Jay Abramson

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9.1 Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions

1.

\begin{matrix} \csc θ \cos θ \tan θ & = & (\frac{1}{\sin θ}) \cos θ (\frac{\sin θ}{\cos θ}) \\ = & \frac{\cos θ}{\sin θ} (\frac{\sin θ}{\cos θ}) \\ = & \frac{\sin θ \cos θ}{\sin θ \cos θ} \\ = & 1 \end{matrix}

2.

\begin{matrix} \frac{\cot θ}{\csc θ} & = & \frac{\frac{\cos θ}{\sin θ}}{\frac{1}{\sin θ}} \\ = & \frac{\cos θ}{\sin θ} \cdot \frac{\sin θ}{1} \\ = & \cos θ \end{matrix}

3.

$\begin{matrix} \frac{\sin^{2} θ - 1}{\tan θ \sin θ - \tan θ} & = & \frac{(\sin θ + 1) (\sin θ - 1)}{\tan θ (\sin θ - 1)} \\ = & \frac{\sin θ + 1}{\tan θ} \end{matrix}$

4.

This is a difference of squares formula: $25 - 9 \sin^{2} θ = (5 - 3 \sin θ) (5 + 3 \sin θ) .$

5.

\begin{matrix} \frac{\cos θ}{1 + \sin θ} (\frac{1 - \sin θ}{1 - \sin θ}) & = & \frac{\cos θ (1 - \sin θ)}{1 - \sin^{2} θ} \\ = & \frac{\cos θ (1 - \sin θ)}{\cos^{2} θ} \\ = & \frac{1 - \sin θ}{\cos θ} \end{matrix}

9.2 Sum and Difference Identities

1.

$\frac{\sqrt{2} + \sqrt{6}}{4}$

2.

$\frac{\sqrt{2} - \sqrt{6}}{4}$

3.

$\frac{1 - \sqrt{3}}{1 + \sqrt{3}}$

4.

$\cos (\frac{5 π}{14})$

5.

\begin{matrix} \tan (π - θ) & = & \frac{\tan (π) - \tan θ}{1 + \tan (π) \tan θ} \\ = & \frac{0 - \tan θ}{1 + 0 \cdot \tan θ} \\ = & - \tan θ \end{matrix}

9.3 Double-Angle, Half-Angle, and Reduction Formulas

1.

$\cos (2 α) = \frac{7}{32}$

2.

$\cos^{4} θ - \sin^{4} θ = (\cos^{2} θ + \sin^{2} θ) (\cos^{2} θ - \sin^{2} θ) = \cos (2 θ)$

3.

$\cos (2 θ) \cos θ = (\cos^{2} θ - \sin^{2} θ) \cos θ = \cos^{3} θ - \cos θ \sin^{2} θ$

4.

$\begin{matrix} 10 \cos^{4} x & = & 10 {(\cos^{2} x)}^{2} \\ = & 10 {[\frac{1 + \cos (2 x)}{2}]}^{2} & {Substitute reduction formula for cos}^{2} x . \\ = & \frac{10}{4} [1 + 2 \cos (2 x) + \cos^{2} (2 x)] \\ = & \frac{10}{4} + \frac{10}{2} \cos (2 x) + \frac{10}{4} (\frac{1 + \cos 2 (2 x)}{2}) & {Substitute reduction formula for cos}^{2} x . \\ = & \frac{10}{4} + \frac{10}{2} \cos (2 x) + \frac{10}{8} + \frac{10}{8} \cos (4 x) \\ = & \frac{30}{8} + 5 \cos (2 x) + \frac{10}{8} \cos (4 x) \\ = & \frac{15}{4} + 5 \cos (2 x) + \frac{5}{4} \cos (4 x) \end{matrix}$

5.

$- \frac{2}{\sqrt{5}}$

9.4 Sum-to-Product and Product-to-Sum Formulas

1.

$\frac{1}{2} (\cos 6 θ + \cos 2 θ)$

2.

$\frac{1}{2} (\sin 2 x + \sin 2 y)$

3.

$\frac{- 2 - \sqrt{3}}{4}$

4.

$2 \sin (2 θ) \cos (θ)$

5.

