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i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | and i 'll be frank , liquids are kind of fascinating because you can never nail them down , i guess is the best way to view them . or a gas . so let 's just draw a water molecule . | at what temperature can a rock be turned into gas ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | and let me write this down . this is in the solid phase , or the solid state of matter . now something very interesting happens . | what is the 4th state of matter ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | i 'll do it in celsius because that 's what we 're familiar with . what happens ? that 's the temperature at which water will vaporize or which water will boil . | what happens in the case of plasma ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | so as i add heat , my temperature will go up . heat is a form of energy . and when i add it to these molecules , as i did in this example , what did it do ? | 0 , does heat affect the energy ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | and this is the temperature . we 'll talk about the states of matter in a second . so heat is often denoted by q . | is bose-einstein condensate a part of the 5 states of matter ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | so this is ice in our example . all solids are n't ice . although , you could think of a rock as solid magma . | should n't you have drawn 2 circles ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | and let me write this down . this is in the solid phase , or the solid state of matter . now something very interesting happens . | may i know the comparision between solid , liquid and gas based on the states of matter ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | and that 's right here . this is the heat of vaporization . and the same idea is happening . | if the molecules have a tendency to `` want '' to be around each other , would there be a horizontal line at the heat of fusion and vaporization if you went from high heat to low heat ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | at least , as far as my neural connections could make it . but enthalpy is closely related to heat . it 's heat content . for our purposes , when you hear someone say change in enthalpy , you should really just be thinking of change in heat . | why ca n't latent heat be measured in the thermometer ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | change in enthalpy is really just change in heat . and just remember , all of these things , whether we 're talking about heat , kinetic energy , potential energy , enthalpy . you 'll hear them in different contexts , and you 're like , i thought i should be using heat and they 're talking about enthalpy . | how can you tell the energy in particles of a substance ? |
i think we 're all reasonably familiar with the three states of matter in our everyday world . at very high temperatures you get a fourth . but the three ones that we normally deal with are , things could be a solid , a liquid , or it could be a gas . and we have this general notion , and i think water is the example t... | so what happens at zero degrees ? which is also 273.15 kelvin . let 's say that 's that line . | how much is a kelvin ? |
- [ lecturer ] in previous videos we talk about the emergence of the maurya empire around 322 bce , shortly after the invasion of alexander the great , as the first truly great indian empire that unifies most of the indian subcontinent . now that empire eventually falls and the next significant empire to emerge , espec... | and all at once is said . '' beyond literature and writing you have significant contributions to science , most notably aryabhata , he 's known for a very accurate approximation of pi , but even more important , a recognition that was an approximation and that he potentially recognized the irrationality of pi , one of ... | even though aryabhatta discovered trigonometry , gravity and pi , why is the credit still given to european scientists and mathematicians ? |
- [ lecturer ] in previous videos we talk about the emergence of the maurya empire around 322 bce , shortly after the invasion of alexander the great , as the first truly great indian empire that unifies most of the indian subcontinent . now that empire eventually falls and the next significant empire to emerge , espec... | the game of chess or the early version of the game of chess was invented , called chaturanga , and they had horsemen , which were the knights , they had infantry , which were the pawns , they had elephants , which eventually turned into bishops , but as it migrated into persia , the muslim world , and then into europe ... | how did the iron pillar not rust ? |
- [ lecturer ] in previous videos we talk about the emergence of the maurya empire around 322 bce , shortly after the invasion of alexander the great , as the first truly great indian empire that unifies most of the indian subcontinent . now that empire eventually falls and the next significant empire to emerge , espec... | this was some of the coinage of the gupta empire . so the big takeaway here is , this was india 's golden age , the classical period of india , a lot of modern hinduism and indian culture can be traced back to this time period . but it is n't just its influence on india , in other videos we talk about the islamic golde... | are there any traces of this dynasty in the current distribution of languages in india ? |
