11.7: More Functions and Identities

8.1 Sum and Difference Formulas

Homework 8-1

1.

\(x_2=x_1, y_2=-y_1\), and \(r_2=r_1\). Thus, \(\cos (-\alpha)=\dfrac{x_2}{r_2}=\dfrac{x_1}{r_1}=\cos \alpha, \sin (-\alpha)=\dfrac{y_2}{r_2}=\dfrac{-y_1}{r_1}= -\sin \alpha\), and \(\tan (-\alpha)=\dfrac{y_2}{x_2}=\dfrac{-y_1}{x_1}=-\tan \alpha\)

3. \(\dfrac{-(\sqrt{2}+\sqrt{6})}{4}\)

5. \(\cos (0.3-2 x)=0.24, \sin (0.3-2 x)=0.97\)

7. \(\cos \left(45^{\circ}+45^{\circ}\right)=\cos \left(90^{\circ}\right)=0\), but \(\cos 45^{\circ}+\cos 45^{\circ}=\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{2}}=\sqrt{2}\)

9. \(\tan \left(87^{\circ}-29^{\circ}\right) \approx 1.600\), but \(\tan 87^{\circ}-\tan 29^{\circ} \approx 18.527\)

11. The curves are different.

13.

a \(\dfrac{63}{65}\)

b \(\dfrac{-16}{65}\)

c \(\dfrac{-16}{63}\)

15.

a \(\dfrac{44}{117}\)

b \(\dfrac{4}{3}\)

17.

a \(\dfrac{36}{85}\)

b \(\dfrac{-13}{84}\)

19.

a \(\dfrac{-16}{65}\)

b \(\dfrac{63}{65}\)

c \(\dfrac{-16}{63}\)

d

21. \(\cos 15^{\circ}=\dfrac{\sqrt{6}+\sqrt{2}}{4}, \tan 15^{\circ}=2-\sqrt{3}\)

23. \(\dfrac{6 \sqrt{2}+1}{10}\)

25. \(\cos \theta\)

27. \(\dfrac{\sqrt{3}}{2} \cos t-\dfrac{1}{2} \sin t\)

29. \(\dfrac{\sqrt{3} \tan \beta-1}{\sqrt{3}+\tan \beta}\)

31. No

33. No

35. \(1=2\left(\dfrac{1}{\sqrt{2}}\right)\left(\dfrac{1}{\sqrt{2}}\right)\)

37. \(\dfrac{1}{2}=\left(\dfrac{\sqrt{3}}{2}\right)^2-\left(\dfrac{1}{2}\right)^2\)

39. False, but \(\cos 2 \alpha=2(0.32)^2-1\)

41. False, but \(2 \theta=\sin ^{-1}(h)\)

43. \(\sin 68^{\circ}\)

45. \(\cos \dfrac{\pi}{8}\)

47. \(\cos 6\theta\)

49. \(\sin 10 t\)

51. \(\tan 128^{\circ}\)

53. \(\cos 4 \beta\)

55.

a \(\dfrac{5}{6}\)

b \(\dfrac{\sqrt{11}}{6}\)

c \(\dfrac{5}{\sqrt{11}}\)

d \(\dfrac{5 \sqrt{11}}{18}\)

e \(\dfrac{-7}{18}\)

f \(\dfrac{-5 \sqrt{11}}{7}\)

57.

a \(\dfrac{1}{\sqrt{w^2+1}}\)

b \(\dfrac{w}{\sqrt{w^2+1}}\)

c \(\dfrac{1}{w}\)

d \(\dfrac{2 w}{w^2+1}\)

e \(\dfrac{w^2-1}{w^2+1}\)

f \(\dfrac{2 w}{w^2-1}\)

59.

a \(\dfrac{-5}{13}\)

b \(\dfrac{-120}{169}\)

c \(\dfrac{119}{169}\)

d \(\dfrac{-120}{119}\)

e

61.

a \(\dfrac{8}{15}\)

b \(\dfrac{-15}{17}\)

c \(\dfrac{-8}{17}\)

63.

a \(2 \sin \theta \cos \theta+\sqrt{2} \cos \theta=0\)

b \(\dfrac{\pi}{2}, \dfrac{5 \pi}{4}, \dfrac{3 \pi}{2}, \dfrac{7 \pi}{4}\)

65.

