Stewart Precalc 6e Chapter 7 Review

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Stewart Precalc 6e Chapter 7 Review 0/80
1 Verify Identity · Level 2
Verify the identity: \(\sin \theta (\cot \theta + \tan \theta) = \sec \theta\).
2 Verify Identity · Level 2
Verify the identity: \((\sec \theta - 1)(\sec \theta + 1) = \tan^2 \theta\).
3 Verify Identity · Level 2
Verify the identity: \(\cos^2 x \csc x - \csc x = - \sin x\).
4 Verify Identity · Level 2
Verify the identity: \(1/(1 - \sin^2 x) = 1 + \tan^2 x\).
5 Verify Identity · Level 2
Verify the identity: \((\cos^2 x - \tan^2 x)/\sin^2 x = \cot^2 x - \sec^2 x\).
6 Verify Identity · Level 2
Verify the identity: \((1 + \sec x)/\sec x = \sin^2 x/(1 - \cos x)\).
7 Verify Identity · Level 2
Verify the identity: \(\cos^2 x/(1 - \sin x) = \cos x/(\sec x - \tan x)\).
8 Verify Identity · Level 2
Verify the identity: \((1 - \tan x)(1 - \cot x) = 2 - \sec x \csc x\).
9 Verify Identity · Level 2
Verify the identity: \(\sin^2 x \cot^2 x + \cos^2 x \tan^2 x = 1\).
10 Verify Identity · Level 2
Verify the identity: \((\tan x + \cot x)^2 = \csc^2 x \sec^2 x\).
11 Verify Identity · Level 2
Verify the identity: \(\sin 2 x/(1 + \cos 2 x) = \tan x\).
12 Verify Identity · Level 3
Verify the identity: \(\cos\dfrac{x + y}{\cos x \sin y} = \cot y - \tan x\).
13 Verify Identity · Level 2
Verify the identity: \(\tan\left(\dfrac{x}{2}\right) = \csc x - \cot x\).
14 Verify Identity · Level 3
Verify the identity: \(\dfrac{\sin(x+y) + \sin(x-y)}{\cos(x+y) + \cos(x-y)} = \tan x\).
15 Verify Identity · Level 3
Verify the identity: \(\sin(x + y) \sin(x - y) = \sin^2 x - \sin^2 y\).
16 Verify Identity · Level 2
Verify the identity: \(\csc x \tan\left(\dfrac{x}{2}\right) = \cot x\).
17 Verify Identity · Level 3
Verify the identity: \(1 + \tan x \tan\left(\dfrac{x}{2}\right) = \sec x\).
18 Verify Identity · Level 3
Verify the identity: \(\dfrac{\sin 3 x + \cos 3 x}{\cos x - \sin x} = 1 + 2 \sin 2 x\).
19 Verify Identity · Level 2
Verify the identity: \((\cos\left(\dfrac{x}{2}\right) - \sin\left(\dfrac{x}{2}\right))^2 = 1 - \sin x\).
20 Verify Identity · Level 3
Verify the identity: \(\dfrac{\cos 3 x - \cos 7 x}{\sin 3 x + \sin 7 x} = \tan 2 x\).
21 Verify Identity · Level 3
Verify the identity: \(\sin 2 \dfrac{x}{\sin} x - \cos 2 \dfrac{x}{\cos} x = \sec x\).
22 Verify Identity · Level 3
Verify the identity: \((\cos x + \cos y)^2 + (\sin x - \sin y)^2 = 2 + 2 \cos(x + y)\).
23 Verify Identity · Level 2
Verify the identity: \(\tan\left(x + \dfrac{\pi}{4}\right) = \dfrac{1 + \tan x}{1 - \tan x}\).
24 Verify Identity · Level 3
Verify the identity: \(\dfrac{\sec x - 1}{\sin x \sec x} = \tan\left(\dfrac{x}{2}\right)\).
25 Graph and Prove · Level 2
(a) Graph \(f(x) = 1 - (\cos\left(\dfrac{x}{2}\right) - \sin\left(\dfrac{x}{2}\right))^2\) and \(g(x) = \sin x\). (b) Do the graphs suggest \(f(x) = g(x)\) is an identity? Prove your answer.
26 Graph and Prove · Level 2
(a) Graph \(f(x) = \sin x + \cos x\) and \(g(x) = \sqrt{\sin^2 x + \cos^2 x}\). (b) Do the graphs suggest \(f(x) = g(x)\) is an identity? Prove your answer.
27 Graph and Prove · Level 2
(a) Graph \(f(x) = \tan x \tan\left(\dfrac{x}{2}\right)\) and \(g(x) = \dfrac{1}{\cos} x\). (b) Do the graphs suggest the identity? Prove your answer.