$\begin{matrix} \tan θ \cot θ - \cos^{2} θ & = & (\frac{\sin θ}{\cos θ}) (\frac{\cos θ}{\sin θ}) - \cos^{2} θ \\ = & 1 - \cos^{2} θ \\ = & \sin^{2} θ \end{matrix}$

9.5 Solving Trigonometric Equations

1.

$x = \frac{7 π}{6}, \frac{11 π}{6}$

2.

$\frac{π}{3} \pm π k$

3.

$θ \approx 1.7722 \pm 2 π k$ and $θ \approx 4.5110 \pm 2 π k$

4.

$\cos θ = - 1, θ = π$

5.

$\frac{π}{2}, \frac{2 π}{3}, \frac{4 π}{3}, \frac{3 π}{2}$

9.1 Section Exercises

1.

All three functions, $F$ , $G$ , and $H$ , are even.

This is because $F (- x) = \sin (- x) \sin (- x) = (- \sin x) (- \sin x) = \sin^{2} x = F (x)$ , $G (- x) = \cos (- x) \cos (- x) = \cos x \cos x = \cos^{2} x = G (x)$ and $H (- x) = \tan (- x) \tan (- x) = (- \tan x) (- \tan x) = \tan^{2} x = H (x) .$

3.

When $\cos t = 0,$ then $\sec t = \frac{1}{0},$ which is undefined.

5.

$\sin x$

7.

$\sec x$

9.

$\csc t$

11.

$−1$

13.

$\sec^{2} x$

15.

$\sin^{2} x + 1$

17.

$\frac{1}{\sin x}$

19.

$\frac{1}{\cot x}$

21.

$\tan x$

23.

$- 4 \sec x \tan x$

25.

$\pm \sqrt{\frac{1}{\cot^{2} x} + 1}$

27.

$\frac{\pm \sqrt{1 - \sin^{2} x}}{\sin x}$

29.

Answers will vary. Sample proof:

$\begin{matrix} \cos x - \cos^{3} x & = & \cos x (1 - \cos^{2} x) \\ = & \cos x \sin^{2} x \end{matrix}$

31.

Answers will vary. Sample proof:
$\frac{1 + \sin^{2} x}{\cos^{2} x} = \frac{1}{\cos^{2} x} + \frac{\sin^{2} x}{\cos^{2} x} = \sec^{2} x + \tan^{2} x = \tan^{2} x + 1 + \tan^{2} x = 1 + 2 \tan^{2} x$

33.

Answers will vary. Sample proof:
$\cos^{2} x - \tan^{2} x = 1 - \sin^{2} x - (\sec^{2} x - 1) = 1 - \sin^{2} x - \sec^{2} x + 1 = 2 - \sin^{2} x - \sec^{2} x$

35.

False

37.

False

39.

Proved with negative and Pythagorean identities

41.

True $3 \sin^{2} θ + 4 \cos^{2} θ = 3 \sin^{2} θ + 3 \cos^{2} θ + \cos^{2} θ = 3 (\sin^{2} θ + \cos^{2} θ) + \cos^{2} θ = 3 + \cos^{2} θ$

9.2 Section Exercises

1.

The cofunction identities apply to complementary angles. Viewing the two acute angles of a right triangle, if one of those angles measures $x,$ the second angle measures $\frac{π}{2} - x .$ Then $\sin x = \cos (\frac{π}{2} - x) .$ The same holds for the other cofunction identities. The key is that the angles are complementary.

3.

$\sin (- x) = - \sin x,$ so $\sin x$ is odd. $\cos (- x) = \cos (0 - x) = \cos x,$ so $\cos x$ is even.

5.

$\frac{\sqrt{2} + \sqrt{6}}{4}$

7.

$\frac{\sqrt{6} - \sqrt{2}}{4}$

9.

$- 2 - \sqrt{3}$

11.

$- \frac{\sqrt{2}}{2} \sin x - \frac{\sqrt{2}}{2} \cos x$

13.

$- \frac{1}{2} \cos x - \frac{\sqrt{3}}{2} \sin x$

15.

$\csc θ$

17.

$\cot x$

19.

$\tan (\frac{x}{10})$

21.

