Chapter 8 More About the Trigonometric Functions

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1 Relationships Among Trigonometric Functions Section 8. 8 Chapter 8 More About the Trigonometric Functions Section 8. Relationships Among Trigonometric Functions. The amplitude of the graph of cos is while the amplitude of cos is. The frequenc of cos is while the frequenc of cos is. Thus cos cos.. The amplitude of the graph of cos is. The maimum and minimum values of cos are and, respectivel, so the amplitude of the graph of cos is. Furthermore, since the point (0, belongs to both graphs, it is reasonable to assume that cos cos.. Evaluating sin cos at, we get , not. So, sin cos is not an identit. 4. Evaluating cos at, we get Evaluating cos cos cos sin. sin at, we get Thus 5. sin sin cos cos sin sin 6. ( cos θ( cos θ cos θ sin θ 7. Taking,, we have sin ( sin 0, whereas sin. Since the two sides of the equation fail to agree at, the equation sin sin is not an identit cos θ sin θ sinθ sinθ ( sin θ( sin θ sinθ sinθ cosα cosα cosα sinα sinα cosα cos α sin α( cos α sin α sin α( cos α sinα cosα 0. Taking β 0, we find that cos ( 0, while cos 0. Since the two sides of the equation fail to agree at β 0, the equation cos β cos β is not an identit.. sin cos6 sin cos( sin cos sin cos. Taking 0, we find that cos evaluates to, while ( sin evaluates to. Since the two sides of the equation fail to agree at 0, the equation is not an identit.. Taking 0, we find that sin(cos0 sin and cos(sin 0 cos0. Since the two sides of the equation fail to agree at 0, the equation sin(cos cos(sin is not an identit. 4. Since substituting 0 into the equation results in the false statement sin ( 0, the equation sin(cos sin cos is not an identit.

2 84 Chapter 8 More About the Trigonometric Functions cos cos( cos cos sin sin cos (cos sin sin (sin cos cos cos sin cos sin cos cos sin cos cos ( cos 4cos cos cos 4 cos( cos ( cos 4 (4 cos 4 cos 4 8cos 8cos 7. cos5 cos( 4 cos cos 4 sin sin 4 4 cos (8cos 8cos sin (sincos 4 cos (8cos 8cos sin [(sin cos (cos sin ] 5 8cos 8cos cos sin (4sincos 4sin cos 5 8cos 8cos cos sin [4sincos ( sin 4sincos ( cos ] 5 8cos 8cos cos 4sin cos ( sin cos 5 8cos 8cos cos 4sin cos (cos 5 8cos 8cos cos 8sin cos 4sin cos 5 8 cos 8 cos cos 8( cos cos 4( cos cos 5 5 8cos 8cos cos 8cos 8cos 4cos 4cos 5 6cos 0cos 5cos 8. One wa to look for patterns is to group the formulas for cos n, where n is even and n is odd. cos cos 4 cos4 8cos 8cos 6 4 cos6 cos 48cos 8cos cos8 8cos 56cos 60cos cos cos 4cos cos 5 cos 5 6 cos 0 cos 5cos 7 5 cos7 64cos cos 56cos 7cos cos9 56cos 576cos 4cos 0cos 9cos n When n is even or odd, the first term of cos n is cos. The degrees of the subsequent terms are alwas less than the degree of the preceding term. When n is even, the last term is, if n is an odd multiple of or the last term is, if n is an even multiple of. n When n is odd, the last term is alwas ( cos or n ( cos. n

3 Relationships Among Trigonometric Functions Section We substitute into Equation (7: 0... a. 4 sin (sin ( [ cos ] ( cos cos cos ( cos 4 cos cos cos cos ( cos 4 sin cos ( cos ( cos ( [cos( ] ( cos4 cos 4 cos( cos cos sin sin cos( cos cos sin sin cos( cos sin. a. sin( / cos sin sin cos cos sin sin 0 cos cos b. ( sin( sin ( sin sin cos cos sin sin ( cos 0 sin

