe their phone was off . in addition to this , they tend to see negative experiences as being stable , so they think that they 'll continue to happen in the future . and they also tend to think that they 're global , so they might assume that one friend not calling them back somehow signifies that none of their friends like them and together these things , these internal attributions , these stable attributions and global attributions , these things form a pessimistic attributional style and it might make certain individuals particularly vulnerable to depression . and there are many other psychological theories about depression . things that have to do with coping style or self-esteem , but it can actually be hard to know whether these things cause depression or if they are the result of it . so does a pessimistic attributional style lead to depression or do people with depression tend to have a pessimistic attributional style ? it is n't always clear . environmental and sociocultural factors can also have a strong influence on depression . having a friend or partner or roommate with depression can actually increase the likelihood that individuals around them will also develop depression . and although we do n't know exactly why that is , some researchers suggest that it might have to do with co-rumination , where friends talk about problems and negative events . but instead of discussing how to solve them , they focus on the negative emotions and dwell on future problems and occurrences . and on some level , this is kind of normal . it is perfectly normal for close friends to take on some of the negative feelings of the other , like being sad when they lose someone who is close to them or being angry if they were dumped by their partner . this is just natural empathy . but the same empathy that allows us to comfort our friends when they 're in distress , might also be the reason that depression seems to spread . we also know that individuals with a low socioeconomic status , especially those living in poverty , are more likely to develop depression , as are those who are struggling to keep a job or have just lost a job . and there are other environmental factors as well . social isolation , child abuse , even prejudice have all been implicated in causing depression . and let 's think about this in terms of prejudice . if someone grows up in a household that has negative feelings about homosexuality and if they grow up and begin to have same sex attractions , they 've probably internalized the prejudice after years of hearing it and this could lead to depression . all right , so stepping back for a second , we said that we had biological factors , psychological factors and sociocultural and environmental factors . and when we put all of these things together , we get what is referred to as a biopsychosocial model of depression and this theory acknowledges that all of these factors play a role . so some people are genetically predisposed to the condition , but it only comes about if the situation is right or if we develop certain patterns of thinking .
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- [ voicover ] major depressive disorder , which is sometimes just referred to as depression , is characterized by prolonged helplessness and discouragement about the future . individuals with this disorder have low self-esteem and very powerful feelings of worthlessness .
|
is being a psycologist considered a doctor ?
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- [ voicover ] major depressive disorder , which is sometimes just referred to as depression , is characterized by prolonged helplessness and discouragement about the future . individuals with this disorder have low self-esteem and very powerful feelings of worthlessness . they also lack the energy to do the things that they used to enjoy , much less the things that they have to do , or the things that they do n't enjoy . they tend to feel socially isolated , and have trouble focusing on important tasks and have trouble making decisions . and this low mood tends to prevade all aspects of their life . there are a number of physical symptoms that also go along with depression . lethargy , so feeling fatigued . individuals with depression also tend to show fluctuations in weight . so either a lot of weight gain or a lot of weight loss . and they might also have trouble sleeping or they might sleep too much . and i think that these physical symptoms are often ignored , because in western cultures , like in the u.s. , we tend to think about depression in terms of moods or in terms of emotional states . but for individuals in some eastern cultures , especially cultures where it might be seen as inappropriate to delve into or talk about feelings and emotions , people in these cultures tend to think about depression and experience depression in terms of these bodily symptoms , and so it 's really important that these are n't discounted . depression or depressive symptoms are the number one reason people seek out mental health services . and , because of this , some people have taken to calling it the common cold of psychological disorders . and i like that term for some reasons and i dislike it for others . what i like about it is that it captures how pervasive this disorder is . it is estimated that 13 % of men and 22 % of women , worldwide , could meet the criteria for depression at least once in their lives . and one study has shown that as high as 31 % of college students might experience this disorder . and these are really high numbers . and so , in that way , i think that this term is appropriate . however , i think that the term common cold does n't really capture the seriousness of this disorder . because depression is n't just feeling down once in awhile . and it 's not feeling sadness or grief at appropriate times , which is just a normal part of life . and i think that this term kind of minimizes that part of the disorder . depression can be triggered by a life event , like a loss or a breakup , but it does n't have to be . it also does n't usually appear alone . it is actually really common for individuals with depression to have other disorders such as anxiety disorders . and i 've been writing in this blue color to kind of signify depression , but i 'm kind of getting bored with it now , so let me switch it up . so there are a number of factors that may be involved in depression and i 'm going to split them up into three categories : biological factors , psychological factors and sociocultural or environmental factors . and let me , let me take a minute to get all of that down . all right . first of all , we know from family and twin studies that there 's a genetic component to depression . and we also know from studies that use functional imaging that individuals with depression show a decreased activation in the prefrontal cortex . and this could be associated with the problems with decision making that people with depression tend to have . as well as their difficulties in generating actions . researchers have also found lower levels of activity in the rewards circuitry in the brain . and this could help to explain why individuals with depression might not find enjoyment in the actions that they once found pleasurable . depression has also been associated with certain neurotransmitters and