Datasets:
jacobcd52
/
hs_math_cleaned

Dataset card Data Studio Files
xet
 Community
1
text
stringlengths
235
3.08k
�csccoscot2sinsectancos2coscsccotsin1sinsectancos1coscsccotsin2cossec1sin2sincsc1cos2seccos1tan2cscsin1cot2tancotsec12cottancsc1224cos4sin4229cossin8�
�32tantansectan252sin1cossin41022sec1tansec23costantansincos4224secsectantancos11seccos11secsin11cscsin11csc1cottan11cottan11tancossin1tancossin�
��tancotseccsccscsincotcoscossectansincostancotcsc 94 The Trigonometric Functions 31. 33. 35. 37. 39. 41. 43. 45. 47. 32. 34. 36. 38. 40. 42. 44. 46. In Exercises 48 – 51, verify the identity. You may need to review the properties of absolute value and logarithms ...
112cotcsccotcsccotcossinsincos1tan1cot1sectansectan1sectansectan1csccotcsccot1csccotcsccot21secsectan1sin21secsectan1sin21csccsccot1cos21csccsccot1cos�
��cos1sin1sincossincsccot1cos21sinsectan1sinlnseclncoslncsclnsinlnsectanlnsectanlncsccotlncsccot 2.5 Beyond the Unit Circle 95 2.5 Beyond the Unit Circle Learning Objectives In this section you will:  Determine the valu...
let be the point on the terminal side of θ which lies on the Unit Circle. Now consider dropping perpendiculars from P and Q to create two triangles, ∆OPA and ∆OQB. These triangles are similar23. Thus, it follows that, from which. We similarly find. Since, by definition, and, we get the coordinates of Q to be and. By r...
�sin22rxycossin,Qxy222xyrcosxrsinyr2222cos= and sin.xxyyrrxyxy221rxycossin4,2Qcossin4x2y22422025r 2.5 Beyond the Unit Circle 97 2. Assuming the Earth is a sphere, a cross-section through the poles produces a circle of radius 3960 miles. Viewing the Equator a...
cos and sin42cos sin2525255cos sin.55xyrr,Qxy3960cos40.7608x3960cos40.76082999,0r0ttt,Qxycosxrsinyrt costxr sintyr cos,sinttrr 98 The Trigonometric Functions Equations for Circular Motion: Suppose an object is traveling on a circular path of radius r cente...
�2999r12 hours costxr sintyr2999cos and 2999sin.1212xtyt 2.5 Beyond the Unit Circle 99 Theorem 2.7. Suppose is the point on the terminal side of an angle θ, plotted in standard position, which lies on the circle. Then the circle has radius r and    , provided, provided.., provided, provided.. Ex...
5r 100 The Trigonometric Functions 2. We look for a point which lies on the terminal side of θ 25, when θ is plotted in standard position. We are given that θ is a Quadrant IV angle, so we know and. Also,. Since, we may choose26 and, from which The five remaining circular function values follow. 25 Again, θ may be an...
�� 2.5 Beyond the Unit Circle 101 We close this section by noting that we have not yet discussed the domains and ranges of the circular functions. In Chapter 3, we will graph the circular functions. This will provide a visual platform for the introduction of the domain and range of each circular function. 102 ...
minutes and 7 seconds. 1,5A3,1B6,2C10,12D7,24P3,4Q5,9R2,11T0t 2.5 Beyond the Unit Circle 103 In Exercises 17 – 30, use the given information to find the exact values of the remaining circular functions of θ. 17. 19. 21. 23. 25. 27. 29. with θ in Quadrant II. with θ in Quadrant I. 18. 20. with ...