4 86 Chapter 8 More About the Trigonometric Functions b. cos( / sin ( cos cos cos sin sin cos 0 sin sin 4. Substituting into sin ( cos, we get sin ( cos. Similarl, cos ( cos can be rewritten as cos ( cos ft 50 ft v0 sin θ Using the formula R, we have g (80 sin θ 50 sin θ θ arcsin(0.5 θ 7.09 or θ θ.55 or θ The nearl vertical trajector should be rejected as unsafe ft 50 s α s 4 ft If we assume that the angle of flight doesn t change much in the last few feet, we are looking for a distance s and an angle α that meet two conditions: The range corresponding to α is 50 s, and in the final s feet, the arrow drops foot, which implies that tan α. If we plot s (80 sin 50 and, tan the intersection should indicate possible angles α. We epect one angle to be close to 0 and one angle to

5 Relationships Among Trigonometric Functions Section be close to 90. Plotting in degrees, the graph on the left has horizontal window from 0 to 0 and vertical window from 0 to 0, indicating an intersection point at about α., s 7.5 feet. The graph on the right has horizontal window from 80 to 90 and vertical window from 0 to, indicating an intersection point at about α 86.5, s 0.06 feet. As we would epect, for the low angle Tell has to aim just a bit lower than the one in Problem 5, and for the high angle he has to aim a tin amount higher than the one in Problem a. To shift the cosine graph 90 to the right to match the corresponding sine graph, we subtract 90 from the argument, giving 5cos t( t b. Using C cos( t D C(cos tcos D sin tsin D, we let C 5, sin D, and cos D. Then D is an angle in the second quadrant with tangent equal to. Since ( 4 arctan 5., D , and our formula is 5cos( t c. The results in parts (a and (b agree. 8. Using our range formula 0 v sin θ R, we have: g (0 sin θ 0 sin θ θ arcsin(0.556 θ 0.8 or θ θ 0.4 or θ a. The vertical shift is 4, the amplitude is, and the frequenc is 4, so a reasonable candidate is cos 4. b. The two graphs appear identical. c. Squaring both sides of sin ( cos and cos ( cos and adding, we get: 4 4 sin cos ( cos ( Using the second half-angle identit with in place of gives cos ( cos 4. Substituting for cos into the right side of Equation ( above, we get: sin cos ( cos 4 cos 4

6 88 Chapter 8 More About the Trigonometric Functions 0. The graph of 6 6 sin cos looks like the following: The vertical shift is 5 8, the amplitude is, and the frequenc is 4, so a reasonable candidate is cos 4. Graphing show that this is correct. To prove this identit, we ll multipl both sides of the identit sin cos cos 4 b sin cos and subtract the terms we don t want on the left, giving us: cos 4 sin cos 4 4 sin cos cos 4 sin cos sin cos cos 4 sin cos (sin cos Using the double-angle formula sin sin cos, we can rewrite the equation above as sin cos cos 4 sin and using our first half-angle identit with in place of, we can replace the last term, getting: sin cos cos 4 ( cos 4 cos 4. The graph in Problem (a of Section 7. on depicts a function that is even. The graph in Problem (b of Section 7. depicts a function that is odd. The graph in Problem (c of Section 7. depicts a function that is even. The graph in Problem (d of Section 7. depicts a function that is neither even nor odd. The graph in Problem (e of Section 7. depicts a function that is neither even nor odd. The graph in Problem (f of Section 7. depicts an even function. The graph in Problem (g of Section 7. depicts a function that is neither even nor odd. The graph in Problem (h of Section 7. depicts a constant function. Constant functions are alwas even. The graph in Problem (i of Section 7. depicts a function that is neither even nor odd. The graph in Problem (j of Section 7. depicts a function that is neither even nor odd.. a. We know that for values of a for which < a <, the formula n a a a a. a This formula provides a convenient and valid method for evaluating the sum. Thus as long as we eclude values of for which sin ±, that is as long as ±/, ±/, ±5/,, we can substitute sin for a and write sin sin sin sin n /( sin. b. Provided that sin, we have n n sin sin sin sin sin. sin