a \(2 \cos ^2 t-5 \cos t+2=0\)

b \(\dfrac{\pi}{3}, \dfrac{5 \pi}{3}\)

67.

a \(\dfrac{2 \tan \beta}{1-\tan ^2 \beta}+2 \sin \beta=0\)

b \(0, \dfrac{\pi}{3}, \pi, \dfrac{5 \pi}{3}\)

69.

a \(3 \cos \phi-\cos \phi=\sqrt{3}\)

b \(\dfrac{\pi}{6}, \dfrac{11 \pi}{6}\)

71.

a \(\sin 3 \phi=1\)

b \(\dfrac{\pi}{6}, \dfrac{5 \pi}{6}, \dfrac{3 \pi}{2}\)

73.

a \(\cos \left(\theta+90^{\circ}\right)=-\sin \theta\)

b \(\sin \left(\theta+90^{\circ}\right)=\cos \theta\)

75.

a \(\cos \left(\dfrac{\pi}{2}-\theta\right)=\cos \dfrac{\pi}{2} \cos \theta+\sin \dfrac{\pi}{2} \sin \theta=\sin \theta\)

b \(\sin \left(\dfrac{\pi}{2}-\theta\right)=\sin \dfrac{\pi}{2} \cos \theta-\cos \dfrac{\pi}{2} \sin \theta=\cos \theta\)

77.

\begin{aligned}
\sin 2 \theta & =\sin (\theta+\theta) \\
& =\sin \theta \cos \theta+\sin \theta \cos \theta \\
& =2 \sin \theta \cos \theta
\end{aligned}

79.

a Not an identity

b \(\beta=\pi\) (many answers possible)

81. Identity

83.

a Not an identity

b \(\theta = 0\) (many answers possible)

85. Identity

87. Identity

89.

a \(l_1=\sin \alpha, l_2=\cos \alpha\)

b \(\theta_1\) and \(\beta\) are both complements of \(\phi ; \theta_2\) and \(\alpha+\beta\) are alternate interior angles

c \(s_1=\cos (\alpha+\beta), s_2=\sin (\alpha+\beta)\)

d \(s_3=\sin \alpha \sin \beta, s_4=\sin \alpha \cos \beta\)

e \(s_5=\cos \alpha \cos \beta, s_6=\cos \alpha \sin \beta\)

f \(\sin (\alpha+\beta)=\sin \alpha \cos \beta+\cos \alpha \sin \beta, \cos (\alpha+\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta\)

91.

a \((A B)^2=2-2 \cos (\alpha-\beta)\)

b \((A B)^2=(\cos \alpha-\cos \beta)^2+(\sin \alpha-\sin \beta)^2\)

c

\begin{aligned}
& 2-2 \cos (\alpha-\beta)=(\cos \alpha-\cos \beta)^2+(\sin \alpha-\sin \beta)^2 \\
& 2-2 \cos (\alpha-\beta)=\cos ^2-2 \cos \alpha \cos \beta+\cos ^2 \beta +\sin ^2 \alpha-2 \sin \alpha \sin \beta+\sin ^2 \beta \\
& 2-2 \cos (\alpha-\beta)=1+1-2(\cos \alpha \cos \beta-\sin \alpha \sin \beta) -2 \cos (\alpha-\beta)=-2(\cos \alpha \cos \beta-\sin \alpha \sin \beta) \\
& \cos (\alpha-\beta)=\cos \alpha \cos \beta-\sin \alpha \sin \beta \\
\end{aligned}

8.2 Inverse Trigonometric Functions

Homework 8-2

1. No inverse: Some horizontal lines intersect the curve in more than one point.

3. Inverse exists: The function is 1-1.

5. No inverse

7. No inverse

9. \(16.5^{\circ}\)

11. \(46.4^{\circ}\)

13. \(=51.9^{\circ}\)

15. \(\dfrac{3 \pi}{4}\)

17. \(\dfrac{-\pi}{6}\)

19. \(\dfrac{\pi}{6}\)

21.

a \(h=500 \tan \theta\)

b \(\theta=\tan ^{-1}\left(\dfrac{h}{500}\right)\)

c \(\theta=\tan ^{-1}(2)\), so the angle of elevation is \(\tan ^{-1} 2 \approx 63.4^{\circ}\) when the rocket is 1000 yd high

23.

a \(d=\dfrac{50}{\tan \theta}\)

b \(\theta=\tan ^{-1}\left(\dfrac{50}{d}\right)\)

c \(\theta=\tan ^{-1}(0.25)\); the billboard subtends an angle of \(\tan ^{-1}(0.25) \approx 14^{\circ}\) at a distance of 200 \(\mathrm{ft}\).