�csc52tan1032sec253220t,0r00t,0r 0()costtxr 0()sinttyr 104 CHAPTER 3 GRAPHS OF THE TRIGONOMETRIC FUNCTIONS Chapter Outline 3.1 Graphs of the Cosine and Sine Functions 3.2 Properties of the Graphs of Sinusoids 3.3 Graphs of the Tangent and Cotangent Functions 3.4 Graphs of t...
s of the Cosine and Sine Functions Learning Objectives In this section you will:  Graph the cosine and sine functions and their transformations. Identify the period.  Learn the properties of the cosine and sine functions, including domain and range.  Determine whether a function is even or odd. We return to our disc...
scale graph of showing several cycles with the fundamental cycle plotted thicker than the others. cosyx,cosxxcosyx,cosxx0,1542252,424222,42323,022,02742272,42342232,422,1,1cosyx,cosxxcosyx0,2cosyxcosyxcosyx 3.1 Graphs of th...
��,sinxxsinyx,sinxx0,0542252,424222,42323,122,12742272,42342232,422,0,0sinyx 108 Graphs of the Trigonometric Functions As with the graph of, we can provide an accurately scaled graph of with the fundamental cycle highlighted. An accurately scaled ...
find the smallest positive real number p so that for all real numbers x or, said differently, the smallest positive real number p such that for all real numbers x. We know that for all real numbers x but the question remains if any smaller real number will do the trick. Suppose and for all real numbers x. Then, in c...
its graph is symmetric about the origin. Observe the symmetry of the ‘accurately scaled’ graphs of the cosine and sine functions from earlier in this section. The graph of is symmetric about the y-axis. As will be proved algebraically in Section 4.1, the cosine function is an even function. The graph of is symmetric a...
��singxxtcosfttsingtt, 110 Graphs of the Trigonometric Functions Looking at the graphs of the cosine and sine functions, we see that the range includes all real numbers between –1 and 1, inclusive. Revisiting the Unit Circle, and represent x- and y-coordinates, respectively, of points on the Unit C...
sinyx 3.1 Graphs of the Cosine and Sine Functions 111 Theorem 3.2. Transformations of Periodic Functions: Suppose f is a periodic function. If and, then to graph 1. Divide the period of the function by ω to determine the period of the transformed function g. For sine and cosine, the period is 2π, so is the period of g...
0cosyxsinyx023222sin31fxxsinyx0,0,12,03,122,00y 112 Graphs of the Trigonometric Functions The steps in Theorem 3.2 help us determine transformations for graphing. Step 1: We first determine the period. The period for is 2π. For coefficient of x is 3, so the graph of th...
��2sin31fxxsinyx1y2sin31fxxsinyx2sin31fxx2sin31fxx0x2361y2sin31fxxxy0 sinyx 3.1 Graphs of the Cosine and Sine Functions 113 The graph of will maintain the shape of the sine function. With an amplitude of 2, the local maximums and minimums will occur 2 units above a...
xyxy 0,1,13 2,13,36,12 2sin31fxx 114 Graphs of the Trigonometric Functions Step 1: The period of is 2π. For, the coefficient of x is 2 so the period is Step 2: The vertical shift of is 0. Thus, the baseline will remain at. Step 3: The amplitude is. Since, th...
0y3cos23cos22fxxx22cosyx22x40y0y3cos2fxxcosyxxy01-1 cosyx 3.1 Graphs of the Cosine and Sine Functions 115 Above, we have graphed one complete period of the function. This graph could easily be extended in either direction. In the next example we sketch the graph of ...
xyxy 3,04 ,3 5,04 3,323cos2fxx,32 116 Graphs of the Trigonometric Functions 1. The period is, so the distance between quarter marks will be. 2. There is a vertical shift of –2, resulting in a baseline of. 3. The amplitude is 4. 4. There is no horizontal shift, so the...
��xy 3.1 Graphs of the Cosine and Sine Functions 117 3.1 Exercises 1. Why are the cosine and sine functions called periodic functions? 2. How does the graph of compare with the graph of? Explain how you could horizontally translate the graph of to obtain the graph of. 3. For the function, what constants affect the r...