$\begin{matrix} \sin (a - b) & = & (\frac{4}{5}) (\frac{1}{3}) - (\frac{3}{5}) (\frac{2 \sqrt{2}}{3}) & = & \frac{4 - 6 \sqrt{2}}{15} \\ \cos (a + b) & = & (\frac{3}{5}) (\frac{1}{3}) - (\frac{4}{5}) (\frac{2 \sqrt{2}}{3}) & = & \frac{3 - 8 \sqrt{2}}{15} \end{matrix}$

23.

$\frac{\sqrt{2} - \sqrt{6}}{4}$

25.

$\sin x$

27.

$\cot (\frac{π}{6} - x)$

Graph of y=cot(pi/6 - x) from -2pi to pi - in comparison to the usual y=cot(x) graph, this one is reflected across the x-axis and shifted by pi/6.

29.

$\cot (\frac{π}{4} + x)$

Graph of y=cot(pi/4 + x) - in comparison to the usual y=cot(x) graph, this one is shifted by pi/4.

31.

$\frac{\sin x}{\sqrt{2}} + \frac{\cos x}{\sqrt{2}}$

Graph of y = sin(x) / rad2 + cos(x) / rad2 - it looks like the sin curve shifted by pi/4.

33.

They are the same.

35.

They are the different, try $g (x) = \sin (9 x) - \cos (3 x) \sin (6 x) .$

37.

They are the same.

39.

They are the different, try $g (θ) = \frac{2 \tan θ}{1 - \tan^{2} θ} .$

41.

They are different, try $g (x) = \frac{\tan x - \tan (2 x)}{1 + \tan x \tan (2 x)} .$

43.

$- \frac{\sqrt{3} - 1}{2 \sqrt{2}}, or - 0.2588$

45.

$\frac{1 + \sqrt{3}}{2 \sqrt{2}},$ or 0.9659

47.

$\begin{matrix} \tan (x + \frac{π}{4}) & = \\ \frac{\tan x + \tan (\frac{π}{4})}{1 - \tan x \tan (\frac{π}{4})} & = \\ \frac{\tan x + 1}{1 - \tan x (1)} & = & \frac{\tan x + 1}{1 - \tan x} \end{matrix}$

49.

$\begin{matrix} \frac{\cos (a + b)}{\cos a \cos b} & = \\ \frac{\cos a \cos b}{\cos a \cos b} - \frac{\sin a \sin b}{\cos a \cos b} & = & 1 - \tan a \tan b \end{matrix}$

51.

$\begin{matrix} \frac{\cos (x + h) - \cos x}{h} & = \\ \frac{\cos x \cosh - \sin x \sinh - \cos x}{h} & = \\ \frac{\cos x (\cosh - 1) - \sin x \sinh}{h} & = & \cos x \frac{\cos h - 1}{h} - \sin x \frac{\sin h}{h} \end{matrix}$

53.

True

55.

True. Note that $\sin (α + β) = \sin (π - γ)$ and expand the right hand side.

9.3 Section Exercises

1.

Use the Pythagorean identities and isolate the squared term.

3.

$\frac{1 - \cos x}{\sin x}, \frac{\sin x}{1 + \cos x},$ multiplying the top and bottom by $\sqrt{1 - \cos x}$ and $\sqrt{1 + \cos x},$ respectively.

5.

a) $\frac{3 \sqrt{7}}{32}$ b) $\frac{31}{32}$ c) $\frac{3 \sqrt{7}}{31}$

7.

a) $\frac{\sqrt{3}}{2}$ b) $- \frac{1}{2}$ c) $- \sqrt{3}$

9.

$\cos θ = - \frac{2 \sqrt{5}}{5}, \sin θ = \frac{\sqrt{5}}{5}, \tan θ = - \frac{1}{2}, \csc θ = \sqrt{5}, \sec θ = - \frac{\sqrt{5}}{2}, \cot θ = - 2$

11.

$2 \sin (\frac{π}{2})$

13.

$\frac{\sqrt{2 - \sqrt{2}}}{2}$

15.

$\frac{\sqrt{2 - \sqrt{3}}}{2}$

17.

$2 + \sqrt{3}$

19.