- [ lecturer ] in previous videos we talk about the emergence of the maurya empire around 322 bce , shortly after the invasion of alexander the great , as the first truly great indian empire that unifies most of the indian subcontinent . now that empire eventually falls and the next significant empire to emerge , espec... | beyond the sciences , and once again , this is just a sample of all that happened during this period , you have the significant hindu epics , the mahabharata , the ramayana , the puranas , get written down and formalized , you can say they were canonized . the game of chess or the early version of the game of chess was... | i thought that elephants became rooks and camels became bishops ? |
- [ lecturer ] in previous videos we talk about the emergence of the maurya empire around 322 bce , shortly after the invasion of alexander the great , as the first truly great indian empire that unifies most of the indian subcontinent . now that empire eventually falls and the next significant empire to emerge , espec... | now , let 's zoom in on our timeline to get a deeper appreciation of the gupta empire . it 's believed that its start was with sri gupta , he started the gupta dynasty around 240 and it 's disputed where they emerged , it mighta been in that region or in that region , there 's different accounts of where the gupta dyna... | what present day countries were part of the gupta dynasty ? |
- [ lecturer ] in previous videos we talk about the emergence of the maurya empire around 322 bce , shortly after the invasion of alexander the great , as the first truly great indian empire that unifies most of the indian subcontinent . now that empire eventually falls and the next significant empire to emerge , espec... | this was some of the coinage of the gupta empire . so the big takeaway here is , this was india 's golden age , the classical period of india , a lot of modern hinduism and indian culture can be traced back to this time period . but it is n't just its influence on india , in other videos we talk about the islamic golde... | was india the only one ? |
so let 's see if we can find the derivative with respect to x , with either x times the cosine of x . and like always , pause this video and give it a go on your own before we work through it . so when you look at this you might say , `` well , i know how to find `` the derivative with e to the x , '' that 's infact j... | we know a few things . we know the derivative with respect to x of e to the x. e to the x is e to the x . we know how to find the derivative cosine of x . the derivative with respect to x of cosine of x is equal to negative sine of x . but , how do we find the derivative of their product ? | why is the 1st derivative of ln ( x ) = 1/x ? |
so let 's see if we can find the derivative with respect to x , with either x times the cosine of x . and like always , pause this video and give it a go on your own before we work through it . so when you look at this you might say , `` well , i know how to find `` the derivative with e to the x , '' that 's infact j... | we know how to find the derivative cosine of x . the derivative with respect to x of cosine of x is equal to negative sine of x . but , how do we find the derivative of their product ? | when using the product rule , do both functions need to be functions of x ( when i 'm taking the derivative with respect to x ) ? |
so let 's see if we can find the derivative with respect to x , with either x times the cosine of x . and like always , pause this video and give it a go on your own before we work through it . so when you look at this you might say , `` well , i know how to find `` the derivative with e to the x , '' that 's infact j... | we know how to find the derivative cosine of x . the derivative with respect to x of cosine of x is equal to negative sine of x . but , how do we find the derivative of their product ? | what if i need to differentiate xy with respect to x ( where y is a function of x ) ? |
so let 's see if we can find the derivative with respect to x , with either x times the cosine of x . and like always , pause this video and give it a go on your own before we work through it . so when you look at this you might say , `` well , i know how to find `` the derivative with e to the x , '' that 's infact j... | or , if you want , you could factor out an e to the x . this is the same thing as e to the x times cosine of x minus sine of x. cosine of x minus sine of x . so hopefully this makes the product rule a little bit more tangible . | i get how y is a function , but how is x a function ? |
( piano music ) steven : this is steven zucker . juliana : and juliana kreinik , and we are discussing lyonel feininger 's cathedral of the future from 1919 . steven : this was one of the primary images of the bauhaus right ? juliana : it was the starting image . steven : was its intent actually marketing ? it was a br... | what does the word bauhaus even mean ? steven : well the bauhaus refers to the small building or workshop that would be just outside of a large medieval building campaign perhaps for a cathedral in fact . so , you have actually the shop where the masons are doing their work outside and it is a place , i think , as it w... | is there any meaning to the cube shape on the pediment in the bottom-center of the building ? |