neurotransmitter regulation . and i 'll abbreviate that here by writing nt for neurotransmitter . research has suggested that individuals with depression might have fewer receptors for serotonin and norepinephrine . and i think that all of this research is amazing . i think that it 's really important . and it 's also really compelling , in a way that research findings that include neuroscience typically are . but with that said , i really want to caution you against oversimplifying these biological factors . to give an example about why i 'm saying this , i wan na talk about the relationship between a certain serotonin transporter gene and depression . and this gene is known as 5-httlpr , and i 'll write that down . and a lot of findings have shown that this gene is involved in depression , but in actuality , it is only associated with depression if the individual with the gene is in a stressful environment . but the story does n't actually end there , it 's actually even more complicated . because it turns out that if an individual with this genetic feature is placed in a warm and positive environment , they actually show a decreased risk for depression . and this is something that we do n't totally understand yet , we 're still trying to figure out why this might be the case . but importantly , i think that this really shows us how complicated biological factors can be . let 's move on to some psychological factors that might influence depression . one theory is based on the concept of learned helplessness . and this theory supposes that if an individual is exposed to aversive situations over and over again , without any power to change or control them , they might begin to feel powerless in a way that might lead to depression . so if someone is exposed to prolonged stress due to family life or bullying or some other cause that they do n't have control over , their helplessness could spiral out of control and they might stop trying to change their situation because they perceive it to be completely helpless . and that 's a behavioral theory or way of thinking about depression , but there are cognitive theories about it as well . and these theories tend to focus on thoughts or beliefs that , with repetition , could trigger depression . and while it 's true that everyone has negative or self-destructive thoughts every once in awhile , generally we are able to step back from them , and we 're able to realize that what we 're thinking is n't completely logical . but sometimes people can get trapped in these thought patterns and they might put too much emphasis on negative thoughts and actions and experiences . and when they ruminate on these things , when they turn them over and over in their minds , it 's possible that these cognitive distortions might lead to depression . another cognitive theory about depression focuses on the concept of attribution or explanatory style . now as we go about our daily lives , we naturally try to understand and explain the events that go on around us . and when we do this , we can either attribute the things we see to internal or external causes . so is it something that i did ? or is it something that happened because of something that is completely out of my control ? did i get a bad grade on a test because i did n't study ? that would be an internal cause . or did i get a bad grade because the teacher made a really unfair test ? which would be an external cause . individuals with depression tend to attribute negative experiences to internal causes . so maybe they 'll think that a friend did n't call or text back because they are unlikeable or unlovable and not because they were at the movies with their family and maybe their phone was off . in addition to this , they tend to see negative experiences as being stable , so they think that they 'll continue to happen in the future . and they also tend to think that they 're global , so they might assume that one friend not calling them back somehow signifies that none of their friends like them and together these things , these internal attributions , these stable attributions and global attributions , these things form a pessimistic attributional style and it might make certain individuals particularly vulnerable to depression . and there are many other psychological theories about depression . things that have to do with coping style or self-esteem , but it can actually be hard to know whether these things cause depression or if they are the result of it . so does a pessimistic attributional style lead to depression or do people with depression tend to have a pessimistic attributional style ? it is n't always clear . environmental and sociocultural factors can also have a strong influence on depression . having a friend or partner or roommate with depression can actually increase the likelihood that individuals around them will also develop depression . and although we do n't know exactly why that is , some researchers suggest that it might have to do with co-rumination , where friends talk about problems and negative events . but instead of discussing how to solve them , they focus on the negative emotions and dwell on future problems and occurrences . and on some level , this is kind of normal . it is perfectly normal for close friends to take on some of the negative feelings of the other , like being sad when they lose someone who is close to them or being angry if they were dumped by their partner . this is just natural empathy . but the same empathy that allows us to comfort our friends when they 're in distress , might also be the reason that depression seems to spread . we also know that individuals with a low socioeconomic status , especially those living in poverty , are more likely to develop depression , as are those who are struggling to keep a job or have just lost a job . and there are other environmental factors as well . social isolation , child abuse , even prejudice have all been implicated in causing depression . and let 's think about this in terms of prejudice . if someone grows up in a household that has negative feelings about homosexuality and if they grow up and begin to have same sex attractions , they 've probably internalized the prejudice after years of hearing it and this could lead to depression . all right , so stepping back for a second , we said that we had biological factors , psychological factors and sociocultural and environmental factors . and when we put all of these things together , we get what is referred to as a biopsychosocial model of depression and this theory acknowledges that all of these factors play a role . so some people are genetically predisposed to the condition , but it only comes about if the situation is right or if we develop certain patterns of thinking .
|
and that 's a behavioral theory or way of thinking about depression , but there are cognitive theories about it as well . and these theories tend to focus on thoughts or beliefs that , with repetition , could trigger depression . and while it 's true that everyone has negative or self-destructive thoughts every once in awhile , generally we are able to step back from them , and we 're able to realize that what we 're thinking is n't completely logical .
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can depression trigger a person to perform violante events ?
