1.
[latex]\sin \left(\beta + \dfrac{\pi}{4}\right)=\sin \beta + \dfrac{1}{\sqrt{2}}[/latex]
Chapter 8: More Functions and Identities
Exercises: Chapter 8 Review Problems
Chapter Review Suggested Problems
Problems: #6, 14, 18, 24, 26, 30, 36, 42, 48, 56, 62, 64, 72, 76, 80
Exercises for Chapter 8 Review
Exercise Group
For Problems 1–8, answer true or false.
2.
[latex]\cos \left(\dfrac{\pi}{3} - t\right)=\dfrac{1}{2}-\cos t[/latex]
3.
[latex]\tan (z-w)=\dfrac{\sin (z-w)}{\cos (z-w)}[/latex]
4.
[latex]\sin 2\phi = 1 - \cos 2\phi[/latex]
5.
[latex]\sin \left(\dfrac{\pi}{2}-x\right)=1-\sin x[/latex]
6.
[latex]\sin (\pi - x)=\sin x[/latex]
7.
[latex]\cos^2 \alpha - \sin^2 \alpha = -1[/latex]
8.
[latex]\tan^{-1} s = \dfrac{1}{\tan s}[/latex]
9.
If [latex]\sin x = -0.4[/latex] and [latex]\cos x \gt 0,[/latex] find an exact value for [latex]\cos \left(x + \dfrac{3\pi}{4}\right)\text{.}[/latex]
10.
If [latex]\cos x = -0.75[/latex] and [latex]\sin x \lt 0,[/latex] find an exact value for [latex]\cos \left(x - \dfrac{4\pi}{3}\right)\text{.}[/latex]
11.
If [latex]\cos \theta = \dfrac{-3}{8},~ \pi \lt \theta \lt \dfrac{3\pi}{2},[/latex] and [latex]\sin \phi = \dfrac{1}{4},~ \dfrac{\pi}{2} \lt \phi \lt \pi,[/latex] find exact values for
- [latex]\displaystyle \sin (\theta + \phi)[/latex]
- [latex]\displaystyle \tan (\theta + \phi)[/latex]
12.
If [latex]\sin \rho = \dfrac{5}{6},~ \dfrac{\pi}{2} \lt \rho \lt \pi,[/latex] and [latex]\cos \mu = \dfrac{-1}{3},~ \dfrac{\pi}{2} \lt \mu \lt \pi,[/latex] find exact values for
- [latex]\displaystyle \cos (\rho - \mu)[/latex]
- [latex]\displaystyle \tan (\rho - \mu)[/latex]
13.
If [latex]\tan (x + y) =2[/latex] and [latex]\tan y = \dfrac{1}{3},[/latex] find [latex]\tan x\text{.}[/latex]
14.
If [latex]\tan (x - y) =\dfrac{1}{4}[/latex] and [latex]\tan x = 4,[/latex] find [latex]\tan y\text{.}[/latex]
Exercise Group
For Problems 15-16, use the sum and difference formulas to expand each expression.
15.
[latex]\tan \left(t - \dfrac{5\pi}{3}\right)[/latex]
16.
[latex]\cos \left(s+ \dfrac{7\pi}{4}\right)[/latex]
Exercise Group
For Problems 17–18, use the figure to find the trigonometric ratios.
17.
- [latex]\displaystyle \sin \theta[/latex]
- [latex]\displaystyle \cos \theta[/latex]
- [latex]\displaystyle \tan \theta[/latex]
- [latex]\displaystyle \sin 2\theta[/latex]
- [latex]\displaystyle \cos 2\theta[/latex]
- [latex]\displaystyle \tan 2\theta[/latex]
18.
- [latex]\displaystyle \sin \phi[/latex]
- [latex]\displaystyle \cos \phi[/latex]
- [latex]\displaystyle \tan \phi[/latex]
- [latex]\displaystyle \sin 2\phi[/latex]
- [latex]\displaystyle \cos 2\phi[/latex]
- [latex]\displaystyle \tan 2\phi[/latex]
Exercise Group
For Problems 19–24, use identities to simplify each expression.
19.
[latex]\sin 4x \cos 5x + \cos 4x \sin 5x[/latex]
20.
[latex]\cos 3\beta \cos 1.5 - \sin 3\beta \sin 1.5[/latex]
21.
[latex]\dfrac{\tan 2\phi - \tan 2}{1 + \tan 2\phi \tan 2}[/latex]
22.
[latex]\dfrac{\tan \dfrac{5\pi}{9} - \tan \dfrac{2\pi}{9}}{1 + \tan \dfrac{5\pi}{9} \tan \dfrac{2\pi}{9}}[/latex]
23.
[latex]2\sin 4\theta \cos 4\theta[/latex]
24.
[latex]1-2\sin^2 3\phi[/latex]
Exercise Group
For Problems 25–26,
- Use identities to rewrite the equation in terms of a single angle.