( piano music ) steven : this is steven zucker . juliana : and juliana kreinik , and we are discussing lyonel feininger 's cathedral of the future from 1919 . steven : this was one of the primary images of the bauhaus right ? juliana : it was the starting image . steven : was its intent actually marketing ? it was a br... | we could really talk about what the cathedral represents to the bauhaus and really what is it about building ? what does the word bauhaus even mean ? steven : well the bauhaus refers to the small building or workshop that would be just outside of a large medieval building campaign perhaps for a cathedral in fact . so ,... | it almost looks like the cube has replaced a cross , perhaps suggesting that the bauhaus is creating a church of creation rather than a church of devotion ? |
( piano music ) man : we 're in the museum of the duomo in florence and we 're looking at a donatello . it 's not marble . it 's not bronze . it 's wood . it looks so frail . it 's a sculpture of mary magdalene . woman : it 's a very difficult sculpture to look at because it 's ugly . mary magdalene is shown as a hermi... | in fact , even her belt is actually her hair wrapped around her . i think it 's so interesting the choice of materials . i started out by saying this is wood . | do you think her worried looking face along with her hopeful posture was being sculpted because it was during the moment jesus was being crucified ? |
( piano music ) man : we 're in the museum of the duomo in florence and we 're looking at a donatello . it 's not marble . it 's not bronze . it 's wood . it looks so frail . it 's a sculpture of mary magdalene . woman : it 's a very difficult sculpture to look at because it 's ugly . mary magdalene is shown as a hermi... | it 's not bronze . it 's wood . it looks so frail . | so how is the wood still there ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | so the short side here is 3 . the shortest side here is 9 square roots of 3 . so we want to see whether the ratio of 3 to 9 square roots of 3 is equal to the next longest side over here , is 3 square roots of 3 over the next longest side over here , which is 27 . | would that be considered an `` improper '' fraction because there are square roots in the denominators ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | this is actually saying , this is 1 over 3 root 3 , which is the same thing as square root of 3 over 9 , which is this right over here , which is the same thing as 1 over 3 root 3 . so actually , these are similar triangles . so we can actually say it , and i 'll make sure i get the order right . | why is it important to write the names of similar triangles in a certain order ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | there are some shared angles . this guy -- they both share that angle , the larger triangle and the smaller triangle . so there could be a statement of similarity we could make if we knew that this definitely was a right angle . | is n't triangle abc the same as triangle cab ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | there are some shared angles . this guy -- they both share that angle , the larger triangle and the smaller triangle . so there could be a statement of similarity we could make if we knew that this definitely was a right angle . | do the letters matter in any triangle or can you put any letter on a triangle ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | blue and magenta side -- that is h. and then we go along the blue side to f , go along the blue side to i , and then you went along the orange side to g , and then you go along the orange side to j . so triangle efj -- efg is similar to triangle hij by side-side-side similarity . they 're not congruent sides . | 6 is n't 27 the longest side of the triangle ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | you definitely want to say , what am i given , and what am i not given ? if these were n't labeled parallel , we would n't be able to make the statement , even if they looked parallel . one thing we do have is that we have this angle right here that 's common to the inner triangle and to the outer triangle , and they '... | what would that squiggly line be called ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | there are some shared angles . this guy -- they both share that angle , the larger triangle and the smaller triangle . so there could be a statement of similarity we could make if we knew that this definitely was a right angle . | if it is proved by sss similarity postulate that triangle abc is similar to triangle edf then does it mean that angle a is equal to angle e ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | there are some shared angles . this guy -- they both share that angle , the larger triangle and the smaller triangle . so there could be a statement of similarity we could make if we knew that this definitely was a right angle . | if it is proved by sss similarity postulate that triangle abc is similar to triangle edf then does it mean that angle a is equal to angle e ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | blue and magenta side -- that is h. and then we go along the blue side to f , go along the blue side to i , and then you went along the orange side to g , and then you go along the orange side to j . so triangle efj -- efg is similar to triangle hij by side-side-side similarity . they 're not congruent sides . | aka if another triangle is double of another triangle does that mean their similar 100 % ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | so the ratio between the shorter sides on either side of the angle and the longer sides on either side of the angle , for both triangles , the ratio is the same . so by sas we know that the two triangles are congruent . but we have to be careful on how we state the triangles . | that means that the triangles must be congruent , so if two triangles are congruent are n't they also similar ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | we have an angle that 's congruent to another angle right over there , and we have two sides . and so it might be tempting to use side-angle-side , because we have side-angle-side here . and even the ratios look kind of tempting , because 4 times 2 is 8 . | ( besides identifying the labeled point/line segments ) , i mean is the hypotenuse the only side with an appelation ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | so the ratio between the shorter sides on either side of the angle and the longer sides on either side of the angle , for both triangles , the ratio is the same . so by sas we know that the two triangles are congruent . but we have to be careful on how we state the triangles . | on the third problem , why cant you use the two small triangles and say segment bd is the same because of they are on the same triangle ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | at first they do n't look equal , but we can actually rationalize this denominator right over here . we can show that 1 over 3 square roots of 3 , if you multiply it by square root of 3 over square root of 3 , this actually gives you in the numerator square root of 3 over square root of 3 times square root of 3 is 3 , ... | how is 3 root 3 ( fg ) considered as the longest side ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | at first they do n't look equal , but we can actually rationalize this denominator right over here . we can show that 1 over 3 square roots of 3 , if you multiply it by square root of 3 over square root of 3 , this actually gives you in the numerator square root of 3 over square root of 3 times square root of 3 is 3 , ... | how did 3 root 3 became bigger than 6 ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | at first they do n't look equal , but we can actually rationalize this denominator right over here . we can show that 1 over 3 square roots of 3 , if you multiply it by square root of 3 over square root of 3 , this actually gives you in the numerator square root of 3 over square root of 3 times square root of 3 is 3 , ... | and how did 18 root 3 became bigger than 27 ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | this is two . xt is 3 plus 1 is 4 . xz is 3 , and xs is 6 . so you have 2 over 4 , which is 1/2 , which is the same thing as 3/6 . so the ratio between the shorter sides on either side of the angle and the longer sides on either side of the angle , for both triangles , the ratio is the same . | sal multiplies the 1/3root3 by root3/root3 , how is he able to do that ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | and it 's a little confusing , because we 've kind of flipped which side , but i 'm just thinking about the shorter side on either side of this angle in between , and then the longer side on either side of this angle . so these are the shorter sides for the smaller triangle and the larger triangle . these are the longe... | what is a good way to find the corresponding sides of one triangle to another ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | let me do this in a neutral color . so this becomes 1 over 3 square roots of 3 . this becomes 1 over root 3 over 9 , which seems like a different number , but we want to be careful here . | why is 27 < 18 square roots of 3 ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . | how did sal come to that conclusion ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | it 's inside of triangle aec . they both share this angle right over there , so that gives us one angle . we need two to get to angle-angle , which gives us similarity . and we know that these two lines are parallel . | 9 , how would knowing , that an angle in the smaller triangle is a right angle , help us make statements about similarity ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | there are some shared angles . this guy -- they both share that angle , the larger triangle and the smaller triangle . so there could be a statement of similarity we could make if we knew that this definitely was a right angle . | why do you have to make sure that triangle ace ~ triangle bcd are triangle ace ~ triangle bcd and not triangle cea ~ triangle dbc ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | this is actually saying , this is 1 over 3 root 3 , which is the same thing as square root of 3 over 9 , which is this right over here , which is the same thing as 1 over 3 root 3 . so actually , these are similar triangles . so we can actually say it , and i 'll make sure i get the order right . | how are the triangles similar ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | blue and magenta side -- that is h. and then we go along the blue side to f , go along the blue side to i , and then you went along the orange side to g , and then you go along the orange side to j . so triangle efj -- efg is similar to triangle hij by side-side-side similarity . they 're not congruent sides . | why is side xt taken in the triangle xst in the above video ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | we have an angle that 's congruent to another angle right over there , and we have two sides . and so it might be tempting to use side-angle-side , because we have side-angle-side