$- 1 - \sqrt{2}$

21.

a) $\frac{3 \sqrt{13}}{13}$ b) $- \frac{2 \sqrt{13}}{13}$ c) $- \frac{3}{2}$

23.

a) $\frac{\sqrt{10}}{4}$ b) $\frac{\sqrt{6}}{4}$ c) $\frac{\sqrt{15}}{3}$

25.

$\frac{120}{169}, - \frac{119}{169}, - \frac{120}{119}$

27.

$\frac{2 \sqrt{13}}{13}, \frac{3 \sqrt{13}}{13}, \frac{2}{3}$

29.

$\cos (74°)$

31.

$\cos (18 x)$

33.

$3 \sin (10 x)$

35.

$- 2 \sin (- x) \cos (- x) = - 2 (- \sin (x) \cos (x)) = \sin (2 x)$

37.

$\begin{matrix} \frac{\sin (2 θ)}{1 + \cos (2 θ)} \tan^{2} θ & = & \frac{2 \sin (θ) \cos (θ)}{1 + \cos^{2} θ - \sin^{2} θ} \tan^{2} θ = \\ \frac{2 \sin (θ) \cos (θ)}{2 \cos^{2} θ} \tan^{2} θ & = & \frac{\sin (θ)}{\cos θ} \tan^{2} θ = \\ \cot (θ) \tan^{2} θ & = & \tan^{3} θ \end{matrix}$

39.

$\frac{1 + \cos (12 x)}{2}$

41.

$\frac{3 + \cos (12 x) - 4 \cos (6 x)}{8}$

43.

$\frac{2 + \cos (2 x) - 2 \cos (4 x) - \cos (6 x)}{32}$

45.

$\frac{3 + \cos (4 x) - 4 \cos (2 x)}{3 + \cos (4 x) + 4 \cos (2 x)}$

47.

$\frac{1 - \cos (4 x)}{8}$

49.

$\frac{3 + \cos (4 x) - 4 \cos (2 x)}{4 (\cos (2 x) + 1)}$

51.

$\frac{(1 + \cos (4 x)) \sin x}{2}$

53.

$4 \sin x \cos x (\cos^{2} x - \sin^{2} x)$

55.

$\begin{array}{l} \frac{2 \tan x}{1 + \tan^{2} x} = \frac{\frac{2 \sin x}{\cos x}}{1 + \frac{\sin^{2} x}{\cos^{2} x}} = \frac{\frac{2 \sin x}{\cos x}}{\frac{\cos^{2} x + \sin^{2} x}{\cos^{2} x}} = \\ \frac{2 \sin x}{\cos x} . \frac{\cos^{2} x}{1} = 2 \sin x \cos x = \sin (2 x) \end{array}$

57.

$\frac{2 \sin x \cos x}{2 \cos^{2} x - 1} = \frac{\sin (2 x)}{\cos (2 x)} = \tan (2 x)$

59.

$\begin{matrix} \sin (x + 2 x) & = & \sin x \cos (2 x) + \sin (2 x) \cos x \\ = & \sin x (\cos^{2} x - \sin^{2} x) + 2 \sin x \cos x \cos x \\ = & \sin x \cos^{2} x - \sin^{3} x + 2 \sin x \cos^{2} x \\ = & 3 \sin x \cos^{2} x - \sin^{3} x \end{matrix}$

61.

$\begin{matrix} \frac{1 + \cos (2 t)}{\sin (2 t) - \cos t} & = & \frac{1 + 2 \cos^{2} t - 1}{2 \sin t \cos t - \cos t} \\ = & \frac{2 \cos^{2} t}{\cos t (2 \sin t - 1)} \\ = & \frac{2 \cos t}{2 \sin t - 1} \end{matrix}$

63.

$\begin{matrix} (\cos^{2} (4 x) - \sin^{2} (4 x) - \sin (8 x)) (\cos^{2} (4 x) - \sin^{2} (4 x) + \sin (8 x)) & = \\ = & (\cos (8 x) - \sin (8 x)) (\cos (8 x) + \sin (8 x)) \\ = & \cos^{2} (8 x) - \sin^{2} (8 x) \\ = & \cos (16 x) \end{matrix}$

9.4 Section Exercises

1.

Substitute $α$ into cosine and $β$ into sine and evaluate.

3.

