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11.4: Equations and Identities

Last updated

Mar 4, 2023
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- 11.3: Trigonometric Functions
- 11.5: Radians

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5.1 Algebra with Trigonometric Ratios

Homework 5.1

1. -2

3. $\dfrac{1}{\sqrt{2}}$

5. 6

7. $\dfrac{1}{2}$

9. 4

11. 2

13. 1

15. 0

17.

a 0.7660

b 0.8164

c 0.7660

19.

a 0.6691

b 1.8271

c 0.6691

21.

a 1

b 1

c 1

23.

a $-2 x^2-x$

b $-2 \cos ^2 \theta-\cos \theta$

25.

a $4 S C$

b $4 \sin \theta \cos \theta$

27.

a $5 C^2 S^3$

b $5 \cos ^2 \theta \sin ^3 \theta$

29. $-2 \cos t+2 \cos t \sin t ; 0.6360$

31. $\tan \theta-\tan \phi ;-56.91$

33. $2 \sin x \cos x-2 \sin (2 x) ; 0$

35. No

37. No

39. Yes

41. No

43. No

45.

a $2 x^2-x$

b $2 \sin ^2 A-\sin A$

47.

a $a b-3 a^2$

b $\tan A \tan B-3 \tan ^2 A$

49.

a $2 C^2+C-1$

b $2 \cos ^2 \phi+\cos \phi-1$

51.

a $a^2-b^2$

b $\cos ^2 \theta-\cos ^2 \phi$

53.

a $1-2 T+T^2$

b $1-2 \tan \theta+\tan ^2 \theta$

55.

a $T^4-4$

b $\tan ^4 \theta-4$

57.

a $3(3 m+5 n)$

b $3(3 \cos \alpha+5 \cos \beta)$

59.

a $5 r(r-2 q)$

b $5 \tan C(\tan C-2 \tan B)$

61.

a $(3 C+1)(3 C-1)$

b $(3 \cos \beta+1)(3 \cos \beta-1)$

63.

a $2 T^2(3 T-4)$

b $2 \tan ^2 A(3 \tan A-4)$

65.

a $(t-5)(t+4)$

b $(\tan \theta-5)(\tan \theta+4)$

67.

a $(3 c-1)(c+1)$

b $(3 \cos B-1)(\cos B+1)$

5.2 Solving Equations

Homework 5.2

1. $70^{\circ}$

3. $40^{\circ}$

5. I: $18^{\circ}$ ; II: $162^{\circ}$ ; III: $198^{\circ} ; \mathrm{IV}: 342^{\circ}$

7. I: $52^{\circ}$ ; II: $128^{\circ}$ ; III: $232^{\circ} ; \mathrm{IV}: 308^{\circ}$

a 0, 4, 2, 0, 4

b -1 or 2

11.

a $1, \dfrac{\sqrt{3}+1}{2}, \sqrt{2}, \dfrac{\sqrt{3}+1}{2}$

b $45^{\circ}$

13.

a $0, \dfrac{2-\sqrt{2}}{2}, \dfrac{1-\sqrt{3}}{2},-1$

b $270^{\circ}$

15. $x = 5, -3$

17. $x = -3, 1, 2$

19. $\theta = 30^{\circ}$ or $\theta = 210^{\circ}$

21. $\theta=60^{\circ}$ or $\theta=300^{\circ}$

23. $\theta=210^{\circ}$ or $\theta=330^{\circ}$

25. $\theta=225^{\circ}$ or $\theta=315^{\circ}$

27. $\theta=0^{\circ}$ or $\theta=180^{\circ}$

29. $\theta=60^{\circ}, \theta=120^{\circ}, \theta=240^{\circ}$ , or $\theta=300^{\circ}$

31. $\theta=45^{\circ}, \theta=135^{\circ}, \theta=225^{\circ}$ , or $\theta=315^{\circ}$

33. $\theta=104.04^{\circ}$ or $\theta=284.04^{\circ}$

35. $\theta=53.13^{\circ}$ or $\theta=306.87^{\circ}$

37. $\theta=188.21^{\circ}$ or $\theta=351.79^{\circ}$

39. $A=135^{\circ}$ or $A=315^{\circ}$

41. $\phi=210^{\circ}$ or $\phi=330^{\circ}$

43. $B=90^{\circ}$ or $B=270^{\circ}$

45. $\theta=210^{\circ}$ or $\theta=330^{\circ}$

47. $B=90^{\circ}$ or $B=270^{\circ}$

49. $\theta=210^{\circ}$ or $\theta=330^{\circ}$

51. $\phi=146^{\circ}$ or $\phi=214^{\circ}$

53. $\theta=54.74^{\circ}, \theta=125.26^{\circ}, \theta=234.74^{\circ}$ , or $\theta=305.26^{\circ}$