2sinfxx2cos3fxx3sinfxx4sinfxx2cosfxxcos2fxx12sin2fxx4cosfxx63cos5fxx3sin845yx2sin3214yx5sin5202yx 118 Graphs of the Trigonometric Functions 28. Show that a constant function f is periodic by showing that for all real numbers x. Then show that f has no pe...
be characterized by four properties: period, amplitude, phase shift, and vertical shift. 1. We have already discussed period, that is, how long it takes for the sinusoid to complete one cycle. The standard period of both and is 2π, but horizontal scalings will change the period of the resulting sinusoid. 2. The amplit...
Sinusoids The proof of Theorem 3.3 is left to the reader. The parameter ω, which is stipulated to be positive, is called the (angular) frequency of the sinusoid and is the number of cycles the sinusoid completes over a 2π interval. We can always ensure using the even property of the cosine function, or the odd propert...
��1B 3.2 Properties of the Graphs of Sinusoids 121  The period of f is.  The amplitude is.  The phase shift is, indicating a shift to the right 1 unit.  The vertical shift is, indicating a shift up 1 unit, and a baseline of. The graph shows one cycle of. Using the period, amplitude, phase shift and vertical shift,...
Since, the graph of must be reflected about the baseline. 13sin22213sin22213sin2 2213sin222gxxxxxfrom odd property of sine12A232B2211222223213sin222gxx12A0Aygxxy ygx sinyx 3.2 Properties of the Graphs of Sinusoid...
sine function whose graph matches the graph of. Solution. 1. We fit the data to a function of the form by determining A, ω, ϕ and B.  Since one cycle is graphed over the interval, its period is. According to Theorem 3.3,, so that. ygx2x32xyfxyfxyfxyfxcosCxAxB1,5516263...
 The trickier part is finding the phase shift. To that end, we imagine extending the graph of the given sinusoid as in the figure below so that we can identify a cycle beginning at. 35,221212y51222A11312B12cos332CxxsinSxAxB32A12B71,22xybaseline 7...
�� 71,22 32,2 3.2 Properties of the Graphs of Sinusoids 125 Taking the phase shift to be, we get, or Hence, our answer is. Note that each of the answers given in Example 3.2.3 is one choice out of many possible answers. For example, when fitting a sine function to the data, we could have chosen to start at...
� 126 Graphs of the Trigonometric Functions Solution. Sketching the height, we note that it will start 1 foot above the ground, then increase up to 7 feet above the ground, and continue to oscillate 3 feet above and below the center value of 4 feet. Although we could use a transformation of either the cosine or s...
wheel has a radius of 67.5 meters. The height will oscillate with amplitude 67.5 meters above and below the horizontal center of the wheel. Passengers board 2 meters above the ground level, so the center of the wheel must be located meters above ground level. The horizontal midline of the oscillation will be at 69.5 m...
mass is the ‘slug’. In the SI system, the unit of force is ‘Newtons’ (N) and the associated unit of mass is the ‘kilogram’ (kg).  We convert between mass and weight using the formula29. Here, w is the weight of the object, m is the mass and g is the acceleration due to gravity. In the English system, and in the SI sy...
the spring force, then Physics tells us that gravity and the spring force will battle each other forever and the object will oscillate indefinitely. In this case, we describe the motion as ‘free’ (meaning there is no external force causing the motion) and ‘undamped’ (meaning we ignore friction caused by surrounding me...
the object,. When does the object first pass through the equilibrium position? Is the object heading upwards or downwards at this instant? 2. If the object is attached to the spring and released 3 feet below the equilibrium position with an upward velocity of 8 feet per second, find the equation of motion of the objec...
��2msinxtAt822km from Theorem 3.4 since =8 lbs./ft. and =2 slugs km03x00v22200220323vAx from Theorem 3.403x3A0sin3sin3sin12Ax from Theorem 3.43sin22xtt 132 Graphs of the Trigonometric Functions Next, to find when the object passes through the equilibrium positi...