here . and even the ratios look kind of tempting , because 4 times 2 is 8 . | why dont we take the side xz ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | so they 've given us the three sides of both triangles . so let 's just figure out if the ratios between corresponding sides are a constant . so let 's start with the short side . | how do you figure out if the ratios of corresponding sides are constant ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | at first they do n't look equal , but we can actually rationalize this denominator right over here . we can show that 1 over 3 square roots of 3 , if you multiply it by square root of 3 over square root of 3 , this actually gives you in the numerator square root of 3 over square root of 3 times square root of 3 is 3 , ... | what does `` sas '' mean ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | so triangle efj -- efg is similar to triangle hij by side-side-side similarity . they 're not congruent sides . they all have just the same ratio or the same scaling factor . | what is congruent and ratio ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | there are some shared angles . this guy -- they both share that angle , the larger triangle and the smaller triangle . so there could be a statement of similarity we could make if we knew that this definitely was a right angle . | how would we prove that triangle xyz ~ triangle xts in a two-column statement/reason proof ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | this is actually saying , this is 1 over 3 root 3 , which is the same thing as square root of 3 over 9 , which is this right over here , which is the same thing as 1 over 3 root 3 . so actually , these are similar triangles . so we can actually say it , and i 'll make sure i get the order right . | how to determine similar triangles by their angles ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | this is actually saying , this is 1 over 3 root 3 , which is the same thing as square root of 3 over 9 , which is this right over here , which is the same thing as 1 over 3 root 3 . so actually , these are similar triangles . so we can actually say it , and i 'll make sure i get the order right . | i mean in the case where the triangles are turned or flipped ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | there are some shared angles . this guy -- they both share that angle , the larger triangle and the smaller triangle . so there could be a statement of similarity we could make if we knew that this definitely was a right angle . | is triangle abc~def is same as acb~dfe ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | we have an angle that 's congruent to another angle right over there , and we have two sides . and so it might be tempting to use side-angle-side , because we have side-angle-side here . and even the ratios look kind of tempting , because 4 times 2 is 8 . | 4 , would n't it be possible to use pythagorean theorem to determine the other side to check for similarity ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | they 're not congruent sides . they all have just the same ratio or the same scaling factor . now let 's do this last one , right over here . | so if the triangles have the same scaling factor does that mean they 're automatically sss ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | so what we want to compare is the ratio between -- let me write it this way . we want to see , is xy over xt equal to the ratio of the longer side ? or if we 're looking relative to this angle , the longer of the two , not necessarily the longest of the triangle , although it looks like that as well . | why is 18root3 longer than 27 ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . | what does this symbol mean -- -- - > ~ ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | at first they do n't look equal , but we can actually rationalize this denominator right over here . we can show that 1 over 3 square roots of 3 , if you multiply it by square root of 3 over square root of 3 , this actually gives you in the numerator square root of 3 over square root of 3 times square root of 3 is 3 , ... | how did 3 x sqrt3 x sqrt3 become a 3 ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | xt is 3 plus 1 is 4 . xz is 3 , and xs is 6 . so you have 2 over 4 , which is 1/2 , which is the same thing as 3/6 . | can someone explain how the three fractions are equal ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . | how do you solve the proportion involving similar figures using variables ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | and even the ratios look kind of tempting , because 4 times 2 is 8 . 5 times 2 is 10 . but it 's tricky here , because they are n't the same corresponding sides . | how do we write that 2 lines are proportional in a proof ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | we have one angle in triangle aec that is congruent to another angle in bdc , and then we have this angle that 's obviously congruent to itself that 's in both triangles . so both triangles have a pair of corresponding angles that are congruent , so they must be similar . so we can write , triangle ace is going to be s... | do n't shapes have to have to have the same angles to be similar ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | this is two . xt is 3 plus 1 is 4 . xz is 3 , and xs is 6 . | 9 how is xy/xt=x3/x5 i kind of did n't know what happen ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | blue and magenta side -- that is h. and then we go along the blue side to f , go along the blue side to i , and then you went along the orange side to g , and then you go along the orange side to j . so triangle efj -- efg is similar to triangle hij by side-side-side similarity . they 're not congruent sides . | how is efg similiar to hig ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | this is actually saying , this is 1 over 3 root 3 , which is the same thing as square root of 3 over 9 , which is this right over here , which is the same thing as 1 over 3 root 3 . so actually , these are similar triangles . so we can actually say it , and i 'll make sure i get the order right . | do these similarity postulates only apply to triangles or to other shapes as well ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . | what is the basic proportionality theorem ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | it 's inside of triangle aec . they both share this angle right over there , so that gives us one angle . we need two to get to angle-angle , which gives us similarity . and we know that these two lines are parallel . | 9 , how would knowing , that an angle in the smaller triangle is a right angle , help us make statements about similarity ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | this is actually saying , this is 1 over 3 root 3 , which is the same thing as square root of 3 over 9 , which is this right over here , which is the same thing as 1 over 3 root 3 . so actually , these are similar triangles . so we can actually say it , and i 'll make sure i get the order right . | with the second problem at around are n't the different sides of the two triangles going up by different factors , meaning that they 're not actually similar ... ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | at first they do n't look equal , but we can actually rationalize this denominator right over here . we can show that 1 over 3 square roots of 3 , if you multiply it by square root of 3 over square root of 3 , this actually gives you in the numerator square root of 3 over square root of 3 times square root of 3 is 3 , ... | how is 18 root 3 larger than 27 ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | and then this triangle over here also shares another side , but that also does n't do anything . so we really ca n't make any statement here about any kind of similarity . so there 's no similarity going on here . there are some shared angles . | what is the difference between similarity and congruence ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | blue and magenta side -- that is h. and then we go along the blue side to f , go along the blue side to i , and then you went along the orange side to g , and then you go along the orange side to j . so triangle efj -- efg is similar to triangle hij by side-side-side similarity . they 're not congruent sides . | can you explain the side-splitter and triangle-angle-bisector theorems ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | so the short side here is 3 . the shortest side here is 9 square roots of 3 . so we want to see whether the ratio of 3 to 9 square roots of 3 is equal to the next longest side over here , is 3 square roots of 3 over the next longest side over here , which is 27 . | would that be considered an `` improper '' fraction because there are square roots in the denominators ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | this is two . xt is 3 plus 1 is 4 . xz is 3 , and xs is 6 . | so would ace ~ dcb be incorrect ? |
what i want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar , using some of the postulates that we 've set up . so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one ... | so over here , i have triangle bdc . it 's inside of triangle aec . they both share this angle right over there , so that gives us one angle . we need two to get to angle-angle , which gives us similarity . and we know that these two lines are parallel . | doesnt the square for the 3rd triangle tell you its a right angle ? |
: we have 4 graphs here and then 4 function definitions . what i want you to do is pause this video and think about which of these graphs map up to which of these function definitions . i 'm assuming you 've given a go at it . let 's go through each of these and think about what their graphs would look like . one thin... | this one is clearly that one . then finally , just deductive reasoning , you could say that this function is represented by that graph , but let 's reason through a little bit . this is where order of operations really matter . | what is the deductive reasoning in 4th graph ? |
: we have 4 graphs here and then 4 function definitions . what i want you to do is pause this video and think about which of these graphs map up to which of these function definitions . i 'm assuming you 've given a go at it . let 's go through each of these and think about what their graphs would look like . one thin... | well , 1/3 to the 1,000 power , that 's going to be a very , very , very small number . we 're multiplying 1/3 times 1/3 , 1/3 . you think of 1,000 1/3s and multiplying them together , i 'm sure you 're going to get a number very , very close to 0 . | what would the graph be if the equation were the other way : y = ( -3 ) ^x ? |