55. $\theta=0^{\circ}, \theta=180^{\circ}, \theta=191.54^{\circ}$ , or $\theta=348.46^{\circ}$

57. $\theta=60^{\circ}, \theta=180^{\circ}$ , or $\theta=300^{\circ}$

59. $\theta=26.57^{\circ}, \theta=161.57^{\circ}, \theta=206.57^{\circ}$ , or $\theta=341.57^{\circ}$

61. $\theta=78.69^{\circ}, \theta=108.43^{\circ}, \theta=258.69^{\circ}$ , or $\theta=288.43^{\circ}$

63. $\theta=0^{\circ}$

65. $17.22^{\circ}$

67. $35.66^{\circ}$

5.3 Trigonometric Identities

Homework 5.3

1. not an identity

3. not an identity

5. identity

7. not an identity

9. not an identity

11. not an identity

13. identity

15. identity

17. $(1+\sin w)(1-\sin w)=1-\sin ^2 w=\cos ^2 w$

19.

$\begin{aligned} (\cos \theta-\sin \theta)^2 & =\cos ^2 \theta-2 \cos \theta \sin \theta+\sin ^2 \theta \\ & =\left(\cos ^2 \theta+\sin ^2 \theta\right)-2 \sin \theta \cos \theta=1-2 \sin \theta \cos \theta \end{aligned}$

21. $\tan \theta \cos \theta=\dfrac{\sin \theta}{\cos \theta} \cdot \cos \theta=\sin \theta$

23.

$\begin{aligned} \cos ^4 x-\sin ^4 x&=\left(\cos ^2 x-\sin ^2 x\right)\left(\cos ^2 x+\sin ^2 x\right) \\ &=\left(\cos ^2 x-\sin ^2 x\right)(1)=\cos ^2 x-\sin ^2 x \end{aligned}$

25. $\dfrac{\sin u}{1+\cos u} \cdot \dfrac{1-\cos u}{1-\cos u}=\dfrac{\sin u(1-\cos u)}{1-\cos ^2 u}= \dfrac{\sin u(1-\cos u)}{\sin ^2 u}=\dfrac{1-\cos u}{\sin u}$

27. 1

29. 1

31. $\sin ^2 A$

33. $\tan ^2 z$

35. 3

37. 1

39. 6

41. $\cos 2 \theta$

43. $\cos \theta$

45. $\sin 2 t$

47. $1 + 2 \sin \theta + \sin ^2 \theta$

49. $3 \cos ^2 \phi - 2$

51. $\theta=90^{\circ}, \theta=180^{\circ}, \theta=270^{\circ}$

53. $\theta=90^{\circ}, \theta=210^{\circ}, \theta=330^{\circ}$

55. $\theta=210^{\circ}, \theta=330^{\circ}$

57. $\theta=18.43^{\circ}, \theta=198.43^{\circ}$

59. $\sin A=\dfrac{-5}{13}, \tan A=\dfrac{-5}{12}$

61. $\cos \phi=\dfrac{-4 \sqrt{3}}{7}, \tan \phi=\dfrac{-1}{4 \sqrt{3}}$

63. $\sin \theta=\dfrac{-1}{\sqrt{5}}, \quad \cos \theta=\dfrac{2}{\sqrt{5}}$

65. $\sin \theta=\dfrac{-3}{5}, \quad \cos \theta=\dfrac{-4}{5}$

67. $\sin \theta=\dfrac{\sqrt{3}}{2}, \quad \cos \theta=\dfrac{-1}{2}, \quad \tan \theta=\sqrt{3}$

69. $\sin \beta=\dfrac{2}{\sqrt{5}}, \quad \cos \beta=\dfrac{-1}{\sqrt{5}}, \quad \tan \beta=-2$

71.