� after the usual analysis0.784txt0txt222xt0,0xt0xt4t203x08vtx 3sin22xtt 3.2 Properties of the Graphs of Sinusoids 133 We use and to determine ϕ. We will need to identify ϕ using the arcsine since is not a common angle. With the sine being positive, ϕ is in Quadra...
v0cos(5)(2)cos84cos5Av3arcsin535sin2arcsin5xtt5xt35sin2arcsin553sin2arcsin15tt 134 Graphs of the Trigonometric Functions Going through the usual machinations, we get for integers k. The smallest of these values is when, t...
. 7. 10. 2. 5. 8. 11. 3. 6. 9. 12. 13. Write an equation of the form for the sine function whose graph is shown below. 14. Write an equation of the form for the cosine function whose graph is shown below. 3sinyxsin3yx2cosyxcos2yxsin3yxsin2yx11cos323yxcos324yxsin24yx2cos413...
� 3.2 Properties of the Graphs of Sinusoids 137 18. Write an equation of the form for the sine function whose graph is shown below. 19. Write an equation of the form for the cosine function whose graph is shown below. 20. Write an equation of the form for the sine function whose graph is shown below. sinSxAxB...
��22cossin1xxcossin2xxcos3sinfxxxsinxfxxsinfxxxyx1sinfxx0xhththtlbs.ft. 3.2 Properties of the Graphs of Sinusoids 139 c. Find the equation of motion of the object if it is released from 6 inches above the equilibrium position with a downward velocity of 2 feet per second. Find ...
��sintancosxxx,tanxx01000,0422221,1420134222213,14100,054222215,14320174222217,1421002,0 142 To determine the behavior of the graph of when x is close to or, we will look at some values for on both sides of and on both sides of.  The following chart shows som...
��tanx22x1.5712tanxAs, tan.2xx22tanxAs, tan2xx1.5712tanx324.712tanx3234.7122tanx3As, tan.2xx 143 x 4.72 4.73 4.75 4.8 undefined –131 –57 –27 –11 Thus we have vertical asymptotes at and at. We also know something about the behavior of the graph as it approaches these vertic...
which is undefined. Thus, the domain of is all real numbers x, excluding for any integer k. The range of, as observed from the graph, includes all real numbers. Graph of the Cotangent Function It should be no surprise that the graph of the cotangent function behaves similarly to the graph of the tangent function. Plot...
interval are, and. We take as one fundamental cycle the interval. A more complete graph of is below, with the fundamental cycle highlighted. The graph of. We see, from the graph, the apparent domain of the cotangent function is all real numbers x except for. These are the x-values where and are the only numbers for wh...
the period of transformations will be.  The vertical scaling is, but tangent and cotangent do not have amplitude since they do not possess the wavelike characteristics of the sine and cosine functions.  The vertical shift is B, but a baseline is not defined for transformations of the tangent and cotangent functions....
��3tan2xfx3cot142gxx3tan2xfxtanyxfx1tan032fxxtanyx2,2tanyx2x40422x2x4,10,04,11tan032fxx1221222xy tanyx 148  The vertical scaling is, so the graph will not be stretched in the vertical direction. However, tells us ...
�xyxy 149 2. We graph the function using transformations of. The fundamental cycle of the cotangent function is on the domain. We use the quarter marks,,, and. Before proceeding, the function can be written as follows.  The period of is, resulting in quarter marks 0, 1, 2, 3 and 4, shown...
��xy 1,1 3,1xy cotyxxy 1,3 3,3 2,0 150 One cycle of. Since the number of classical applications involving sinusoids far outnumber those involving tangent and cotangent functions, we omit the more extensive coverage here that was given to sinusoidal functions. The ambitiou...
1tan213yxcot6yx111cot5yx13cot2132yxtanfxxcotfxx2tan432fxxtan2fxxtan2fxx4tanfxxtan4fxxtanfxx3cot2fxx 17. 152 18. Verify the identity by using technology to graph the right and left hand sides. 19. Graph the function with the help of technolog...