7 Relationships Among Trigonometric Functions Section Substitute /6 into the formula of Problem (b: for n for n for n 4 ( sin 6 sin sin sin ( sin 6 sin sin sin sin ( sin 6 sin sin sin sin sin.98 6 Using the infinite sum formula, we know that sin sin. Looking at the 6 6 sin 6 different partial sums, we see that it takes a partial sum of terms to achieve three-decimal place accurac. For n, the partial sum equals Since sin 0.5 and sin , the powers 6 of sin get smaller much faster than the powers of sin. So we would epect that ou will need more 6 terms to achieve the same three-place accurac. The infinite sum when is Calculating the partial sum with terms, sin sin sin, is onl So ou need more than terms to achieve the three-decimal place accurac of a. b. tanθ sinθ tanθ cosθ sinθ cosθ sinθ cosθ cosθ sinθ sin θ cos θ sinθ cosθ sinθ cosθ

8 90 Chapter 8 More About the Trigonometric Functions 5. tan sin tan sin cos cos cos sin sin cos cos sin sin cos cos sin cos cos sin sin cos tan 6. tan tan tan tan tan tan tan tan tan 7. a. Using the double-angle formulas for sine and cosine, we have: tan sin cos sincos cos sin sin cos cos cos sin cos sin cos sin ( cos tan tan b. Using the addition formula for sine and cosine, we have:

9 Relationships Among Trigonometric Functions Section 8. 9 sin( tan( cos( sin cos cos sin cos cos sin sin sin cos cos sin cos cos cos cos sin sin cos cos sin sin cos cos sin sin cos cos tan tan tan tan 8. cos θ sin θ cos θ tan θ sin θ sin θ sin θ sin θ cos θ 9. Taking θ the equation gives tan tan, or Since this is a false statement, the equation is not an identit. sin 40. tan sin sin cos sin sin cos cos sin ( cos cos sin sin tan sin (tan sin cos 4. tan sin cos sincos cos sin sincos cos sin sin cos sin ( cos tan tan cos cos

10 9 Chapter 8 More About the Trigonometric Functions 4. First, we consider cos θ. sin θ cosθ ( sin θ sin θ sinθ tan θ. sin θ sinθ cosθ sinθ cosθ cosθ We now know that cos θ tanθ is an identit. So we can substitute θ α/, to get the identit: sin θ tan α cosα sin α 4. Letting 0 in tan tan gives tan 0 tan 0, or 0 0. Since this 0 is a false statement, the equation is not an identit. 44. cos ( sin sin sin cos sin cos sin sin tan cos cos 45. Substituting into the equation tan (sin tan sin, we get , we find that the equation is not an identit. 46. Substituting into the equation cos (tan tan (cos, we get , thereb establishing that the equation is not an identit. 47. The epression cos(tan means the cosine of the value tan. That is, the epression involves the composition of two functions, not the product. So the epression cos(tan does not mean cos tan. Therefore, the epression cos sin ( cos does not mean cos sin, and so we cannot cancel cos. cos Section 8. Approimating Sine and Cosine with Polnomials. With T (!, we have: T (0 cos 0 T ( cos T ( cos T ( cos T ( cos T ( cos T ( cos T ( cos With Error( cos T (, we have: Error(0 0 Error( Error( Error( Error( Error( Error( The fourth differences are fairl constant, which suggests that the errors ma be growing as a fourth-degree polnomial.

11 Properties of Comple Numbers Section T ( sin T ( sin a, b. / /4 /6 0 /6 /4 / T ( T / sin sin ( / c. 5 The function has a kind of smooth stair-step graph; it is not periodic. 5 d. The cubic function gives a reasonabl good fit for between and. e. The graph looks essentiall cubic ecept that it is ver flat around the origin sin The cubic regression equation is : sin ( ( ; it comes ver close to the cubic approimation sin / If T5 ( is a good approimation to sin, then ( ( 5 T ( T should be a good approimation to sin(. 5 5