$\begin{aligned} \sin C & =\dfrac{1}{\sqrt{5}}, \quad \cos C=\dfrac{2}{\sqrt{5}}, \quad \tan C=\dfrac{1}{2} \\ \text { or } \quad \sin C & =\dfrac{1}{\sqrt{5}}, \quad \cos C=\dfrac{-2}{\sqrt{5}}, \quad \tan C=\dfrac{-1}{2} \end{aligned}$

73. $\dfrac{\tan \alpha}{1+\tan \alpha}=\dfrac{\frac{\sin \alpha}{\cos \alpha}}{1+\frac{\sin \alpha}{\cos \alpha}} \cdot \dfrac{\cos \alpha}{\cos \alpha}=\dfrac{\sin \alpha}{\sin \alpha+\cos \alpha}$

75. $\dfrac{1+\tan ^2 \beta}{1-\tan ^2 \beta}=\dfrac{\frac{1}{\cos ^2 \beta}}{1-\frac{\sin ^2 \beta}{\cos ^2 \beta}} \cdot \dfrac{\cos ^2 \beta}{\cos ^2 \beta}= \dfrac{1}{\cos ^2 \beta-\sin ^2 \beta}$

77. $Screen Shot 2023-02-14 at 11.29.42 PM.png$

a By the distance formula, $\sqrt{x^2+y^2}=r$ , or $x^2+y^2=r^2$ .

b $\dfrac{x^2}{r^2}+\dfrac{y^2}{r^2}=1$

c $\left(\dfrac{x}{r}\right)^2+\left(\dfrac{y}{r}\right)^2=1$

d $(\cos \theta)^2+(\sin \theta)^2=1$

5.4 Chapter 5 Summary and Review

Chapter 5 Review Problems

1. $\dfrac{-3}{4 \sqrt{2}}$

3. $\dfrac{1}{\sqrt{6}}$

a 0.8660

b 0.9848; No

a 1.4821

b 1.4821; Yes

9. $5 \sin x-2 \sin x \cos y-\cos y$

11. $2 \tan \theta-10 \tan ^2 \theta$

13. Not equivalent

15. Equivalent

17. $2 \cos ^2 \alpha+\cos \alpha-6$

19. $\tan ^2 \phi-2 \tan \phi \cos \phi+\cos ^2 \phi$

21. $6(2 \sin 3 x-\sin 2 x)$

23. $(1+3 \tan \theta)(1-3 \tan \theta)$

25. $\cos \alpha+\sin \alpha$

27. $\dfrac{3}{2}$

29. $\dfrac{3 \tan C+2}{\tan C-2}$

31. $51.32^{\circ}, 308.68^{\circ}$

33. $90^{\circ}, 270^{\circ}, 120^{\circ}, 240^{\circ}$

35. $90^{\circ}, 210^{\circ}, 330^{\circ}$

37. $30^{\circ}, 150^{\circ}, 210^{\circ}, 330^{\circ}$

39. $0^{\circ}, 120^{\circ}, 240^{\circ}$

41. $57.99^{\circ}, 237.99^{\circ}$

43. $90^{\circ}, 270^{\circ}$

45. $33.17^{\circ}$

47. Identity

49. Not an identity

51. Not an identity

53. Identity

55. $\dfrac{1-\cos ^2 \alpha}{\tan \alpha}=\sin ^2 \alpha \cdot \dfrac{\cos \alpha}{\sin \alpha}=\sin \alpha \cos \alpha$

57.

$\begin{aligned} \dfrac{\frac{\sin \theta}{\cos \theta}-\sin \theta \cos \theta}{\sin \theta \cdot \frac{\sin \theta}{\cos \theta}}&=\dfrac{\sin \theta-\sin \theta \cos ^2 \theta}{\sin ^2 \theta} \\ & =\dfrac{\sin \theta\left(1-\cos ^2 \theta\right)}{\sin ^2 \theta}=\dfrac{\sin \theta \sin ^2 \theta}{\sin ^2 \theta}=\sin \theta \end{aligned}$

59. $\dfrac{1}{\sin \theta \cos \theta}$

61. 1

63. 0

65. 1

67. $\dfrac{1}{\cos ^2 \beta}$

69. $2+\cos t-2 \cos ^2 t$

71. $\sin \beta=\dfrac{-6}{\sqrt{85}}, \quad \cos \beta=\dfrac{-7}{\sqrt{85}}, \quad \tan \beta=\dfrac{6}{7}$

73. $\sin \alpha=\dfrac{\sqrt{21}}{5}, \cos \alpha=\dfrac{-2}{5}, \quad \tan \alpha=\dfrac{-\sqrt{21}}{2}$

75. $0^{\circ}, 180^{\circ}, 270^{\circ}$

77. $135^{\circ}, 315^{\circ}$

79. $0^{\circ}, 60^{\circ}, 180^{\circ}, 300^{\circ}$

81. $0^{\circ}, 180^{\circ}$