1f2.5f 153 3.4 Graphs of the Secant and Cosecant Functions In this section you will: Learning Objectives  Graph the secant and cosecant functions and their transformations. Identify the period and vertical asymptotes.  Learn the properties of the secant and cosecant functions, including domain and range; determine...
of and on both sides of.  The following chart shows some values for when x is less than, but close to. We note that and include approximate values of the secant for the indicated radian measures of x. x 1.5 1.55 1.56 1.57 14 48 93 1256 undefined We note that values of are positive, getting larger and larger, as x app...
of the first. The graph of over. The graph of, with fundamental cycle highlighted. In the above illustration, the dotted graph of is included for reference. It is helpful to graph the secant function by starting with a graph of the cosine function and sketching vertical asymptotes at each x-value for which. The points...
is included for reference. x undefined undefined undefined 1y1y1ycscyxsinyx0xx2xcscyxcscyxsinyxsinx1cscsinxx,cscxx004222,24211,12342223,240542225,2432113,12742227,2420 157 The fundamental cycle of. The graph of. Since and are...
��secyxsecyxcscyx 158 Theorem 3.5. Properties of the Secant and Cosecant Functions:  The function – has domain – has range is continuous and smooth on its domain is even – – – has period 2π  The function – has domain – has range is continuous and smooth on its domain is odd – – – has period 2π Graphs of Trans...
2yx12cos2yx2cos201yx221y 159  The amplitude is and, since, the graph will be reflected about the baseline.  There is no horizontal shift. One cycle of. Next, to graph, we observe the following.  Points where the graph of the cosine function crosses the baseline are vertical asymptotes of the secant f...
��csc53xgxsin53xysin5315sin3315sin3315sin33xyxxx from odd property of sine15sin33yx sinyx225353yxybaseline 161  The amplitude is and, since, the graph will be reflected about the baseline.  Since, the graph will be shifted to...
�csc53xgxxybaseline 12cos2yx 51,3 3,22 52,3 54,23 53,3 53yxybaseline 12cos2yx 162 We find the period to be. While real world applications of secants and cosecants are limited, at least in comparison to the large number of available sinusoidal appl...
csc2yxsec324yxcsc24yxsecfxxcscfxx2sec14fxx6csc3fxx2cscfxx1csc4fxx4sec3fxx7sec5fxx9csc10fxx2csc14fxxsec23fxx7csc54fxx 164 20. 21. 22. 165 23. Standing on the shore of a lake, a fisherman sights a boat far in the distance to...
ptotes on the graph of. d. Calculate and interpret. Round to the nearest hundredth. e. Calculate and interpret. Round to the nearest hundredth. f. What is the minimum distance between the fisherman and the boat? When does this occur? 24. A laser rangefinder is locked on a comet approaching Earth. The distance, in kilom...
identities, the half angle formulas will help us obtain exact values for trigonometric functions of some ‘non-common’ angles. The product to sum and sum to product formulas introduced in Section 4.5 will allow further manipulation of trigonometric expressions. Finally, Section 4.6 revisits the formulas of sinusoids fi...
the even/odd properties of the circular functions are so named because they identify cosine and secant as even functions, while the remaining four circular functions are odd. sinsincsccsccoscossecsectantancotcotcoscossinsincossin000200...
required. Simplifying Expressions The even/odd identities are readily demonstrated using any of the common angles noted in Section 2.2. Their true utility, however, lies not in computation, but in simplifying expressions involving the circular functions. Example 4.1.1. Use identities to fully simplify the expression:....
add the even/odd identities to the Pythagorean, reciprocal and quotient identities as tools in proving that identities are true. Example 4.1.2. Verify the identity:. Solution. We begin with the left, more complicated, side. 2222 sinsin 1s cosin1sin1sin1sin1s1sinincosxxxxxxxxxx...
os sincossincos1coseven/odd identitiesdifference of squares sin 170 4.1 Exercises 1. We know is an even function, and and are odd functions. What about, and? Are they even, odd or neither? Why? 2. Examine the graph of on the interval. How can we tell whether...