25.

a \(\alpha=\tan ^{-1}\left(\dfrac{1}{x}\right)\)

b \(\beta=\tan ^{-1}\left(\dfrac{5}{x}\right)-\tan ^{-1}\left(\dfrac{1}{x}\right)\)

c \(\beta=45^{\circ}-\tan ^{-1}\left(\dfrac{1}{5}\right)\), so the painting subtends an angle of \(45^{\circ}-\tan ^{-1}\left(\dfrac{1}{5}\right) \approx 33.7^{\circ}\) when Martin is 5 meters from the wall.

27. \(t=\dfrac{1}{2 \pi \omega}\left(\sin ^{-1} \dfrac{V}{V_0}-\phi\right)\)

29. \(A=\sin ^{-1}\left(\dfrac{a \sin B}{b}\right)\)

31. \(\theta=\pm \cos ^{-1}\left(\dfrac{k}{P R^4}\right)\)

33. \(\dfrac{2}{\sqrt{5}}\)

35. \(\dfrac{1}{\sqrt{5}}\)

37. \(\dfrac{5}{7}\)

39. \(\dfrac{\sqrt{1-x^2}}{x}\)

41. \(\sqrt{1-h^2}\)

43. \(\dfrac{2 t}{\sqrt{4 t^2+1}}\)

45.

\(x\)	-1	\(\dfrac{-\sqrt{3}}{2}\)	\(\dfrac{-\sqrt{2}}{2}\)	\(\dfrac{-1}{2}\)	0	\(\dfrac{1}{2}\)	\(\dfrac{\sqrt{2}}{2}\)	\(\dfrac{\sqrt{3}}{2}\)	1
\(\cos ^{-1} x\)	\(\pi\)	\(\dfrac{5\pi}{6}\)	\(\dfrac{3\pi}{4}\)	\(\dfrac{2\pi}{3}\)	\(\dfrac{\pi}{2}\)	\(\dfrac{\pi}{3}\)	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{6}\)	0

47.

\(x\)	\(-\sqrt{3}\)	-1	\(\dfrac{-1}{\sqrt{3}}\)	0	\(\dfrac{1}{\sqrt{3}}\)	1	\(\sqrt{3}\)
\(\cos ^{-1} x\)	\(\dfrac{-\pi}{2}\)	\(\dfrac{-\pi}{3}\)	\(\dfrac{-\pi}{6}\)	0	\(\dfrac{\pi}{6}\)	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{3}\)

49.

a-b.

c. No

51.

a.

c. No

53. \(\dfrac{8}{17}\)

55. \(\dfrac{16}{65}\)

57. \(\dfrac{4 \sqrt{2}}{7}\)

59.

a \(\dfrac{-63}{65}\)

b \(\dfrac{16}{65}\)

c \(\dfrac{-33}{65}\)

d \(\dfrac{56}{65}\)

61. 1

63.

a \(\dfrac{2 x}{x^2+1}\)

b \(1-x^2\)

65. \(\sin 2 \theta=\dfrac{2 x \sqrt{25-x^2}}{25}, \cos 2 \theta=\dfrac{25-2 x^2}{25}\)

67. \(\arctan \dfrac{x}{3}+\dfrac{3 x}{2\left(x^2+9\right)}\)

69.

a \(-1 \leq x \leq 1\)

b Yes.

c All

d \(x<\dfrac{-\pi}{2}\) or \(x>\dfrac{\pi}{2}\)

71.

a Domain: \(-1 \leq x \leq 1\), range: \(\left\{\dfrac{\pi}{2}\right\}\)

73.

a \(\dfrac{\theta}{2}\)

b \(t=\sin \theta\)

c \(\dfrac{1}{2} \arcsin t\)

8.3 The Reciprocal Functions

Homework 8-3

1. 2.203

3. 0.466

5. 5.883

7. 1.203

9. 2

11. 1

13. \(\dfrac{-2 \sqrt{3}}{3}\)

15. \(\sqrt{2}\)

17.