Answers will vary. There are some equations that involve a sum of two trig expressions where when converted to a product are easier to solve. For example: $\frac{\sin (3 x) + \sin x}{\cos x} = 1.$ When converting the numerator to a product the equation becomes: $\frac{2 \sin (2 x) \cos x}{\cos x} = 1$

5.

$8 (\cos (5 x) - \cos (27 x))$

7.

$\sin (2 x) + \sin (8 x)$

9.

$\frac{1}{2} (\cos (6 x) - \cos (4 x))$

11.

$2 \cos (5 t) \cos t$

13.

$2 \cos (7 x)$

15.

$2 \cos (6 x) \cos (3 x)$

17.

$\frac{1}{4} (1 + \sqrt{3})$

19.

$\frac{1}{4} (\sqrt{3} - 2)$

21.

$\frac{1}{4} (\sqrt{3} - 1)$

23.

$\cos (80°) - \cos (120°)$

25.

$\frac{1}{2} (\sin (221°) + \sin (205°))$

27.

$\sqrt{2} \cos (31°)$

29.

$2 \cos (66.5°) \sin (34.5°)$

31.

$2 \sin (−1.5°) \cos (0.5°)$

33.

$\begin{array}{l} 2 \sin (7 x) - 2 \sin x = 2 \sin (4 x + 3 x) - 2 \sin (4 x - 3 x) = \\ 2 (\sin (4 x) \cos (3 x) + \sin (3 x) \cos (4 x)) - 2 (\sin (4 x) \cos (3 x) - \sin (3 x) \cos (4 x)) = \\ 2 \sin (4 x) \cos (3 x) + 2 \sin (3 x) \cos (4 x)) - 2 \sin (4 x) \cos (3 x) + 2 \sin (3 x) \cos (4 x)) = \\ 4 \sin (3 x) \cos (4 x) \end{array}$

35.

$\begin{matrix} \sin x + \sin (3 x) & = & 2 \sin (\frac{4 x}{2}) \cos (\frac{- 2 x}{2}) = \\ 2 \sin (2 x) \cos x & = & 2 (2 \sin x \cos x) \cos x = \\ 4 \sin x \cos^{2} x \end{matrix}$

37.

$\begin{array}{l} 2 \tan x \cos (3 x) = \frac{2 \sin x \cos (3 x)}{\cos x} = \frac{2 (.5 (\sin (4 x) - \sin (2 x)))}{\cos x} = \\ \frac{1}{\cos x} (\sin (4 x) - \sin (2 x)) = \sec x (\sin (4 x) - \sin (2 x)) \end{array}$

39.

$2 \cos (35°) \cos (23°), 1.5081$

41.

$- 2 \sin (33°) \sin (11°), - 0.2078$

43.

$\frac{1}{2} (\cos (99°) - \cos (71°)), −0.2410$

45.

It is an identity.

47.

It is not an identity, but $2 \cos^{3} x$ is.

49.

$\tan (3 t)$

51.

$2 \cos (2 x)$

53.

$- \sin (14 x)$

55.

Start with $\cos x + \cos y .$ Make a substitution and let $x = α + β$ and let $y = α - β,$ so $\cos x + \cos y$ becomes $\begin{array}{l} \cos (α + β) + \cos (α - β) = \cos α \cos β - \sin α \sin β + \cos α \cos β + \sin α \sin β = \\ 2 \cos α \cos β \end{array}$

Since $x = α + β$ and $y = α - β,$ we can solve for $α$ and $β$ in terms of x and y and substitute in for $2 \cos α \cos β$ and get $2 \cos (\frac{x + y}{2}) \cos (\frac{x - y}{2}) .$

57.

$\frac{\cos (3 x) + \cos x}{\cos (3 x) - \cos x} = \frac{2 \cos (2 x) \cos x}{- 2 \sin (2 x) \sin x} = - \cot (2 x) \cot x$

59.

$\begin{matrix} \frac{\cos (2 y) - \cos (4 y)}{\sin (2 y) + \sin (4 y)} & = & \frac{- 2 \sin (3 y) \sin (- y)}{2 \sin (3 y) \cos y} = \\ \frac{2 \sin (3 y) \sin (y)}{2 \sin (3 y) \cos y} & = & \tan y \end{matrix}$

61.