�,secfxxsincoscscxxxcsccoscotxxxcottansecttt323sincsccos2coscosttttttancotxxsincosseccsctancotxxxxxxsin32sin23cos5cos544tt22tan1tan1tt55csccscsec6sec6tt9779cotcot112cotcsc1cos1cosxxxx...
2tansincossecxxxxsecsintancotxxxx1sincoscos1sinxxxx 171 4.2 The Sum and Difference Identities In this section you will: Learning Objectives  Learn the sum and difference identities for cosine, sine and tangent.  Use the sum and difference identities to find values of trigonometric fun...
0000 172 Since the angles POQ and AOB are congruent, the distance between P and Q is equal to the distance between A and B.1 Using the distance formula to determine distances QP and BA, we have or, after squaring both sides: Expanding the left hand side, then using the Pythagorean identities, we get the following...
2200cossin12200cossin1220000coscossinsin2222000000002222000000000000cos2coscoscossin2sinsinsincossincossin2coscos2sinsin22coscos2sinin.s220000cossin1222200000000002200000000co...
1cossin2co.s22cos00000022coscos2sinsi2cn2os00000000000000000022cos22coscos2sinsin2cos2coscos2sinsincoscoscossinsin after swapping sides after subtracting 2 from each side after dividing thro...
of cosine, we get from which it follows that. To verify the sum identity for cosine, we use the difference identity along with the even/odd identities: We put these newfound identities to good use in the following example. Example 4.2.1. 1. Find the exact value of. 2. Verify the identity. Solution. 1. In order to use ...
�coscoscoscossinsincoscossinsincoscossinsincoscossinsin. difference identity for cosine even identity of cosine odd identity of sinecos15cossin2cos15154530 174 2. This is a straightforward application of Theorem 4.2. The identity verified in Example 4.2.1, namely, is t...
�coscoscossinsin2220cos1sinsincossin2sincos2sincos222cos 175 Theorem 4.3. Cofunction Identities: For all applicable angles θ,       With the cofunction identities in place, we are now in the position to derive the sum and difference ident...
��seccsc2cscsec2tancot2cottan2 ssincos2cos2coscossinsin22incos2from difference identity for cosine sinsinsinsincoscossinsinsincoscossin
19sin125sin13tan2sintantantan Solution. 176 1. As in Example 4.2.1, we need to write the angle as a sum or difference of common angles. The denominator of 12 suggests a combination of angles with denominators 3 and 4. One such combination is from which we have 2. Using the difference identit...
sin13coscossincos5sin13222225cos113144cos16912cos1312cos1cossin13  from since is a Quadrant II angle tan2cos 177 We need to determine, knowing that and. We now have all the pieces needed to find. 3. We can start expanding. 22222 112secsec5sec515co...
 from the quotient identity sinsinsincoscossin511221313552913529565tan 178 The last step is to replace with and with. Naturally, this result is limited to those cases where all of the tangents are defined. The Sum and Difference Identities for the ...
sinsincosc1sincoscossincoscos1coscossinsincoscossincoscossincoscoscooscoscoscossinsincoscoscoscossinsincoscossins1coss for cosine and sine goal: & sincossincostansincostan
tantantan1tantantantantan 179 Theorem 4.5. Sum and Difference Identities: For all applicable angles α and β,    In the statement of Theorem 4.5, we have combined the cases for the sum ‘+’ and difference ‘–‘ of angles into one formula. The convention is that if you want the formula for ...
tantantan1tantantantantan1tantanftpfttanJxxtantantan1tantantan01tan0tanJxxxxxxxJxtanxtantanxpx0xtantan0tan0tantan0xpxpp from We leave it to the reader to prove algebraically that the period of the cotang...
the quotient, reciprocal or even/odd identities as well. 1. 4. 7. 10. 13. 2. 5. 8. 11. 14. 3. 6. 9. 12. 15. 16. If α is a Quadrant IV angle with, and, where, find (a) (d) (b) (e) (c) (f) 17. If, where, and β is a Quadrant II angle with, find (a) (d) (b) (e) (c) (f) 18. If, where, and, where, find (a) (b) (c) 19. If, w...