\(\theta\)	0	\(\dfrac{\pi}{6}\)	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{3}\)	\(\dfrac{\pi}{2}\)	\(\dfrac{2\pi}{3}\)	\(\dfrac{3\pi}{4}\)	\(\dfrac{5\pi}{6}\)	\(\pi\)
\(\sec \theta\)	1	\(\dfrac{2\sqrt{3}}{3}\)	\(\sqrt{2}\)	2	undefined	-2	\(-\sqrt{2}\)	\(-\dfrac{2\sqrt{3}}{3}\)	-1
\(\csc \theta\)	undefined	2	\(\sqrt{2}\)	\(\dfrac{2\sqrt{3}}{3}\)	1	\(\dfrac{2\sqrt{3}}{3}\)	\(\sqrt{2}\)	2	undefined
\(\cot \theta\)	undefined	\(\sqrt{3}\)	1	\(\dfrac{\sqrt{3}}{3}\)	0	\(\dfrac{-\sqrt{3}}{3}\)	-1	\(-\sqrt{3}\)	undefined

19.

a 0.980

b 1.020

c 1.369

d 1.020

e 0.284

f 1.020

21. \(\sin \theta=\dfrac{4}{5}, \quad \cos \theta=\dfrac{3}{5}, \quad \tan \theta=\dfrac{4}{3}, \quad \sec \theta=\dfrac{5}{3}, \quad \csc \theta=\dfrac{5}{4}, \quad \cot \theta=\dfrac{3}{4}\)

23. \(\sin \theta=\dfrac{4}{\sqrt{41}}, \quad \cos \theta=\dfrac{5}{\sqrt{41}}, \quad \tan \theta=\dfrac{4}{5}, \quad \sec \theta=\dfrac{\sqrt{41}}{5}, \quad \csc \theta=\dfrac{\sqrt{41}}{4}, \quad \cot \theta=\dfrac{5}{4}\)

25. \(\sin \theta=\dfrac{5}{\sqrt{74}}, \quad \cos \theta=\dfrac{-7}{\sqrt{74}}, \quad \tan \theta=\dfrac{-5}{7}, \quad \sec \theta=\dfrac{-\sqrt{74}}{7}, \quad \csc \theta=\dfrac{\sqrt{74}}{5}, \quad \cot \theta=\dfrac{-7}{5}\)

27. \(\sin \theta=\dfrac{-5}{8}, \quad \cos \theta=\dfrac{\sqrt{39}}{8}, \quad \tan \theta=\dfrac{5}{\sqrt{39}}, \sec \theta=\dfrac{-8}{\sqrt{39}}, \quad \csc \theta=\dfrac{-8}{5}, \quad \cot \theta=\dfrac{\sqrt{39}}{5}\)

29.

a \(d=h \csc \theta\)

b 155.572 miles

31.

a \(0.78 \mathrm{sec}\)

b \(l=8 t^2 \sin 2 \theta\)

33. \(\sin \theta = \dfrac{7}{\sqrt{x^2 + 49}}, \cos \theta = \dfrac{x}{\sqrt{x^2 + 49}}, \tan \theta = \dfrac{7}{x}, \sec \theta = \dfrac{\sqrt{x^2 + 49}}{x}, \csc \theta = \dfrac{\sqrt{x^2 + 49}}{7}, \cot \theta = \dfrac{x}{7}\)

35. \(\sin \theta = S, \cos \theta = \sqrt{1 - S^2}, \tan \theta = \dfrac{S}{\sqrt{1 - S^2}}, \sec \theta = \dfrac{1}{\sqrt{1 - S^2}}, \csc \theta = \dfrac{1}{S}, \cot \theta = \dfrac{\sqrt{1 - S^2}}{S}\)

37. \(\sin \theta = \dfrac{-\sqrt{9 - a^2}}{3}, \cos \theta = \dfrac{a}{3}, \tan \theta = \dfrac{-\sqrt{9 - a^2}}{a}, \sec \theta = \dfrac{3}{a}, \csc \theta = \dfrac{-3}{\sqrt{9 - a^2}}, \cot \theta = \dfrac{-a}{\sqrt{9 - a^2}}\)

39. \(A C, O A, B D, O D, O E, E F\)

41.