$\begin{array}{l} \cos x - \cos (3 x) = - 2 \sin (2 x) \sin (- x) = \\ 2 (2 \sin x \cos x) \sin x = 4 \sin^{2} x \cos x \end{array}$

63.

$\tan (\frac{π}{4} - t) = \frac{\tan (\frac{π}{4}) - \tan t}{1 + \tan (\frac{π}{4}) \tan (t)} = \frac{1 - \tan t}{1 + \tan t}$

9.5 Section Exercises

1.

There will not always be solutions to trigonometric function equations. For a basic example, $\cos (x) = −5.$

3.

If the sine or cosine function has a coefficient of one, isolate the term on one side of the equals sign. If the number it is set equal to has an absolute value less than or equal to one, the equation has solutions, otherwise it does not. If the sine or cosine does not have a coefficient equal to one, still isolate the term but then divide both sides of the equation by the leading coefficient. Then, if the number it is set equal to has an absolute value greater than one, the equation has no solution.

5.

$\frac{π}{3}, \frac{2 π}{3}$

7.

$\frac{3 π}{4}, \frac{5 π}{4}$

9.

$\frac{π}{4}, \frac{5 π}{4}$

11.

$\frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4}$

13.

$\frac{π}{4}, \frac{7 π}{4}$

15.

$\frac{7 π}{6}, \frac{11 π}{6}$

17.

$\frac{π}{18}, \frac{5 π}{18}, \frac{13 π}{18}, \frac{17 π}{18}, \frac{25 π}{18}, \frac{29 π}{18}$

19.

$\frac{3 π}{12}, \frac{5 π}{12}, \frac{11 π}{12}, \frac{13 π}{12}, \frac{19 π}{12}, \frac{21 π}{12}$

21.

$\frac{1}{6}, \frac{5}{6}, \frac{13}{6}, \frac{17}{6}, \frac{25}{6}, \frac{29}{6}, \frac{37}{6}$

23.

$0, \frac{π}{3}, π, \frac{5 π}{3}$

25.

$\frac{π}{3}, π, \frac{5 π}{3}$

27.

$\frac{π}{3}, \frac{3 π}{2}, \frac{5 π}{3}$

29.

$0, π$

31.

$π - \sin^{- 1} (- \frac{1}{4}), \frac{7 π}{6}, \frac{11 π}{6}, 2 π + \sin^{- 1} (- \frac{1}{4})$

33.

$\frac{1}{3} (\sin^{- 1} (\frac{9}{10}))$ , $\frac{π}{3} - \frac{1}{3} (\sin^{- 1} (\frac{9}{10}))$ , $\frac{2 π}{3} + \frac{1}{3} (\sin^{- 1} (\frac{9}{10}))$ , $π - \frac{1}{3} (\sin^{- 1} (\frac{9}{10}))$ , $\frac{4 π}{3} + \frac{1}{3} (\sin^{- 1} (\frac{9}{10}))$ , $\frac{5 π}{3} - \frac{1}{3} (\sin^{- 1} (\frac{9}{10}))$

35.

$0$

37.

$\frac{π}{6}, \frac{5 π}{6}, \frac{7 π}{6}, \frac{11 π}{6}$

39.

$\frac{3 π}{2}, \frac{π}{6}, \frac{5 π}{6}$

41.

$0, \frac{π}{3}, π, \frac{4 π}{3}$

43.

There are no solutions.

45.

$\cos^{- 1} (\frac{1}{3} (1 - \sqrt{7}))$ , $2 π - \cos^{- 1} (\frac{1}{3} (1 - \sqrt{7}))$

47.

$\tan^{- 1} (\frac{1}{2} (\sqrt{29} - 5))$ , $π + \tan^{- 1} (\frac{1}{2} (- \sqrt{29} - 5))$ , $π + \tan^{- 1} (\frac{1}{2} (\sqrt{29} - 5))$ , $2 π + \tan^{- 1} (\frac{1}{2} (- \sqrt{29} - 5))$

49.

There are no solutions.

51.

There are no solutions.

53.

$0, \frac{2 π}{3}, \frac{4 π}{3}$

55.