12 94 Chapter 8 More About the Trigonometric Functions 5 7. If T5 ( is a good approimation to sin, then T5 ( 6 0 should be a good approimation to g( sin. In fact, as the following graph shows, there is a good match for between 0.5 and sin( 5 8. a. If T5 ( is a good approimation to sin, then ( 5 T should be a good approimation to h( sin. b. To approimate sin, we take twice the product of these two approimations: c. 4 ( sin This result will get closer and closer to the value when 0 if ou use higher degree approimations. 0. Substituting θ / into cos θ cos θ sin θ, we get cos cos ( / sin ( /. Substituting the approimation cos( / ( / and sin( / ( / ( /, we get the equation: 6 ( ( cos Substituting θ / into cos θ cos θ, we get cos cos ( /. Substituting the approimation cos( / ( /, we get the equation: 4 4 ( cos 4 64

13 Properties of Comple Numbers Section a (sin / T 4 ( T( T5( b.,, and ; each of these approimations gives at 0. sin( sin. Approimating sin b its linear polnomial in cubic Talor approimation of sine gives: gives the value of. Substituting the sin( sin ( ( /6 ( /6 ( ( ( ( /6 /6 / ( / ( /6 ( 6 Setting 0, we get /. This suggests that for small values of, [ sin( sin( ]/ is well approimated b /. (Curiousl, / is the quadratic approimation to cos. 4. Talor Approimation / / /! / /! 4 /4! / /! 4 /4! 5 /5! / /! 4 /4! 5 /5! 6 /6! / /! 4 /4! 5 /5! 6 /6! 7 /7! Section 8. Properties of Comple Numbers z 4 ( 5, θ arctan(/ z 5, θ arctan(/5.760 z ( 5, θ arctan(5/ z ( 5 0 5, θ arctan(4 /.40 z 64 ( , θ arctan(9/ z 8 ( , θ arctan(/ z ( , θ arctan(7 / 5.905

14 96 Chapter 8 More About the Trigonometric Functions z ( , arctan ( 8 / θ z ( 8 ( , θ arctan ( / (cos( i sin( (cos(.760 i sin(.760. (cos( i sin( (cos(.40 i sin( (cos( 0.59 i sin( (cos( i sin( (cos(.905 i sin( (cos( i sin( (cos(.548 i sin( z (4 i (4 i 6 i i 9i 6 4i 9 7 4i. 0. z (5 i (5 i 5 60i 60i 44i 5 0i i.. z ( 5i ( 5i 44 60i 60i 5i 44 0i 5 9 0i.. z ( 5 0i ( 5 0i 5 600i 400i 5 600i i For Problems through, all equalities are approimate since we are using approimate values for the angles associated with each z.. z ( 5(cos( isin( (cos((( i sin((( (cos(.8700 i sin( i 4. z ( (cos(.760 isin( (cos(((.760 i sin((( (cos(.50 i sin( i 5. z ( (cos( isin( (cos((( i sin((( (cos( i sin( i 6. z ( 5(cos(.40 isin(.40 65(cos(((.40 isin(((.40 65(cos( i sin( i

15 Properties of Comple Numbers Section ( z 59(cos( 0.59 isin( (cos((( 0.59 i sin((( (cos(.0478 i sin( i 8. ( z 7(cos( isin( (cos((( i sin((( (cos( isin( i 9. ( z 74(cos(.905 isin( (cos(((.905 i sin((( (cos(4.80 i sin( i 0. ( z 7(cos( isin( (cos((( i sin((( (cos(.594 i sin( i. ( z 67(cos(.548 isin( (cos(((.548 isin((( (cos( i sin( i We use the binomial identit ( in the following four problems.. z (4 i 64 ((6( i ((4(9i 7i 64 44i 08 7i 44 7i. z (5 i 5 ((5(i ((5(44i 78i 5 900i 60 78i 05 88i 4. z ( 5i 78 ((44(5i (((5i 5i 78 60i 900 5i 88 05i 5. z ( 5 0i 75 ((5(0i (( 5(400i 8000i i i 4, i 6. z ( 5(cos( isin( (cos((( i sin((( (cos(.9050 i sin( i

