Chapter 7

Try It

7.1 Simplifying and Verifying Trigonometric Identities

\begin{array}{l} \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{array}

\begin{array}{l} \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{array}

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

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

\begin{array}{l} \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{array}

7.2 Sum and Difference Identities

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

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

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

$\cos (\frac{5 π}{14})$

\begin{array}{l} \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{array}

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

$\cos (2 α) = \frac{7}{32}$

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

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

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

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

7.4 Sum-to-Product and Product-to-Sum Formulas

$\frac{1}{2} (\cos 6 θ + \cos 2 θ)$

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

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

$2 \sin (2 θ) \cos (θ)$

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

7.5 Solving Trigonometric Equations

$x = \frac{7 π}{6}, \frac{11 π}{6}$

$\frac{π}{3} \pm π k$

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

$\cos θ = - 1, θ = π$

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

7.6 Modeling with Trigonometric Functions

The amplitude is $3,$ and the period is $\frac{2}{3} .$

x	$3 \sin (3 x)$
0	0
$\frac{π}{6}$	3
$\frac{π}{3}$	0
$\frac{π}{2}$	$−3$
$\frac{2 π}{3}$	0

Graph of y=3sin(3x) using the five key points: intervals of equal length representing 1/4 of the period. Here, the points are at 0, pi/6, pi/3, pi/2, and 2pi/3.

$y = 8 \sin (\frac{π}{12} t) + 32$
The temperature reaches freezing at noon and at midnight.

Graph of the function y=8sin(pi/12 t) + 32 for temperature. The midline is at 32. The times when the temperature is at 32 are midnight and noon.

initial displacement =6, damping constant = -6, frequency = $\frac{2}{π}$

$y = 10 e^{- 0.5 t} \cos (π t)$

$y = 5 \cos (6 π t)$

7.1 Section Exercises

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 $H (- x) = \tan (- x) \tan (- x) = (- \tan x) (- \tan x) = \tan^{2} x = H (x) .$

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

$\sin x$

$\sec x$

$\csc t$

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:

$\cos x - \cos^{3} x = \cos x (1 - \cos^{2} x)$
$= \cos x \sin^{2} x$

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

7.2 Section Exercises

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.

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

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

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

$- 2 - \sqrt{3}$

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.

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

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.

7.3 Section Exercises

Use the Pythagorean identities and isolate the squared term.

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

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

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

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

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.

$\cos (74^{\circ})$

31.

$\cos (18 x)$

33.

$3 \sin (10 x)$

35.

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

37.

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

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.

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

57.

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

59.

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

61.

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

63.

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

7.4 Section Exercises

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

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: $\frac{2 \sin (2 x) \cos x}{\cos x} = 1$

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

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

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

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.

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

37.

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

39.

$2 \cos (35^{\circ}) \cos (23^{\circ}), 1 .5081$

41.

$- 2 \sin (33^{\circ}) \sin (11^{\circ}), - 0.2078$

43.

$\frac{1}{2} (\cos (99^{\circ}) - \cos (71^{\circ})), - 0.2410$

45.

It is and 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
$\cos (α + β) + \cos (α - β) = \cos α \cos β - \sin α \sin β + \cos α \cos β + \sin α \sin β = 2 \cos α \cos β$

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.

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

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.

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

7.5 Section Exercises

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

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.

$\frac{π}{3}, \frac{2 π}{3}$

$\frac{3 π}{4}, \frac{5 π}{4}$

$\frac{π}{4}, \frac{5 π}{4}$

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.

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

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.

${36.9}^{\circ}$

7.6 Section Exercises

Physical behavior should be periodic, or cyclical.

Since cumulative rainfall is always increasing, a sinusoidal function would not be ideal here.

$y = - 3 \cos (\frac{π}{6} x) - 1$

$5 \sin (2 x) + 2$

$y = 4 - 6 \cos (\frac{x π}{2})$

10.

$y = \tan (\frac{x π}{8})$

12.

$\tan (\frac{x π}{12})$

13.

A line graph with an x-axis ranging from 0 to 12 and a y-axis ranging from 0 to 120. The curve begins at approximately (0, 30), rises to a plateau around y=90 between x=4 and x=8, and then descends to approximately (12, 30).

15.

A line graph with an x-axis labeled from 0 to 80, marked in increments of 8. The y-axis is labeled from 0 to 60, marked in increments of 20. A dark blue wavy line starts near y=10 at x=0 and generally increases, with oscillations, reaching approximately y=50 at x=80.

17.

75 °F

19.

8 a.m.

21.

2:49

23.

From June 15 through November 16

25.

From day 41 through day 68

27.

Floods: April 16 to July 15. Drought: October 16 to January 15.

29.

Amplitude: 8, period: $\frac{1}{3},$ frequency: 3 Hz

31.

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

33.

$P (t) = - 19 \cos (\frac{π}{6} t) + 800 + \frac{160}{12} t P (t) = - 19 \cos (\frac{π}{6} t) + 800 + \frac{40}{3} t$

35.

$P (t) = - 33 \cos (\frac{π}{6} t) + 900 + {(1.07)}^{t}$

37.

$D (t) = 10 {(0.85)}^{t} \cos (36 π t)$

39.

$D (t) = 17 {(0.9145)}^{t} \cos (28 π t)$

41.

6 years

43.

15.4 seconds

45.

Spring 2 comes to rest first after 7.3 seconds.

47.

234.3 miles, at 72.2°

49.

$y = 6 {(4)}^{x} + 5 \sin (\frac{π}{2} x)$

51.

$y = 4 {(- 2)}^{x} + 8 \sin (\frac{π}{2} x)$

53.

$y = 3 {(2)}^{x} \cos (\frac{π}{2 x}) + 1$

Review Exercises

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

$\frac{7 π}{6}, \frac{11 π}{6}$

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

$1$

Yes

11.

$- 2 - \sqrt{3}$

13.

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

15.

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

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.

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

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.

$1.3694$ , $1.9106$ , $4.3726$ , $4.9137$

51.

$3 \sin (\frac{x π}{2}) - 2$

53.

${71.6}^{\circ}$

55.

$P (t) = 950 - 450 \sin (\frac{π}{6} t)$

57.

Amplitude: 3, period: 2, frequency: $\frac{1}{2}$ Hz

59.

$C (t) = 20 \sin (2 π t) + 100 {(1.4427)}^{t}$

Practice Test

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

$- 2 - \sqrt{3}$

$0, π$

$\frac{π}{2}, \frac{3π}{2}$

11.

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

13.

$x = {cos}^{–1} (\frac{1}{5})$

15.

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

17.

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

19.

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

21.

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

23.

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

25.

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