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what we 're going to talk about in this video are the origins of the russian people . and in particular , we 're going to talk about the eastern slavs who not just modern russians , but also ukrainians and belorussians view as their ancestors . so let 's think about the world in the ninth century . the early ninth century , we see the reign of charlemagne that we talk about in some depth in other videos . as we get into the 10th century , you see the reign of otto the great , holy roman emperor over the germanic kingdoms . the ninth century is also the time of tang china . you have the abbasid muslim caliphate in control over most of the middle east and north africa . and it is also the viking age . so we have here in this magenta color , this would be modern day sweden , but it was also the home of the varangians , or whom we later would refer to as the vikings , and we know them to be great seafarers . in western europe , they 're viewed as raiders of towns along the coast . but you have to remember , these histories are often written by the western europeans , not by the vikings themselves . but they were also known as traders . what you see here are two of the major centers of power and trade in the ninth century . you have constantinople , the capital of the byzantine empire , and you have baghdad , the capital of the abbasid caliphate . you also see these major waterways in eastern europe , in particular , the dnieper and the volga rivers . and so you have these significant trade routes going from the baltic sea either via the dnieper , crossing the black sea to constantinople , or going from the baltic to the volga all the way to the caspian sea and eventually making their way to baghdad . and this is well documented . there is archeological evidence of viking jewelry along these routes . there 's evidence of artifacts from these far off lands in viking territory , and we believe what the varangians traded were first they hunted in this area of northern europe . now , the people who lived in this area were known as the slavs . and there were several broad groups of slavs that you will hear historians refer to . you have the western slavs , who you could view as some of the ancestors of modern poles , czechs , and slovaks . you have the southern slavs in what would eventually be referred to as the balkans . and then you have the eastern slavs in what will eventually be russia , belarus , and ukraine . now to be clear , they were n't unified groups . there were many many many , for example , eastern slavic tribes . our best account of the early history , especially in the period as we get into the ninth century comes from what is known as the russian primary chronicle . and keep in mind , this was written at a much later period . it was written in the early 12th century . it is sometimes ascribed to the kievan monk nestor . so at previous times , it was known as nestor 's chronicles , but we do n't even have surviving accounts of this . we have surviving accounts of copies of this , or what we believe are copies of this . what i 'll share is a version of the russian primary chronicle known as the laurentian text from 1377 . and this is , of course , an english translation . it gives us some of the earliest accounts of the relationship between the varangians and the eastern slavs and how what we have come to identify as the russian people and the ukrainian people and the belorussian people , how they got their start . so right before this passage , it talks about how the varangians tried to get tribute from some of the eastern slavic tribes . and it says the tributaries of the varangians drove them back beyond the sea and , refusing them further tribute , set out to govern themselves . so they pushed them back beyond what we now call the baltic sea , and they decided to govern themselves . there was no law among them . tribe rose against tribe , and they began to war one against another . they said to themselves , let us seek a prince who may rule over us and judge us according to the law . they accordingly went overseas to the varangian russes . so they went back to the russes , and they said , these warring eastern slavic tribes said , our land is great and rich , but there is no order in it . come to rule and reign over us . they thus selected three brothers . the oldest , rurik , located himself in novgorod . right over here . novgorod literally means new town . gorod means town . the district of novgorod became known as the land of rus ' . so a lot of really interesting things going on . the varangians , first , are trying to get tribute from these tribes , which is a way of saying tax them , making them subservient to these vikings . and even though these eastern slavs were able to push them back , according to the primary chronicle , they said hey we need your help . we want you to rule over us . there 's very few times in history where people are asking a foreign group to rule over them . and so this is an interesting question . remember , this history is written under the rule of one of the descendants of rurik . so do you think it was actually this way ? or do you think the varangians maybe forced themselves on the eastern slavs and later created this narrative , that they were invited to come in ? but according to the primary chronicle , we have rurik coming from scandinavia to novgorod and establishing the land of rus ' . now the word rus is really interesting . most historians believe it to be the source of what we now say russia , or even belarus , which means white rus . some historians think it comes from the name of sweden at the time . some believe that the rus were a subgroup of varangians , of vikings . some believe that the word is derived from those who row . but either way , the primary chronicle goes on to tell us from 870 to 879 , on his deathbed , rurik bequeathed his realm to oleg , who belonged to his kin , and entrusted to oleg 's hands his son igor , for he was very young . and then from 880 to 882 , oleg set himself up as prince in kiev and declared that it should be the mother of russian cities . so rurik 's immediate successor is oleg . and in the early 880s , he goes and establishes himself in kiev , expanding the land of rus ' . this is kiev right over here , and because oleg was able to take kiev , the state that emerges from rurik and oleg not only is it known as the land of the rus ' , but it 's also known as the kievan state , and they 're often known as the kievan rus ' . and you can see here how that state expands over the next few hundred years . as we get to the year 900 , you have this off-white color , and you can see , it is in control of both novgorod and kiev . as you get to 1015 , it 's taken even more territory , and by 1113 , which is near the peak of the kievan state , you see that it has taken control of a good chunk of eastern europe . and as the state expands , its character changes as well . as you get to the end of the 10th century , you have a major event in one of rurik 's descendants , vladimir , often known as vladimir the great , he decides to convert to eastern orthodox christianity . and in a future video , i might talk about his rationales , or what historians view as his rationales for conversion . and as we will see , over time , and because of not only his conversion , but essentially the conversion of the entire kievan state , over time , especially with the eventual decline of the byzantine empire , what would eventually be russia becomes a center of eastern orthodox christianity . now the kievan state lasts as an independent state until we get to the 13th century . and from many other videos , you might be guessing what happens in the 13th century . you have genghis khan and then his descendants emerge out of central and eastern asia , and in 1240 , you have the mongol invasion , at which point , many of the principalities within the land of rus ' become tributaries to the mongolians . and they would be so for the next , roughly , 200 years until ivan the great comes along and is able to exert independence from the mongols for the rus , but we will cover that in a future video .
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and then you have the eastern slavs in what will eventually be russia , belarus , and ukraine . now to be clear , they were n't unified groups . there were many many many , for example , eastern slavic tribes .
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how closely related are these slavic groups ?