16 98 Chapter 8 More About the Trigonometric Functions 7. z ( (cos(.760 isin( (cos(((.760 i sin((( (cos(.580 i sin( i 8. z ( (cos( isin( (cos((( i sin((( (cos(.847 i sin( i 9. ( z 5(cos(.40 isin(.40 5,65(cos(((.40 i sin(((.40 5,65(cos( i sin( , i 40. z ( 59(cos( 0.59 isin( 0.59 / ( i 59 (cos((( 0.59 sin((( , (cos(.577 i sin( i 4. z ( 7(cos( isin( / ( i 7 (cos((( sin((( (cos(.076 isin( i 4. z ( 74(cos(.905 isin(.905 / ( i 74 (cos(((.905 sin((( (cos(6.575 i sin( i 4. z ( 7(cos( isin( / ( i 7 (cos((( sin((( (cos(.679 i sin( i 44. z ( 67(cos(.548 isin(.548 / ( i 67 (cos(((.548 sin((( (cos( i sin( i

17 Properties of Comple Numbers Section ( i 0 ( i i ( i 4i ( i i ( i 4 7 4i Im Re 0 0 Powers of i 46. ( i 0 ( i i ( i i ( i i ( i i Im Powers of ( i Re 47. z 4 z z ( z (cos θ i sin θ ( z (cosθ i sin θ ( cos θ cosθ sin θ sin θ (sin θ cosθ cos θ sin θ 4 z i i ([cosθ cosθ sin θsin θ] i[sin θ cosθ cosθsin θ] z z (cos( θ θ isin( θ θ z (cos 4θ isin 4 θ 48. The proof is b induction on n. We have seen that DeMoivre s theorem is true for n,,, and 4. Let n be an number for which DeMoivre s theorem is true. We show that DeMoivre s theorem is true for n and thus b induction for all positive integers: n n z z z n ( z (cos( n θ isin( n θ ( z (cosθ isin θ n n ( cos( θ cosθ sin( θsin θ (sin( θ cosθ cos( θsin θ z n i n i n n z (cos(( n θ θ isin(( n θ θ n z (cos nθ isin nθ 49. a. zw ( a bi( c di ( ac bd ( ad bc i

18 00 Chapter 8 More About the Trigonometric Functions b. If θ and θ are the angles associated with w and z, then zw z (cosθ i sin θ w (cosθ i sin θ z w (cosθ cosθ sinθsin θ i(sinθcosθ cosθsin θ z w (cos( θ θ isin( θ θ c. This provides the etension of DeMoivre s theorem to products in trigonometric form: zw z w (cos( θ θ isin( θ θ d. (i z i and w. i Since arctan(.07 radians, the trigonometric forms for z and w are z 5(cos(.07 isin(.07 and w 5(cos(.07 isin(.07. Then zw 5 5(cos(. 7 (.07 isin(.07 (.07 5(cos(0 i sin(0 5. (ii z i and w i. Since arctan( and arctan, the trigonometric forms 6 for z and w are z cos( isin( and w cos( isin (. 6 6 Then zw ( cos ( isin 6 ( 6 cos ( isin ( i a. DeMoivre s theorem holds for an integer n. Using n we can write z / was z w. If z z i (cosθ sin θ and w w θ i θ (cos sin then z z (cos( θ θ isin( θ θ. w w b. (i z iand w. i Since arctan(.07 radians, the trigonometric forms for z and w are z 5(cos(.07 isin(.07 and w 5(cos(.07 isin(.07. Then (ii z 5 (cos(.07 (.07 i sin(.07 (.07 w 5 (cos(.4 isin( i. z i and w i. Since arctan( and arctan, the trigonometric forms 6 for z and w are z cos( isin( and w cos( isin (. 6 6 z cos ( i sin w 6 ( 6 cos ( isin ( i. 5. a. If the angle associated with z is θ z z θ ( i θ (,then cos sin. Like nonzero real numbers, nonzero comple numbers have two square roots, generated b the formula z z cos θ ( isin θ ( and z z cos θ ( isin θ (.