$\frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4}$

57.

$\sin^{- 1} (\frac{3}{5}), \frac{π}{2}, π - \sin^{- 1} (\frac{3}{5}), \frac{3 π}{2}$

59.

$\cos^{- 1} (- \frac{1}{4}), 2 π - \cos^{- 1} (- \frac{1}{4})$

61.

$\frac{π}{3}$ , $\cos^{- 1} (- \frac{3}{4})$ , $2 π - \cos^{- 1} (- \frac{3}{4})$ , $\frac{5 π}{3}$

63.

$\cos^{- 1} (\frac{3}{4})$ , $\cos^{- 1} (- \frac{2}{3})$ , $2 π - \cos^{- 1} (- \frac{2}{3})$ , $2 π - \cos^{- 1} (\frac{3}{4})$

65.

$0, \frac{π}{2}, π, \frac{3 π}{2}$

67.

$\frac{π}{3}$ , $\cos^{−1} (- \frac{1}{4})$ , $2 π - \cos^{−1} (- \frac{1}{4})$ , $\frac{5 π}{3}$

69.

There are no solutions.

71.

$π + \tan^{−1} (−2)$ , $π + \tan^{−1} (- \frac{3}{2})$ , $2 π + \tan^{−1} (−2)$ , $2 π + \tan^{−1} (- \frac{3}{2})$

73.

$2 π k + 0.2734, 2 π k + 2.8682$

75.

$π k - 0.3277$

77.

$0.6694, 1.8287, 3.8110, 4.9703$

79.

$1.0472, 3.1416, 5.2360$

81.

$0.5326, 1.7648, 3.6742, 4.9064$

83.

$\sin^{- 1} (\frac{1}{4}), π - \sin^{- 1} (\frac{1}{4}), \frac{3 π}{2}$

85.

$\frac{π}{2}, \frac{3 π}{2}$

87.

There are no solutions.

89.

$0, \frac{π}{2}, π, \frac{3 π}{2}$

91.

There are no solutions.

93.

${7.2}^{\circ}$

95.

${5.7}^{\circ}$

97.

${82.4}^{\circ}$

99.

${31.0}^{\circ}$

101.

${88.7}^{\circ}$

103.

${59.0}^{\circ}$

105.

${36.9}^{\circ}$

Review Exercises

1.

$\sin^{- 1} (\frac{\sqrt{3}}{3})$ , $π - \sin^{- 1} (\frac{\sqrt{3}}{3})$ , $π + \sin^{- 1} (\frac{\sqrt{3}}{3})$ , $2 π - \sin^{- 1} (\frac{\sqrt{3}}{3})$

3.

$\frac{7 π}{6}, \frac{11 π}{6}$

5.

$\sin^{- 1} (\frac{1}{4}), π - \sin^{- 1} (\frac{1}{4})$

7.

$1$

9.

Yes

11.

$- 2 - \sqrt{3}$

13.

$\frac{\sqrt{2}}{2}$

15.

$\begin{matrix} \cos (4 x) - \cos (3 x) \cos x & = & \cos (2 x + 2 x) - \cos (x + 2 x) \cos x \\ = & \cos (2 x) \cos (2 x) - \sin (2 x) \sin (2 x) - \cos x \cos (2 x) \cos x + \sin x \sin (2 x) \cos x \\ = & {(\cos^{2} x - \sin^{2} x)}^{2} - 4 \cos^{2} x \sin^{2} x - \cos^{2} x (\cos^{2} x - \sin^{2} x) + \sin x (2) \sin x \cos x \cos x \\ = & {(\cos^{2} x - \sin^{2} x)}^{2} - 4 \cos^{2} x \sin^{2} x - \cos^{2} x (\cos^{2} x - \sin^{2} x) + 2 \sin^{2} x \cos^{2} x \\ = & \cos^{4} x - 2 \cos^{2} x \sin^{2} x + \sin^{4} x - 4 \cos^{2} x \sin^{2} x - \cos^{4} x + \cos^{2} x \sin^{2} x + 2 \sin^{2} x \cos^{2} x \\ = & \sin^{4} x - 4 \cos^{2} x \sin^{2} x + \cos^{2} x \sin^{2} x \\ = & \sin^{2} x (\sin^{2} x + \cos^{2} x) - 4 \cos^{2} x \sin^{2} x \\ = & \sin^{2} x - 4 \cos^{2} x \sin^{2} x \end{matrix}$

17.