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what we 're going to talk about in this video are the origins of the russian people . and in particular , we 're going to talk about the eastern slavs who not just modern russians , but also ukrainians and belorussians view as their ancestors . so let 's think about the world in the ninth century . the early ninth century , we see the reign of charlemagne that we talk about in some depth in other videos . as we get into the 10th century , you see the reign of otto the great , holy roman emperor over the germanic kingdoms . the ninth century is also the time of tang china . you have the abbasid muslim caliphate in control over most of the middle east and north africa . and it is also the viking age . so we have here in this magenta color , this would be modern day sweden , but it was also the home of the varangians , or whom we later would refer to as the vikings , and we know them to be great seafarers . in western europe , they 're viewed as raiders of towns along the coast . but you have to remember , these histories are often written by the western europeans , not by the vikings themselves . but they were also known as traders . what you see here are two of the major centers of power and trade in the ninth century . you have constantinople , the capital of the byzantine empire , and you have baghdad , the capital of the abbasid caliphate . you also see these major waterways in eastern europe , in particular , the dnieper and the volga rivers . and so you have these significant trade routes going from the baltic sea either via the dnieper , crossing the black sea to constantinople , or going from the baltic to the volga all the way to the caspian sea and eventually making their way to baghdad . and this is well documented . there is archeological evidence of viking jewelry along these routes . there 's evidence of artifacts from these far off lands in viking territory , and we believe what the varangians traded were first they hunted in this area of northern europe . now , the people who lived in this area were known as the slavs . and there were several broad groups of slavs that you will hear historians refer to . you have the western slavs , who you could view as some of the ancestors of modern poles , czechs , and slovaks . you have the southern slavs in what would eventually be referred to as the balkans . and then you have the eastern slavs in what will eventually be russia , belarus , and ukraine . now to be clear , they were n't unified groups . there were many many many , for example , eastern slavic tribes . our best account of the early history , especially in the period as we get into the ninth century comes from what is known as the russian primary chronicle . and keep in mind , this was written at a much later period . it was written in the early 12th century . it is sometimes ascribed to the kievan monk nestor . so at previous times , it was known as nestor 's chronicles , but we do n't even have surviving accounts of this . we have surviving accounts of copies of this , or what we believe are copies of this . what i 'll share is a version of the russian primary chronicle known as the laurentian text from 1377 . and this is , of course , an english translation . it gives us some of the earliest accounts of the relationship between the varangians and the eastern slavs and how what we have come to identify as the russian people and the ukrainian people and the belorussian people , how they got their start . so right before this passage , it talks about how the varangians tried to get tribute from some of the eastern slavic tribes . and it says the tributaries of the varangians drove them back beyond the sea and , refusing them further tribute , set out to govern themselves . so they pushed them back beyond what we now call the baltic sea , and they decided to govern themselves . there was no law among them . tribe rose against tribe , and they began to war one against another . they said to themselves , let us seek a prince who may rule over us and judge us according to the law . they accordingly went overseas to the varangian russes . so they went back to the russes , and they said , these warring eastern slavic tribes said , our land is great and rich , but there is no order in it . come to rule and reign over us . they thus selected three brothers . the oldest , rurik , located himself in novgorod . right over here . novgorod literally means new town . gorod means town . the district of novgorod became known as the land of rus ' . so a lot of really interesting things going on . the varangians , first , are trying to get tribute from these tribes , which is a way of saying tax them , making them subservient to these vikings . and even though these eastern slavs were able to push them back , according to the primary chronicle , they said hey we need your help . we want you to rule over us . there 's very few times in history where people are asking a foreign group to rule over them . and so this is an interesting question . remember , this history is written under the rule of one of the descendants of rurik . so do you think it was actually this way ? or do you think the varangians maybe forced themselves on the eastern slavs and later created this narrative , that they were invited to come in ? but according to the primary chronicle , we have rurik coming from scandinavia to novgorod and establishing the land of rus ' . now the word rus is really interesting . most historians believe it to be the source of what we now say russia , or even belarus , which means white rus . some historians think it comes from the name of sweden at the time . some believe that the rus were a subgroup of varangians , of vikings . some believe that the word is derived from those who row . but either way , the primary chronicle goes on to tell us from 870 to 879 , on his deathbed , rurik bequeathed his realm to oleg , who belonged to his kin , and entrusted to oleg 's hands his son igor , for he was very young . and then from 880 to 882 , oleg set himself up as prince in kiev and declared that it should be the mother of russian cities . so rurik 's immediate successor is oleg . and in the early 880s , he goes and establishes himself in kiev , expanding the land of rus ' . this is kiev right over here , and because oleg was able to take kiev , the state that emerges from rurik and oleg not only is it known as the land of the rus ' , but it 's also known as the kievan state , and they 're often known as the kievan rus ' . and you can see here how that state expands over the next few hundred years . as we get to the year 900 , you have this off-white color , and you can see , it is in control of both novgorod and kiev . as you get to 1015 , it 's taken even more territory , and by 1113 , which is near the peak of the kievan state , you see that it has taken control of a good chunk of eastern europe . and as the state expands , its character changes as well . as you get to the end of the 10th century , you have a major event in one of rurik 's descendants , vladimir , often known as vladimir the great , he decides to convert to eastern orthodox christianity . and in a future video , i might talk about his rationales , or what historians view as his rationales for conversion . and as we will see , over time , and because of not only his conversion , but essentially the conversion of the entire kievan state , over time , especially with the eventual decline of the byzantine empire , what would eventually be russia becomes a center of eastern orthodox christianity . now the kievan state lasts as an independent state until we get to the 13th century . and from many other videos , you might be guessing what happens in the 13th century . you have genghis khan and then his descendants emerge out of central and eastern asia , and in 1240 , you have the mongol invasion , at which point , many of the principalities within the land of rus ' become tributaries to the mongolians . and they would be so for the next , roughly , 200 years until ivan the great comes along and is able to exert independence from the mongols for the rus , but we will cover that in a future video .