�sintancossintan3sin50212cos13322sincostan5sec3224tan732cscseccot 182 21. 23. 25. 27. In Exercises 20 – 32, verify the identity. 20. 22. 24. 26. 28. 29. 30. 31. 32. 33. Verify the cofunction identities for tangent, secant, cosec...
tansincossincostansincossincossinsinsincos1cossinththhtthhhcoscoscos1sincossinththhtthhh2tantantansec1tantanthththhth 183 4.3 Double Angle Identities Learning Objectives In this section you will:  Learn the doubl...
exchange squares of cosine and sine via a Pythagorean identity. We verify that : sinsincoscossincoscoscossinsintantantan.1tantansin22sincos2222cossincos22cos112sin22tantan21tancos22cos212sin 184 Tr...
θ lies when it is plotted in standard position. 2. If for, find an expression for in terms of x. Solution. 1. Using, from Theorem 2.6 in Section 2.5, with and, we find. Hence, and. It follows that and 222222222cos2coscoscossinsincosscos2cossincoin1sinsin12sin.ssin1...
 from double angle identitysin22sincos4325524.25 from double angle identity 185 Since both the cosine and sine of 2θ are negative, the terminal side of 2θ, when plotted in standard position, lies in Quadrant III. 2. If your first reaction to is that x should be the cosine of θ, th...
�cos222222cossin1cos1cos1cos1cos022xxx Pythagorean identity from the problem statement since cosoppositehypotenusesin,1xx222222adjacent lengthadjacent lengthadjacent length111xxx1x  186 This results in the following triangle. From the triangle we see that. Then, ba...
to start with the left side. Beginning with the right side would have required some thinking ahead, possibly working backwards. Try it! When using identities to simplify a trigonometric expression, solve a trigonometric equation or verify a trigonometric identity, there are usually several paths to a desired result. T...
�double angle identitygoal: numerator of 2reciprocal identity for cotangent 188 One last note before we move on to Section 4.4. While double angle identities could be established for secant, cosecant and cotangent, the identities already established in this section may be used in their place. Recall that secant, c...
��32csc423cos5024sin53212cos133225sin132sec5322tan2222tansin21tan221tancos21tan22sincostan22cos12cossin1sin22cossin1sin211tan21tan1tan�
��cottancsc22cossinsec2cossincossin 190 21. 22. 23. Suppose θ is a Quadrant I angle with. Verify the following formulas. (a) (b) (c) 24. Discuss with your classmates how each of the formulas, if any, in Exercise 23 change if we assume θ is a Quadrant II, III or IV angle. 25. Suppose ...
tanx21cos1x2sin1xx22sin21xx221cos21xxsin2x22cos2tan7x22sin2 191 4.4 Power Reduction and Half Angle Formulas Learning Objectives In this section you will:  Learn and apply the power reduction formulas for sine and cosine.  Learn and apply the half angle formulas for sin...
�22sincos 192 Another application of the power reduction formulas is the half angle formulas. Half Angle Formulas To start, we apply the power reduction formula to : We can obtain a formula for by extracting square roots. In a similar fashion, we may obtain a half angle formula for sine. By using a quotient id...
a half angle formula to find the exact value of. 2. Suppose with. Find. 3. Use the identity, verified in Example 4.3.2, to derive the identity. Solution. 1. To use the half angle formula, we note that. 1cossin221coscos221costan21cos2cos1503cos5sin222tansin21tan...
��022sin021cossin223152315525810255 half angle formula for sine22tansin21tan 195 and we will manipulate it into the identity. If we are to use to derive an identity for, it seems reasonable to proceed by replacing each occurrence of θ with. sintan21cos22tansi...
��1cos22sin2tan22sintan1cos2sintan21cos from reciprocal identity for secant from power reduction formula for cosine 196 4.4 Exercises In Exercises 1 – 15, use half angle formulas to find the exact value. You may have need of the quotient, reciprocal or even/odd identities as w...