19 The Road to Chaos Section b. The trigonometric form for z i is cos ( isin (. Then ( ( ( ( c. i ( i i z cos isin i and z cos 7 isin 7 i d. The complete set of nth roots of z is given b cos( sin ( n z θ k i θ k n n for 0,,,...,. k n 5. Multipling an comple number z b a negative number b rotates z through an angle of. If z starts on the negative real ais, it ends up on the positive real ais, that is, it is a positive number. 5. z z ( a bi( a bi a ( bi a b a bi z Section 8.4 The Road to Chaos. a. The limiting value, one of the equilibrium values ± 4 C, will be real whenever C /4. Note however that for some C /4, the equilibrium values ma be the onl starting values for which the iteration f( where f( C converges. b. {0.5, 0.5, 0.5, , 0.5, , 0.8, 0.884, , , 0.70, } c. The sequence of iterates diverges to infinit.. a. If the sequence of iterates has a limit z, it will satisf i. b. The sequence diverges from this starting point. c. The sequence diverges from this starting point. z z 5, which has two solutions, 4 i and. The equilibrium values for the difference equation are the solutions to the equation f(. The equilibrium values, ± 4 C, will be real whenever C /4. 4. For C > /4, the line does not intersect the parabola real solution. 5. For ever choice of C, the graph of intersects the shifted cubic C and the equation C. C has no 6. a. {.0, ,.9509,.45965, , , } b. {5.0, , , , , , } c. {8.0, , , , , , } is converging to. d. {5.0, , , , , } is converging to 5. e. The roots of sin are the integral multiples of. 7. The roots of cos are the half-odd-integer multiples of.

20 0 Chapter 8 More About the Trigonometric Functions Chapter 8 Review Problems. Not an identit. Identit: (. Not an identit 4. Not an identit (sin cos sin sin cos sin sin 5. Identit: 6. Identit: 7. Identit: 8. Identit: (sin cos sin sin cos cos sin sinθ cosθ sin θ ( cos θ(cos θ cosθ cosθ sin θ ( cos θ(sin θ ( cos θ(sin θ sinθ cost cost cos cos ( cos ( cos cos sin t t t t t t 4 4 cos θ sin θ (cos θ sin θ(cos θ sin θ cos θ 9. Not an identit. 0. a. cos( cos( cos cos sin sin cos cos sin sin cos cos b. sin( sin( sin cos cos sin sin cos cos sin sin cos. With T 9 (, we have ( T Below we plot T ( andsin. 9 T ( sin.0 5. With T 9 8 5(, we have 5 ( T Below we plot T 5 ( andsin sin 9 8 T 5 ( The maimum difference for is about 0.06.

21 8 and arctan ( Chapter 8 Review Problems 0 4. a. z 6 8; i ( Since 6 8i is in Quadrant II, the angle for z is Thus z 0(cos.5 isin.5. ( w 5 i; 5 ( 9 and arctan 0.8. Since 5 i is in Quadrant IV, the angle for 5 w is 0.8. Thus w 9(cos( 0.8 isin( 0.8. b. Since.5 ( , zw (0( 9(cos.84 isin (cos.84 i sin i Since.5 ( , For w z z 0 (cos.596 i sin.596 w (cos.596 i sin i we take the reciprocal of the modulus and the negative of the angle, getting: w 9 (cos(.596 i sin(.596 z (cos(.596 isin( i 5. a. z (cos 5 isin 5 b. z i c. 5 5 z (cos 60 i sin i d. z (cos 6 i sin i 6. Period 7. Period

22 04 Chapter 8 More About the Trigonometric Functions 8. Period 9. Period Period Period 4. For integers m and n, the frequenc of sin m cos n is m n, where LCM is the least common LCM( m, n multiple. Thus the period is LCM( m, n. If the two coefficients of are of the form m and m n for n integers m and n, the period is n.

(c) cos Arctan ( 3) ( ) PRECALCULUS ADVANCED REVIEW FOR FINAL FIRST SEMESTER

(c) cos Arctan ( 3) ( ) PRECALCULUS ADVANCED REVIEW FOR FINAL FIRST SEMESTER PRECALCULUS ADVANCED REVIEW FOR FINAL FIRST SEMESTER Work the following on notebook paper ecept for the graphs. Do not use our calculator unless the problem tells ou to use it. Give three decimal places