\begin{aligned}
\sin \theta &= \dfrac{-\sqrt{3}}{2}, \quad \cos \theta = \dfrac{1}{2}, \\
\tan \theta &= -\sqrt{3}, \quad \sec \theta = 2, \\
\csc \theta &= \dfrac{-2\sqrt{3}}{3}, \quad \cot \theta = \dfrac{-\sqrt{3}}{3}
\end{aligned}

43.

\begin{aligned}
\sin \alpha &= \dfrac{1}{3}, \quad \cos \alpha = \dfrac{2\sqrt{2}}{3}, \\
\tan \alpha &= \dfrac{\sqrt{2}}{4}, \quad \sec \alpha = \dfrac{3\sqrt{2}}{4}, \\
\csc \alpha &= 3, \quad \cot \alpha = 2\sqrt{2}
\end{aligned}

45.

\begin{aligned}
& \sin \gamma=\dfrac{-4}{\sqrt{17}}, \quad \cos \gamma= \dfrac{-1}{\sqrt{17}},\\
& \tan \gamma=4, \quad \sec \gamma= -\sqrt{17},\\
& \csc \gamma=\dfrac{-\sqrt{17}}{4}, \quad \cot \gamma=\dfrac{1}{4}
\end{aligned}

47. \(\dfrac{4\sqrt{3}}{3} + 2\sqrt{2}\)

49. \(\dfrac{\sqrt{3}}{3}\)

51. \(\dfrac{4\sqrt{6}}{3} + \dfrac{10}{3}\)

53.

\(x\)	0	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{2}\)	\(\dfrac{3\pi}{4}\)	\(\pi\)	\(\dfrac{5\pi}{4}\)	\(\dfrac{3\pi}{2}\)	\(\dfrac{7\pi}{4}\)	\(2\pi\)
\(\sec x\)	1	\(\sqrt{2}\)	undefined	\(-\sqrt{2}\)	-1	\(-\sqrt{2}\)	undefined	\(\sqrt{2}\)	1

55.

57.

\(x\)	0	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{2}\)	\(\dfrac{3\pi}{4}\)	\(\pi\)	\(\dfrac{5\pi}{4}\)	\(\dfrac{3\pi}{2}\)	\(\dfrac{7\pi}{4}\)	\(2\pi\)
\(\cot x\)	undefined	1	0	-1	undefined	1	0	-1	undefined

59.

\begin{aligned}
\dfrac{\csc x}{\cot x} &= \dfrac{\frac{1}{\sin x}}{\frac{\cos x}{\sin x}} \\
&= \dfrac{1}{\sin x} \div \dfrac{\cos x}{\sin x} \\
&= \dfrac{1}{\sin x} \cdot \dfrac{\sin x}{\cos x} \\
&= \dfrac{1}{\cos x} \\
&= \sec x
\end{aligned}

61.

\begin{aligned}
\dfrac{\sec x \cot x}{\csc x} &= \dfrac{\frac{1}{\cos x} \cdot \frac{\cos x}{\sin x}}{\frac{1}{\sin x}} \\
&= \dfrac{\frac{1}{\sin x}}{\frac{1}{\sin x}} \\
&= 1
\end{aligned}

63.

\begin{aligned}
\tan x \csc x &= \dfrac{\sin x}{\cos x} \cdot \dfrac{1}{\sin x} \\
&= \dfrac{1}{\cos x} =\sec x
\end{aligned}

65. \(\dfrac{\pi}{6}, \dfrac{5 \pi}{6}\)

67. \(\dfrac{3 \pi}{4}, \dfrac{5 \pi}{4}\)

69. \(\dfrac{5 \pi}{6}, \dfrac{11 \pi}{6}\)

71. \(\dfrac{-\sqrt{5}}{5}\)

73. \(\dfrac{\sqrt{a^2-4}}{2}\)

75. \(\dfrac{\sqrt{w^2-1}}{-w}\)

77. \(\sec s = \dfrac{-5}{4}, \quad \csc s = \dfrac{5}{3}, \quad \cot s = \dfrac{-4}{3}\)

79. \(\sec s = \dfrac{1}{\sqrt{1-w^2}}, \quad \csc s = \dfrac{1}{w}, \quad \cot s = \dfrac{\sqrt{1-w^2}}{w}\)

81. \(\dfrac{\sin \theta}{\cos ^2 \theta}\)

83. \(\sec t\)

85. \(\dfrac{1-\sin \beta}{\cos \beta}\)