- Solve. Give exact solutions between [latex]0[/latex] and [latex]2\pi\text{.}[/latex]
25.
[latex]\cos 2\theta - \sin \theta = 1[/latex]
26.
[latex]\tan 2z + \tan z = 0[/latex]
Exercise Group
For Problems 27–28, graph the function and decide if it has an inverse function.
27.
[latex]f(x) = 4x - x^3[/latex]
28.
[latex]g(x) = 5 + \sqrt[3]{x - 2}[/latex]
Exercise Group
For Problems 29–30, give exact values in radians.
29.
- [latex]\displaystyle \tan^{-1}(-\sqrt{3})[/latex]
- [latex]\displaystyle \arccos \left(-\dfrac{1}{2}\right)[/latex]
30.
- [latex]\displaystyle \arcsin (-1)[/latex]
- [latex]\displaystyle \cos^{-1}(-1)[/latex]
31.
An IMAX movie screen is 52.8 feet high.
- If your line of sight is level with the bottom of the screen, write an expression for the angle subtended by the screen when you sit [latex]x[/latex] feet away.
- Evaluate your expression for [latex]x = 20[/latex] feet and for [latex]x = 100[/latex] feet.
32.
Rembrandt’s painting The Night Watch measures 13 feet high by 16 feet wide.
- Write an expression for the angle subtended by the width of the painting if you sit [latex]d[/latex] feet back from the center of the painting.
- Evaluate your expression for [latex]d = 10[/latex] feet and for [latex]d = 25[/latex] feet.
Exercise Group
For Problems 33–34, solve for [latex]\theta\text{.}[/latex]
33.
[latex]v_y = v_0\sin \theta - gt[/latex]
34.
[latex]\Delta W = -q_0 E\cos (\pi - \theta)\Delta l[/latex]
Exercise Group
For Problems 35–36, find exact values without using a calculator.
35.
[latex]\cos\left[\tan^{-1}\left(\dfrac{-\sqrt{5}}{2}\right)\right][/latex]
36.
[latex]\tan\left[\sin^{-1}\left(\dfrac{2}{7}\right)\right][/latex]
Exercise Group
For Problems 37–38, simplify the expression.
37.
[latex]\sin(\cos^{-1}2t)[/latex]
38.
[latex]\tan(\cos^{-1}m)[/latex]
39.
Explain why one of the expressions [latex]\sin^{-1}x[/latex] or [latex]\sin^{-1}\left(\dfrac{1}{x}\right)[/latex] must be undefined.
40.
Does [latex]\sin^{-1}(-x) = -\sin^{-1}(x)\text{?}[/latex] Does [latex]\cos^{-1}(-x) = -\cos^{-1}(x)\text{?}[/latex]
Exercise Group
For Problems 41–42, evaluate. Round answers to 3 decimal places if necessary.
41.
- [latex]\displaystyle \csc 27°[/latex]
- [latex]\displaystyle \sec 108°[/latex]
- [latex]\displaystyle \cot 245°[/latex]
42.
- [latex]\displaystyle \csc 5.3[/latex]
- [latex]\displaystyle \cot 0.98[/latex]
- [latex]\displaystyle \sec 2.17[/latex]
Exercise Group
For Problems 43–50, find all six trigonometric ratios for the angle [latex]\theta\text{.}[/latex]
43.
44.
45.
46.
47.
48.
49.
[latex]6\cos \alpha = -5\text{,}[/latex] [latex]~ 180° \lt \alpha \lt 270°[/latex]
50.
[latex]4\sin \theta = 3,~ \theta[/latex] is obtuse
Exercise Group
For Problems 51–56, write algebraic expressions for the six trigonometric ratios of the angle.
51.
52.
53.
54.
55.
[latex]2\sin \alpha - k = 0,~\dfrac{\pi}{2} \lt \alpha \lt \pi[/latex]
56.
[latex]h\cos \beta - 3 = 0,~\dfrac{3\pi}{2} \lt \beta \lt 2\pi[/latex]
Exercise Group
For Problems 57–58, find all six trigonometric ratios of the arc [latex]\theta\text{.}[/latex] Round to two places.
57.
58.
Exercise Group
For Problems 59–62, evaluate exactly.
59.
[latex]4\cot \dfrac{3\pi}{4} - \sec^2 \dfrac{\pi}{3}[/latex]
60.
[latex]\dfrac{1}{2}\csc \dfrac{2\pi}{3} + \tan^2 \dfrac{5\pi}{6}[/latex]
61.
[latex]\csc \dfrac{7\pi}{6}\cos \dfrac{5\pi}{4}[/latex]
62.
[latex]\sec \dfrac{7\pi}{4}\cot \dfrac{4\pi}{3}[/latex]
Exercise Group
For Problems 63–64, find all solutions between [latex]0[/latex] and [latex]2\pi\text{.}[/latex] Round your solutions to tenths.