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and it is also the viking age . so we have here in this magenta color , this would be modern day sweden , but it was also the home of the varangians , or whom we later would refer to as the vikings , and we know them to be great seafarers . in western europe , they 're viewed as raiders of towns along the coast .
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would a russian have extremely similar genes as someone from croatia or the czech republic ?
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what we 're going to talk about in this video are the origins of the russian people . and in particular , we 're going to talk about the eastern slavs who not just modern russians , but also ukrainians and belorussians view as their ancestors . so let 's think about the world in the ninth century . the early ninth century , we see the reign of charlemagne that we talk about in some depth in other videos . as we get into the 10th century , you see the reign of otto the great , holy roman emperor over the germanic kingdoms . the ninth century is also the time of tang china . you have the abbasid muslim caliphate in control over most of the middle east and north africa . and it is also the viking age . so we have here in this magenta color , this would be modern day sweden , but it was also the home of the varangians , or whom we later would refer to as the vikings , and we know them to be great seafarers . in western europe , they 're viewed as raiders of towns along the coast . but you have to remember , these histories are often written by the western europeans , not by the vikings themselves . but they were also known as traders . what you see here are two of the major centers of power and trade in the ninth century . you have constantinople , the capital of the byzantine empire , and you have baghdad , the capital of the abbasid caliphate . you also see these major waterways in eastern europe , in particular , the dnieper and the volga rivers . and so you have these significant trade routes going from the baltic sea either via the dnieper , crossing the black sea to constantinople , or going from the baltic to the volga all the way to the caspian sea and eventually making their way to baghdad . and this is well documented . there is archeological evidence of viking jewelry along these routes . there 's evidence of artifacts from these far off lands in viking territory , and we believe what the varangians traded were first they hunted in this area of northern europe . now , the people who lived in this area were known as the slavs . and there were several broad groups of slavs that you will hear historians refer to . you have the western slavs , who you could view as some of the ancestors of modern poles , czechs , and slovaks . you have the southern slavs in what would eventually be referred to as the balkans . and then you have the eastern slavs in what will eventually be russia , belarus , and ukraine . now to be clear , they were n't unified groups . there were many many many , for example , eastern slavic tribes . our best account of the early history , especially in the period as we get into the ninth century comes from what is known as the russian primary chronicle . and keep in mind , this was written at a much later period . it was written in the early 12th century . it is sometimes ascribed to the kievan monk nestor . so at previous times , it was known as nestor 's chronicles , but we do n't even have surviving accounts of this . we have surviving accounts of copies of this , or what we believe are copies of this . what i 'll share is a version of the russian primary chronicle known as the laurentian text from 1377 . and this is , of course , an english translation . it gives us some of the earliest accounts of the relationship between the varangians and the eastern slavs and how what we have come to identify as the russian people and the ukrainian people and the belorussian people , how they got their start . so right before this passage , it talks about how the varangians tried to get tribute from some of the eastern slavic tribes . and it says the tributaries of the varangians drove them back beyond the sea and , refusing them further tribute , set out to govern themselves . so they pushed them back beyond what we now call the baltic sea , and they decided to govern themselves . there was no law among them . tribe rose against tribe , and they began to war one against another . they said to themselves , let us seek a prince who may rule over us and judge us according to the law . they accordingly went overseas to the varangian russes . so they went back to the russes , and they said , these warring eastern slavic tribes said , our land is great and rich , but there is no order in it . come to rule and reign over us . they thus selected three brothers . the oldest , rurik , located himself in novgorod . right over here . novgorod literally means new town . gorod means town . the district of novgorod became known as the land of rus ' . so a lot of really interesting things going on . the varangians , first , are trying to get tribute from these tribes , which is a way of saying tax them , making them subservient to these vikings . and even though these eastern slavs were able to push them back , according to the primary chronicle , they said hey we need your help . we want you to rule over us . there 's very few times in history where people are asking a foreign group to rule over them . and so this is an interesting question . remember , this history is written under the rule of one of the descendants of rurik . so do you think it was actually this way ? or do you think the varangians maybe forced themselves on the eastern slavs and later created this narrative , that they were invited to come in ? but according to the primary chronicle , we have rurik coming from scandinavia to novgorod and establishing the land of rus ' . now the word rus is really interesting . most historians believe it to be the source of what we now say russia , or even belarus , which means white rus . some historians think it comes from the name of sweden at the time . some believe that the rus were a subgroup of varangians , of vikings . some believe that the word is derived from those who row . but either way , the primary chronicle goes on to tell us from 870 to 879 , on his deathbed , rurik bequeathed his realm to oleg , who belonged to his kin , and entrusted to oleg 's hands his son igor , for he was very young . and then from 880 to 882 , oleg set himself up as prince in kiev and declared that it should be the mother of russian cities . so rurik 's immediate successor is oleg . and in the early 880s , he goes and establishes himself in kiev , expanding the land of rus ' . this is kiev right over here , and because oleg was able to take kiev , the state that emerges from rurik and oleg not only is it known as the land of the rus ' , but it 's also known as the kievan state , and they 're often known as the kievan rus ' . and you can see here how that state expands over the next few hundred years . as we get to the year 900 , you have this off-white color , and you can see , it is in control of both novgorod and kiev . as you get to 1015 , it 's taken even more territory , and by 1113 , which is near the peak of the kievan state , you see that it has taken control of a good chunk of eastern europe . and as the state expands , its character changes as well . as you get to the end of the 10th century , you have a major event in one of rurik 's descendants , vladimir , often known as vladimir the great , he decides to convert to eastern orthodox christianity . and in a future video , i might talk about his rationales , or what historians view as his rationales for conversion . and as we will see , over time , and because of not only his conversion , but essentially the conversion of the entire kievan state , over time , especially with the eventual decline of the byzantine empire , what would eventually be russia becomes a center of eastern orthodox christianity . now the kievan state lasts as an independent state until we get to the 13th century . and from many other videos , you might be guessing what happens in the 13th century . you have genghis khan and then his descendants emerge out of central and eastern asia , and in 1240 , you have the mongol invasion , at which point , many of the principalities within the land of rus ' become tributaries to the mongolians . and they would be so for the next , roughly , 200 years until ivan the great comes along and is able to exert independence from the mongols for the rus , but we will cover that in a future video .