87. \(-\cos x\)

89.

\begin{aligned}
\cos ^2 \theta+\sin ^2 \theta & =1 \\
\dfrac{\cos ^2 \theta}{\cos ^2 \theta}+\dfrac{\sin ^2 \theta}{\cos ^2 \theta} & =\dfrac{1}{\cos ^2 \theta} \\
1+\tan ^2 \theta & =\sec ^2 \theta
\end{aligned}

91.

a \(\csc \theta=-\sqrt{26}\)

b \(\sin \theta=\dfrac{-\sqrt{26}}{26}, \cos \theta=\dfrac{-5 \sqrt{26}}{26}, \tan \theta=\dfrac{1}{5}, \sec \theta=\dfrac{-\sqrt{26}}{5}\)

93. \(\cos t=\pm \sqrt{1-\sin ^2 t}, \quad \tan t=\dfrac{\pm \sin t}{\sqrt{1-\sin ^2 t}}, \sec t=\dfrac{\pm 1}{\sqrt{1-\sin ^2 t}}, \csc t=\dfrac{1}{\sin t}, \cot t=\dfrac{\pm \sqrt{1-\sin ^2 t}}{\sin t}\)

95.

\begin{aligned}
\dfrac{a}{\sin A} & =\dfrac{b}{\sin B}=\dfrac{c}{\sin C} \\
a \cdot \dfrac{1}{\sin A} & =b \cdot \dfrac{1}{\sin B}=c \cdot \dfrac{1}{\sin C} \\
a \csc A & =b \csc B=c \csc C
\end{aligned}

8.4 Chapter Summary and Review

Review Problems

1. False

3. True

5. False

7. False

9. \(\dfrac{2-\sqrt{21}}{5 \sqrt{2}}\)

11.

a \(\dfrac{5 \sqrt{33}-3}{32}\)

b \(\dfrac{5 \sqrt{33}-3}{\sqrt{5}(3 \sqrt{3}+\sqrt{11})}\)

13. 1

15. \(\dfrac{\tan t+\sqrt{3}}{1-\sqrt{3} \tan t}\)

17.

a \(\dfrac{4}{5}\)

b \(\dfrac{3}{5}\)

c \(\dfrac{4}{3}\)

d \(\dfrac{24}{25}\)

e \(\dfrac{-7}{25}\)

f \(\dfrac{-24}{7}\)

19. \(\sin 9x\)

21. \(\tan (2 \phi-2)\)

23. \(\sin 8 \theta\)

25.

a \(1-2 \sin ^2 \theta-\sin \theta=1\)

b \(0, \pi, \dfrac{7 \pi}{6}, \dfrac{11 \pi}{6}\)

27. No

29.

a \(\dfrac{-\pi}{3}\)

b \(\dfrac{2 \pi}{3}\)

31.

a \(\tan ^{-1}\left(\dfrac{52.8}{x}\right)\)

b \(69.25^{\circ}, 27.83^{\circ}\)

33. \(\theta=\sin ^{-1}\left(\dfrac{v_y+g t}{v_0}\right)\)

35. \(\dfrac{2}{3}\)

37. \(\sqrt{1-4 t^2}\)

39. Because \(|\sin \theta| \leq 1, \sin ^{-1} t\) is undefined for \(|t|>1\). If \(x \neq 0\), then either \(|x|>1\) or \(\left|\frac{1}{x}\right|>1\). If \(x=0\), then \(\frac{1}{x}\) is undefined.

41.

a 2.203

b −3.236

c 0.466

43.

\begin{aligned}
\sin \theta &= \dfrac{13}{\sqrt{313}}, \quad \cos \theta = \dfrac{12}{\sqrt{313}}, \quad \tan \theta = \dfrac{13}{12}, \\
\sec \theta &= \dfrac{\sqrt{313}}{12}, \quad \csc \theta = \dfrac{\sqrt{313}}{13}, \quad \cot \theta = \dfrac{12}{13}
\end{aligned}

45.

\begin{aligned}
\sin \theta &= \dfrac{1}{3}, \quad \cos \theta = \dfrac{-2\sqrt{2}}{3}, \quad \tan \theta = \dfrac{-1}{2\sqrt{2}}, \\
\sec \theta &= \dfrac{-3}{2\sqrt{2}}, \quad \csc \theta = 3, \quad \cot \theta = -2\sqrt{2}
\end{aligned}