$\tan (\frac{5}{8} x)$

19.

$\frac{\sqrt{3}}{3}$

21.

$- \frac{24}{25}, - \frac{7}{25}, \frac{24}{7}$

23.

$\sqrt{2 (2 + \sqrt{2})}$

25.

$\frac{\sqrt{2}}{10}, \frac{7 \sqrt{2}}{10}, \frac{1}{7}, \frac{3}{5}, \frac{4}{5}, \frac{3}{4}$

27.

$\begin{matrix} \cot x \cos (2 x) & = & \cot x (1 - 2 \sin^{2} x) \\ = & \cot x - \frac{\cos x}{\sin x} (2) \sin^{2} x \\ = & - 2 \sin x \cos x + \cot x \\ = & - \sin (2 x) + \cot x \end{matrix}$

29.

$\frac{10 \sin x - 5 \sin (3 x) + \sin (5 x)}{8 (\cos (2 x) + 1)}$

31.

$\frac{\sqrt{3}}{2}$

33.

$- \frac{\sqrt{2}}{2}$

35.

$\frac{1}{2} (\sin (6 x) + \sin (12 x))$

37.

$2 \sin (\frac{13}{2} x) \cos (\frac{9}{2} x)$

39.

$\frac{3 π}{4}, \frac{7 π}{4}$

41.

$0, \frac{π}{6}, \frac{5 π}{6}, π$

43.

$\frac{3 π}{2}$

45.

No solution

47.

$0.2527, 2.8889, 4.7124$

49.

$1.3694, 1.9106, 4.3726, 4.9137$

Practice Test

1.

1

3.

$\sec (θ)$

5.

$\frac{\sqrt{2} - \sqrt{6}}{4}$

7.

$- 2 - \sqrt{3}$

9.

$- \frac{1}{2} \cos θ + \frac{\sqrt{3}}{2} \sin θ$

11.

$\frac{1 - \cos (64^{\circ})}{2}$

13.

$0, π$

15.

$\frac{π}{2}, \frac{3 π}{2}$

17.

$2 cos (3 x) cos (5 x)$

19.

$4 \sin (2 θ) \cos (6 θ)$

21.

$x = {cos}^{–1} (\frac{1}{5})$

23.

$\frac{π}{3}$

25.

$\frac{3}{5}, - \frac{4}{5}, - \frac{3}{4}$

27.

$\begin{matrix} {tan}^{3} x - tan x {sec}^{2} x & = & tan x ({tan}^{2} x - {sec}^{2} x) \\ = & tan x ({tan}^{2} x - (1 + {tan}^{2} x)) \\ = & tan x ({tan}^{2} x - 1 - {tan}^{2} x) \\ = & - tan x = tan (- x) = tan (- x) \end{matrix}$

29.

$\begin{matrix} \frac{sin (2 x)}{sin x} - \frac{cos (2 x)}{cos x} & = & \frac{2 sin x cos x}{sin x} - \frac{2 {cos}^{2} x - 1}{cos x} \\ = & 2 cos x - 2 cos x + \frac{1}{cos x} \\ = & \frac{1}{cos x} = sec x = sec x \end{matrix}$

31.

Amplitude: $\frac{1}{4}$ , period: $\frac{1}{60}$ , frequency: 60 Hz

33.

Amplitude: 8, fast period: $\frac{1}{500}$ , fast frequency: 500 Hz, slow period: $\frac{1}{10}$ , slow frequency: 10 Hz

35.

$D (t) = 20 (0.9086)^{t} cos (4 π t)$ , 31 second

Chapter 9

Try It

9.1 Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions

9.2 Sum and Difference Identities

9.3 Double-Angle, Half-Angle, and Reduction Formulas

9.4 Sum-to-Product and Product-to-Sum Formulas

9.5 Solving Trigonometric Equations

9.1 Section Exercises

9.2 Section Exercises

9.3 Section Exercises

9.4 Section Exercises

9.5 Section Exercises

Review Exercises

Practice Test