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you have the southern slavs in what would eventually be referred to as the balkans . and then you have the eastern slavs in what will eventually be russia , belarus , and ukraine . now to be clear , they were n't unified groups .
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were the `` rus '' eastern orthodox christians since they conquered the eastern slavs ?
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so here i want to talk about the gradient and the context of a contour map . so let 's say we have a multivariable function . a two-variable function f of x , y . and this one is just gon na equal x times y . so we can visualize this with a contour map just on the xy plane . so what i 'm gon na do is i 'm gon na go over here . i 'm gon na draw a y axis and my x axis . all right , so this right here represents x values . and this represents y values . and this is entirely the input space . and i have a video on contour maps if you are unfamiliar with them or are feeling uncomfortable . and the contour map for x times y looks something like this . and each one of these lines represents a constant value . so you might be thinking that you have , you know , let 's say you want a the constant value for f of x times y is equal to two . would be one of these lines . that would be what one of these lines represents . and a way you could think about that for this specific function is you saying hey , when is x times y equal to two ? and that 's kind of like the graph y equals two over x . and that 's where you would see something like this . so all of these lines , they 're representing constant values for the function . and now i want to take a look at the gradient field . and the gradient , if you 'll remember , is just a vector full of the partial derivatives of f. and let 's just actually write it out . the gradient of f , with our little del symbol , is a function of x and y . and it 's a vector-valued function whose first coordinate is the partial derivative of f with respect to x . and the second component is the partial derivative with respect to y . so when we actually do this for our function , we take the partial derivative with respect to x . it takes a look . x looks like a variable . y looks like a constant . the derivative of this whole thing is just equal to that constant , y . and then kind of the reverse for when you take the partial derivative with respect to y. y looks like a variable . x looks like a constant . and the derivative is just that constant , x . and this can be visualized as a vector field in the xy plane as well . you know , at every given point , xy , so you kind of go like x equals two , y equals one , let 's say . so that would be x equals two , y equals one . you would plug in the vector and see what should be output . and at this point , the point is two , one . the desired output kind of swaps those . so we 're looking somehow to draw the vector one , two . so you would expect to see the vector that has an x component of one and a y component of two . something like that . but it 's probably gon na be scaled down because of the way we usually draw vector fields . and the entire field looks like this . so i 'll go ahead and erase what i had going on . since this is a little bit clearer . and remember , we scaled down all the vectors . the color represents length . so red here is super-long . blue is gon na be kind of short . and one thing worth noticing . if you take a look at all of the given points around , if the vector is crossing a contour line , it 's perpendicular to that contour line . wherever you go . you know , you go down here , this vector 's perpendicular to the contour line . over here , perpendicular to the contour line . and this happens everywhere . and it 's for a very good reason . and it 's also super-useful . so let 's just think about what that reason should be . let 's zoom in on a point . so i 'm gon na clear up our function here . clear up all of the information about it . and just zoom in on one of those points . so let 's say like right here . we 'll take that guy and kind of imagine zooming in and saying what 's going on in that region ? so you 've got some kind of contour line . and it 's swooping down like this . and that represents some kind of value . let 's say that represents the value f equals two . and , you know , it might not be a perfect straight line . but the more you zoom in , the more it looks like a straight line . and when you want to interpret the gradient vector . if you remember , in the video about how to interpret the gradient in the context of a graph , i said it points in the direction of steepest descent . so if you imagine all the possible vectors kind of pointing away from this point , the question is , which direction should you move to increase the value of f the fastest ? and there 's two ways of thinking about that . one is to look at all of these different directions and say which one increases x the most ? but another way of doing it would be to get rid of them all and just take a look at another contour line that represents a slight increase . all right , so let 's say you 're taking a look at a contour line , another contour line . something like this . and maybe that represents something that 's right next to it . like 2.1 . that represents , you know , another value that 's very close . and if i were a better artist , and this was more representative , it would look like a line that 's parallel to the original one . because if you change the output by just a little bit , the set of in points that look like it is pretty much the same but just shifted over a bit . so another way we can think about the gradient here is to say of all of the vectors that move from this output of two up to the value of 2.1 . you know , you 're looking at all of the possible different vectors that do that . you know , which one does it the fastest ? and this time , instead of thinking of the fastest as constant-length vectors , what increases it the most , we 'll be thinking , constant increase in the output . which one does it with the shortest distance ? and if you think of them as being roughly parallel lines , it should n't be hard to convince yourself that the shortest distance is n't gon na be , you know , any of those . it 's gon na be the one that connects them pretty much perpendicular to the original line . because if you think about these as lines , and the more you zoom in , the more they pretty much look like parallel lines , the path that connects one to the other is gon na be perpendicular to both of them . so because of this interpretation of the gradient as the direction of steepest descent , it 's a natural consequence that every time it 's on a contour line , wherever you 're looking it 's actually perpendicular to that line . because you can think about it as getting to the next contour line as fast as it can . increasing the function as fast as it can . and this is actually a very useful intepretation of the gradient in different contexts . so it 's a good one to keep in the back of your mind . gradient is always perpendicular to contour lines . great . see you next video .