47.

\begin{aligned}
\sin \theta &= \dfrac{-9}{\sqrt{106}}, \quad \cos \theta = \dfrac{-5}{\sqrt{106}}, \quad \tan \theta = \dfrac{9}{5} \\
\sec \theta &= \dfrac{-\sqrt{106}}, \quad \csc \theta = \dfrac{-\sqrt{106}}{9}, \quad \cot \theta = \dfrac{5}{9}
\end{aligned}

49.

\begin{aligned}
\sin \alpha &= \dfrac{-\sqrt{11}}{6}, \quad \cos \alpha = \dfrac{-5}{6}, \quad \tan \alpha = \dfrac{\sqrt{11}}{5}, \\
\sec \alpha &= \dfrac{-6}{5}, \quad \csc \alpha = \dfrac{-6}{\sqrt{11}}, \quad \cot \alpha = \dfrac{5}{\sqrt{11}}
\end{aligned}

51.

\begin{aligned}
\sin \theta &= \dfrac{s}{4}, \quad \cos \theta = \dfrac{\sqrt{16 - s^2}}{4}, \\
\tan \theta &= \dfrac{s}{\sqrt{16-s^2}}, \quad \sec \theta = \dfrac{4}{\sqrt{16-s^2}}, \\
\csc \theta &= \dfrac{4}{s}, \quad \cot \theta = \dfrac{\sqrt{16 - s^2}}{s}
\end{aligned}

53.

\begin{aligned}
\sin \theta &= \dfrac{w}{\sqrt{w^2+144}}, \quad \cos \theta = \dfrac{-12}{\sqrt{w^2+144}}, \\
\tan \theta &= \dfrac{-w}{12}, \quad \sec \theta = \dfrac{-\sqrt{w^2+144}}{12}, \\
\csc \theta &= \dfrac{\sqrt{w^2+144}}{w}, \quad \cot \theta = \dfrac{-12}{w}
\end{aligned}

55.

\begin{aligned}
\sin \alpha &= \dfrac{k}{2}, \quad \cos \alpha = \dfrac{-\sqrt{4-k^2}}{2}, \\
\tan \alpha &= \dfrac{-k}{\sqrt{4-k^2}}, \quad \sec \alpha = \dfrac{-2}{\sqrt{4-k^2}}, \\
\csc \alpha &= \dfrac{2}{k}, \quad \cot \alpha = \dfrac{-\sqrt{4-k^2}}{k}
\end{aligned}

57. \(\sin \theta=0.3, \cos \theta=-0.4, \tan \theta=-0.75, \sec \theta=-2.5, \csc \theta \approx 3.33, \cot \theta \approx-1.33\)

59. -8

61. \(\sqrt{2}\)

63. \(\theta \approx 2.8, \theta \approx 0.30\)

65. \(y=\csc x\) or \(y=\cot x\)

67. \(y=\sec x\)

69. \(y=\sec x\) or \(y=\csc x\)

71. \(f(x)=\sin x-1\)

73. \(G(x)=\tan x-1\)

75. \(\cos ^2 x\)

77. \(\cos ^2 B\)

79. \(\csc \theta\)

81. \(\sqrt{3} \tan \theta \sin \theta\)

83.

a \(A C=\tan \alpha, D C=\tan \beta, A D=\tan \alpha-\tan \beta\)

b They are right triangles that share \(\angle B\).

c \(\angle A=\angle F, \angle B\) is the complement of \(\angle A\), and \(\angle F D C\) is the complement of \(\angle F\).

d \(\dfrac{C F}{C D}=\tan \alpha\), so \(C F=\tan \alpha \tan \beta\)

e They are right triangles with \(\angle A=\angle F\).

f \(\angle E B D=\alpha-\beta\), so \(\tan (\alpha-\beta)=\dfrac{\text { opp }}{\text { adj }}=\dfrac{D E}{B E} ; \dfrac{D E}{B E}\) and \(\dfrac{A D}{B F}\) are ratios of corresponding sides of similar triangles; \(A D=\tan \alpha-\tan \beta\) by part (a), \(B F=B C+C F=1+\tan \alpha \tan \beta\) by part (d).

85. \(d=25 \csc 112^{\circ}, \alpha=45^{\circ}, a \approx 19.07, b \approx 10.54\)