: we have 4 graphs here and then 4 function definitions . what i want you to do is pause this video and think about which of these graphs map up to which of these function definitions . i 'm assuming you 've given a go at it . let 's go through each of these and think about what their graphs would look like . one thin... | as x becomes more and more and more negative , this expression is going to get closer and closer and closer to 0 . this one is clearly that one . then finally , just deductive reasoning , you could say that this function is represented by that graph , but let 's reason through a little bit . | 3 why does y= ( -3 ) ^x graph go through the y axais at one ? |
: we have 4 graphs here and then 4 function definitions . what i want you to do is pause this video and think about which of these graphs map up to which of these function definitions . i 'm assuming you 've given a go at it . let 's go through each of these and think about what their graphs would look like . one thin... | well , 1/3 to the 1,000 power , that 's going to be a very , very , very small number . we 're multiplying 1/3 times 1/3 , 1/3 . you think of 1,000 1/3s and multiplying them together , i 'm sure you 're going to get a number very , very close to 0 . | what if like the last equation `` y=-3^x '' is then `` y=-3^-x '' ? |
: we have 4 graphs here and then 4 function definitions . what i want you to do is pause this video and think about which of these graphs map up to which of these function definitions . i 'm assuming you 've given a go at it . let 's go through each of these and think about what their graphs would look like . one thin... | as x increases , y increases . this is your classic exponential graph shape . as x approaches more and more negative values , as x approaches more and more negative values , raising 2 to a very large negative value , so imagine when x is ... imagine y of negative 10 . | how would the graph change ? |
i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonomet... | and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . | could somebody walk me through a detailed explanation of this problem ; what is the principal value of sin^-1 ( -1/2 ) ? |
i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonomet... | so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . | what is the difference between range and domain ? |
i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonomet... | you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . | how does sal know the triangle is a 30-60-90 ? |
i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonomet... | i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonomet... | is the relationship between arccos and cos the same as the relationship between logarithms and exponents ? |
i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonomet... | either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . | how did sal know what are the domain and range of arccos x ? |
i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonomet... | and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? | why does sal take theta as an angle outside the triangle and not as 60 itself ? |
i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonomet... | we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . | if the range of arccos is restricted between 0 and pi , how can we get the arccos of 3pi ? |
i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonomet... | we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . | why is the cosine restricted to the upper quadrants and the sine restricted to th right quadrants and tangent as well ? |
i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonomet... | i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonomet... | how does sam decide the restrictions ? |
i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonomet... | and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circl... | how is the x-value of minus 1/2 determined when sal makes a point on that specific part of the circle ? |
i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonomet... | you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . | how does he know that the triangle in `` '' is a 30 60 90 triangle ? |
i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonomet... | so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . | is the arccos function the same as secant function ? |
i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonomet... | what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . | why is the angle supplementary to 180 and not 360 ? |
i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonomet... | that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . | how does sal know that theta is the larger angle on the right and not the smaller angle on the left ? |
i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonomet... | you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . | how does sal know it is a 30 , 60 , 90 triangle ? |
i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonomet... | and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . | 5 , what if there is bigger angle like 100 pi instead of 3pi ? |
i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonomet... | the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about i... | so algebraically speaking , sin ( arcsin x ) = sin ( 1/sin x ) and the sins ' cancel out leaving only x ? |
i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonomet... | let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . | is the value of sin and cos is always between -1 and 1 because the ratio of opp/hyp or adj/hyp is always equal or less than 1 ? |
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