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so all of these lines , they 're representing constant values for the function . and now i want to take a look at the gradient field . and the gradient , if you 'll remember , is just a vector full of the partial derivatives of f. and let 's just actually write it out .
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given a vector field , is it always the gradient field of a function ?
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so here i want to talk about the gradient and the context of a contour map . so let 's say we have a multivariable function . a two-variable function f of x , y . and this one is just gon na equal x times y . so we can visualize this with a contour map just on the xy plane . so what i 'm gon na do is i 'm gon na go over here . i 'm gon na draw a y axis and my x axis . all right , so this right here represents x values . and this represents y values . and this is entirely the input space . and i have a video on contour maps if you are unfamiliar with them or are feeling uncomfortable . and the contour map for x times y looks something like this . and each one of these lines represents a constant value . so you might be thinking that you have , you know , let 's say you want a the constant value for f of x times y is equal to two . would be one of these lines . that would be what one of these lines represents . and a way you could think about that for this specific function is you saying hey , when is x times y equal to two ? and that 's kind of like the graph y equals two over x . and that 's where you would see something like this . so all of these lines , they 're representing constant values for the function . and now i want to take a look at the gradient field . and the gradient , if you 'll remember , is just a vector full of the partial derivatives of f. and let 's just actually write it out . the gradient of f , with our little del symbol , is a function of x and y . and it 's a vector-valued function whose first coordinate is the partial derivative of f with respect to x . and the second component is the partial derivative with respect to y . so when we actually do this for our function , we take the partial derivative with respect to x . it takes a look . x looks like a variable . y looks like a constant . the derivative of this whole thing is just equal to that constant , y . and then kind of the reverse for when you take the partial derivative with respect to y. y looks like a variable . x looks like a constant . and the derivative is just that constant , x . and this can be visualized as a vector field in the xy plane as well . you know , at every given point , xy , so you kind of go like x equals two , y equals one , let 's say . so that would be x equals two , y equals one . you would plug in the vector and see what should be output . and at this point , the point is two , one . the desired output kind of swaps those . so we 're looking somehow to draw the vector one , two . so you would expect to see the vector that has an x component of one and a y component of two . something like that . but it 's probably gon na be scaled down because of the way we usually draw vector fields . and the entire field looks like this . so i 'll go ahead and erase what i had going on . since this is a little bit clearer . and remember , we scaled down all the vectors . the color represents length . so red here is super-long . blue is gon na be kind of short . and one thing worth noticing . if you take a look at all of the given points around , if the vector is crossing a contour line , it 's perpendicular to that contour line . wherever you go . you know , you go down here , this vector 's perpendicular to the contour line . over here , perpendicular to the contour line . and this happens everywhere . and it 's for a very good reason . and it 's also super-useful . so let 's just think about what that reason should be . let 's zoom in on a point . so i 'm gon na clear up our function here . clear up all of the information about it . and just zoom in on one of those points . so let 's say like right here . we 'll take that guy and kind of imagine zooming in and saying what 's going on in that region ? so you 've got some kind of contour line . and it 's swooping down like this . and that represents some kind of value . let 's say that represents the value f equals two . and , you know , it might not be a perfect straight line . but the more you zoom in , the more it looks like a straight line . and when you want to interpret the gradient vector . if you remember , in the video about how to interpret the gradient in the context of a graph , i said it points in the direction of steepest descent . so if you imagine all the possible vectors kind of pointing away from this point , the question is , which direction should you move to increase the value of f the fastest ? and there 's two ways of thinking about that . one is to look at all of these different directions and say which one increases x the most ? but another way of doing it would be to get rid of them all and just take a look at another contour line that represents a slight increase . all right , so let 's say you 're taking a look at a contour line , another contour line . something like this . and maybe that represents something that 's right next to it . like 2.1 . that represents , you know , another value that 's very close . and if i were a better artist , and this was more representative , it would look like a line that 's parallel to the original one . because if you change the output by just a little bit , the set of in points that look like it is pretty much the same but just shifted over a bit . so another way we can think about the gradient here is to say of all of the vectors that move from this output of two up to the value of 2.1 . you know , you 're looking at all of the possible different vectors that do that . you know , which one does it the fastest ? and this time , instead of thinking of the fastest as constant-length vectors , what increases it the most , we 'll be thinking , constant increase in the output . which one does it with the shortest distance ? and if you think of them as being roughly parallel lines , it should n't be hard to convince yourself that the shortest distance is n't gon na be , you know , any of those . it 's gon na be the one that connects them pretty much perpendicular to the original line . because if you think about these as lines , and the more you zoom in , the more they pretty much look like parallel lines , the path that connects one to the other is gon na be perpendicular to both of them . so because of this interpretation of the gradient as the direction of steepest descent , it 's a natural consequence that every time it 's on a contour line , wherever you 're looking it 's actually perpendicular to that line . because you can think about it as getting to the next contour line as fast as it can . increasing the function as fast as it can . and this is actually a very useful intepretation of the gradient in different contexts . so it 's a good one to keep in the back of your mind . gradient is always perpendicular to contour lines . great . see you next video .