63.
[latex]3\csc \theta + 2 = 12[/latex]
64.
[latex]5\cot \theta + 15 = -3[/latex]
Exercise Group
For Problems 65–70, sketch a graph of each function. Then choose the function or functions described by each statement.
[latex]y = \sec x ~~~~~~~~~~~ y = \csc x ~~~~~~~~~~~ y = \cot x[/latex]
[latex]y = \cos^{-1} x ~~~~~~~~ y = \sin^{-1} x ~~~~~~ y = \tan^{-1} x[/latex]
65.
The graph has vertical asymptotes at multiples of [latex]\pi\text{.}[/latex]
66.
The graph has a horizontal asymptote at [latex]\dfrac{\pi}{2}\text{.}[/latex]
67.
The function values are the reciprocals of [latex]y = \cos x\text{.}[/latex]
68.
The function is defined only for [latex]x[/latex]-values between [latex]-1[/latex] and [latex]1\text{,}[/latex] inclusive.
69.
None of the function values lie between [latex]-1[/latex] and [latex]1\text{.}[/latex]
70.
The graph includes the origin.
Exercise Group
For Problems 71–74,
- Graph the function on the interval [latex]-2\pi \le\ x \le 2\pi\text[/latex] and use the graph to write the function in a simpler form.
- Verify your conjecture algebraically.
71.
[latex]f(x)=\tan x(\cos x - \cot x)[/latex]
72.
[latex]g(x)=\csc x - \cot x\cos x[/latex]
73.
[latex]G(x) = \sin x(\sec x - \csc x)[/latex]
74.
[latex]F(x) = \dfrac{1}{2}\left(\dfrac{\cos x}{1+\sin x} + \dfrac{1+ \sin x}{\cos x}\right)[/latex]
Exercise Group
For Problems 75–78, simplify the expression.
75.
[latex]1-\dfrac{\sin x}{\csc x}[/latex]
76.
[latex]\dfrac{\sin x}{\csc x}+\dfrac{\cos x}{\sec x}[/latex]
77.
[latex]\dfrac{2+\tan^2 B}{\sec^2 B} - 1[/latex]
78.
[latex]\dfrac{\csc t}{\tan t + \cot t}[/latex]
Exercise Group
For Problems 79–82, use the suggested substitution to simplify the expression.
79.
[latex]\dfrac{\sqrt{16+x^2}}{x},~~x = 4\tan \theta[/latex]
80.
[latex]x\sqrt{4-x^2},~~x=2\sin \theta[/latex]
81.
[latex]\dfrac{x^2 - 3}{x},~~x=\sqrt{3}\sec \theta[/latex]
82.
[latex]\dfrac{x}{\sqrt{x^2+2}},~~x=\sqrt{2}\tan \theta[/latex]
83.
This problem outlines a geometric proof of difference of angles formula for tangent.
- In the figure below left, [latex]\alpha=\angle ABC[/latex] and [latex]\beta = \angle DBC\text{.}[/latex] Write expressions in terms of [latex]\alpha[/latex] and [latex]\beta[/latex] for the sides [latex]AC,~DC,[/latex] and [latex]AD\text{.}[/latex]
- In the figure above right, explain why [latex]\triangle ABC[/latex] is similar to [latex]\triangle FBE\text{.}[/latex]
- Explain why [latex]\angle FDC = \alpha\text{.}[/latex]
- Write an expression in terms of [latex]\alpha[/latex] and [latex]\beta[/latex] for side [latex]CF\text{.}[/latex]
- Explain why [latex]\triangle FBE[/latex] is similar to [latex]\triangle ADE\text{.}[/latex]
- Justify each equality in the statement
[latex]\tan (\alpha - \beta) = \dfrac{DE}{BE} = \dfrac{AD}{BF} = \dfrac{\tan \alpha - \tan \beta}{1+\tan \alpha \tan \beta}[/latex]
84.
Let [latex]L_1[/latex] and [latex]L_2[/latex] be two lines with slopes [latex]m_1[/latex] and [latex]m_2\text{,}[/latex] respectively, and let [latex]\theta[/latex] be the acute angle formed between the two lines. Use an identity to show that
[latex]\tan \theta = \dfrac{m_2-m_1}{1+m_1m_2}[/latex]
Exercise Group
For Problems 85–86, use the fact that if [latex]\theta[/latex] is one angle of a triangle and [latex]s[/latex] is the length of the opposite side, then the diameter of the circumscribing circle is
[latex]d=s \csc \theta[/latex]
Round your answers to the nearest hundredth.
85.
In the figure above, find the diameter of the circumscribing circle, the angle [latex]\alpha\text{,}[/latex] and the sides [latex]a[/latex] and [latex]b\text{.}[/latex]
86.
A triangle has one side of length 17 cm and the angle opposite is [latex]26°\text{.}[/latex] Find the diameter of the circle that circumscribes the triangle.