11.4: Equations and Identities

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: \(232^{\circ} ; \mathrm{IV}: 308^{\circ}\)

9.

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.

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. \(\dfrac{1}{\sqrt{6}}\)

5.

a 0.8660

b 0.9848; No

7.

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. \(0^{\circ}, 180^{\circ}\)