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but the more you zoom in , the more it looks like a straight line . and when you want to interpret the gradient vector . if you remember , in the video about how to interpret the gradient in the context of a graph , i said it points in the direction of steepest descent .
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does the orthogonality propierty of the gradient vector mantains as we go into higher dimensions ?
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so here i want to talk about the gradient and the context of a contour map . so let 's say we have a multivariable function . a two-variable function f of x , y . and this one is just gon na equal x times y . so we can visualize this with a contour map just on the xy plane . so what i 'm gon na do is i 'm gon na go over here . i 'm gon na draw a y axis and my x axis . all right , so this right here represents x values . and this represents y values . and this is entirely the input space . and i have a video on contour maps if you are unfamiliar with them or are feeling uncomfortable . and the contour map for x times y looks something like this . and each one of these lines represents a constant value . so you might be thinking that you have , you know , let 's say you want a the constant value for f of x times y is equal to two . would be one of these lines . that would be what one of these lines represents . and a way you could think about that for this specific function is you saying hey , when is x times y equal to two ? and that 's kind of like the graph y equals two over x . and that 's where you would see something like this . so all of these lines , they 're representing constant values for the function . and now i want to take a look at the gradient field . and the gradient , if you 'll remember , is just a vector full of the partial derivatives of f. and let 's just actually write it out . the gradient of f , with our little del symbol , is a function of x and y . and it 's a vector-valued function whose first coordinate is the partial derivative of f with respect to x . and the second component is the partial derivative with respect to y . so when we actually do this for our function , we take the partial derivative with respect to x . it takes a look . x looks like a variable . y looks like a constant . the derivative of this whole thing is just equal to that constant , y . and then kind of the reverse for when you take the partial derivative with respect to y. y looks like a variable . x looks like a constant . and the derivative is just that constant , x . and this can be visualized as a vector field in the xy plane as well . you know , at every given point , xy , so you kind of go like x equals two , y equals one , let 's say . so that would be x equals two , y equals one . you would plug in the vector and see what should be output . and at this point , the point is two , one . the desired output kind of swaps those . so we 're looking somehow to draw the vector one , two . so you would expect to see the vector that has an x component of one and a y component of two . something like that . but it 's probably gon na be scaled down because of the way we usually draw vector fields . and the entire field looks like this . so i 'll go ahead and erase what i had going on . since this is a little bit clearer . and remember , we scaled down all the vectors . the color represents length . so red here is super-long . blue is gon na be kind of short . and one thing worth noticing . if you take a look at all of the given points around , if the vector is crossing a contour line , it 's perpendicular to that contour line . wherever you go . you know , you go down here , this vector 's perpendicular to the contour line . over here , perpendicular to the contour line . and this happens everywhere . and it 's for a very good reason . and it 's also super-useful . so let 's just think about what that reason should be . let 's zoom in on a point . so i 'm gon na clear up our function here . clear up all of the information about it . and just zoom in on one of those points . so let 's say like right here . we 'll take that guy and kind of imagine zooming in and saying what 's going on in that region ? so you 've got some kind of contour line . and it 's swooping down like this . and that represents some kind of value . let 's say that represents the value f equals two . and , you know , it might not be a perfect straight line . but the more you zoom in , the more it looks like a straight line . and when you want to interpret the gradient vector . if you remember , in the video about how to interpret the gradient in the context of a graph , i said it points in the direction of steepest descent . so if you imagine all the possible vectors kind of pointing away from this point , the question is , which direction should you move to increase the value of f the fastest ? and there 's two ways of thinking about that . one is to look at all of these different directions and say which one increases x the most ? but another way of doing it would be to get rid of them all and just take a look at another contour line that represents a slight increase . all right , so let 's say you 're taking a look at a contour line , another contour line . something like this . and maybe that represents something that 's right next to it . like 2.1 . that represents , you know , another value that 's very close . and if i were a better artist , and this was more representative , it would look like a line that 's parallel to the original one . because if you change the output by just a little bit , the set of in points that look like it is pretty much the same but just shifted over a bit . so another way we can think about the gradient here is to say of all of the vectors that move from this output of two up to the value of 2.1 . you know , you 're looking at all of the possible different vectors that do that . you know , which one does it the fastest ? and this time , instead of thinking of the fastest as constant-length vectors , what increases it the most , we 'll be thinking , constant increase in the output . which one does it with the shortest distance ? and if you think of them as being roughly parallel lines , it should n't be hard to convince yourself that the shortest distance is n't gon na be , you know , any of those . it 's gon na be the one that connects them pretty much perpendicular to the original line . because if you think about these as lines , and the more you zoom in , the more they pretty much look like parallel lines , the path that connects one to the other is gon na be perpendicular to both of them . so because of this interpretation of the gradient as the direction of steepest descent , it 's a natural consequence that every time it 's on a contour line , wherever you 're looking it 's actually perpendicular to that line . because you can think about it as getting to the next contour line as fast as it can . increasing the function as fast as it can . and this is actually a very useful intepretation of the gradient in different contexts . so it 's a good one to keep in the back of your mind . gradient is always perpendicular to contour lines . great . see you next video .
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the color represents length . so red here is super-long . blue is gon na be kind of short .
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grant said red vectors are super long but they should be super small right ?
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