28 Graph and Prove · Level 3
(a) Graph \(f(x) = 1 - 8 \sin^2 x + 8 \sin^4 x\) and \(g(x) = \cos 4 x\). (b) Prove the identity.
29 Graph and Conjecture · Level 2
(a) Graph \(f(x) = 2 \sin^2 3 x + \cos 6 x\) and make a conjecture. (b) Prove your conjecture.
30 Graph and Conjecture · Level 2
(a) Graph \(f(x) = \sin x \cot\left(\dfrac{x}{2}\right)\) and \(g(x) = \cos x\), then make a conjecture. (b) Prove your conjecture.
31 Solve Equation · Level 1
Solve in \([0, 2 \pi)\): \(4 \sin \theta - 3 = 0\).
32 Solve Equation · Level 1
Solve in \([0, 2 \pi)\): \(5 \cos \theta + 3 = 0\).
33 Solve Equation · Level 2
Solve in \([0, 2 \pi)\): \(\cos x \sin x - \sin x = 0\).
34 Solve Equation · Level 2
Solve in \([0, 2 \pi)\): \(\sin x - 2 \sin^2 x = 0\).
35 Solve Equation · Level 2
Solve in \([0, 2 \pi)\): \(2 \sin^2 x - 5 \sin x + 2 = 0\).
36 Solve Equation · Level 2
Solve in \([0, 2 \pi)\): \(\sin x \cos x \tan x = -1\).
37 Solve Equation · Level 2
Solve in \([0, 2 \pi)\): \(2 \cos^2 x - 7 \cos x + 3 = 0\).
38 Solve Equation · Level 2
Solve in \([0, 2 \pi)\): \(4 \sin^2 x + 2 \cos^2 x = 3\).
39 Solve Equation · Level 2
Solve in \([0, 2 \pi)\): \(\dfrac{1 - \cos x}{1 + \cos x} = 3\).
40 Solve Equation · Level 2
Solve in \([0, 2 \pi)\): \(\sin x = \cos 2 x\).
41 Solve Equation · Level 3
Solve in \([0, 2 \pi)\): \(\tan^3 x + \tan^2 x - 3 \tan x - 3 = 0\).
42 Solve Equation · Level 3
Solve in \([0, 2 \pi)\): \(\cos 2 x \csc^2 x = 2 \cos 2 x\).
43 Solve Equation · Level 4
Solve in \([0, 2 \pi)\): \(\tan\left(\dfrac{x}{2}\right) + 2 \sin 2 x = \csc x\).
44 Solve Equation · Level 3
Solve in \([0, 2 \pi)\): \(\cos 3 x + \cos 2 x + \cos x = 0\).
45 Solve Equation · Level 3
Solve in \([0, 2 \pi)\): \(\tan x + \sec x = \sqrt{3}\).
46 Solve Equation · Level 3
Solve in \([0, 2 \pi)\): \(2 \cos x - 3 \tan x = 0\).
47 Solve Equation · Level 2
Solve in \([0, 2 \pi)\): \(\cos x = x^2 - 1\).
48 Solve Equation · Level 2
Solve in \([0, 2 \pi)\): \(e^{\sin x} = x\).
49 Applications · Level 3
If a projectile is fired with velocity \(v_0\) at angle \(\theta\), the maximum height (in feet) is \(M(\theta) = (v_0^2 \sin^2 \theta)/64\). Suppose \(v_0 = 400\) ft/s. (a) At what angle should the projectile be fired so the max height is 2000 ft? (b) Is it possible to reach 3000 ft? (c) Find the angle for maximum height.
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50 Applications · Level 2
The displacement of a shock absorber is \(f(t) = 2^{-0.2 t} \sin 4 \pi t\). Find the times when \(f(t) = 0\).
51 Exact Value · Level 2
Find the exact value: \(\cos 15^{\circ}\).
52 Exact Value · Level 2
Find the exact value: \(\sin\left(5 \dfrac{\pi}{12}\right)\).
53 Exact Value · Level 2
Find the exact value: \(\tan\left(\dfrac{\pi}{8}\right)\).
54 Exact Value · Level 2
Find the exact value: \(2 \sin\left(\dfrac{\pi}{12}\right) \cos\left(\dfrac{\pi}{12}\right)\).
55 Exact Value · Level 2
Find the exact value: \(\sin 5^{\circ} \cos 40^{\circ} + \cos 5^{\circ} \sin 40^{\circ}\).
56 Exact Value · Level 2
Find the exact value: \(\dfrac{\tan 66^{\circ} - \tan 6^{\circ}}{1 + \tan 66^{\circ} \tan 6^{\circ}}\).
57 Exact Value · Level 2
Find the exact value: \(\cos^2\left(\dfrac{\pi}{8}\right) - \sin^2\left(\dfrac{\pi}{8}\right)\).
58 Exact Value · Level 2
Find the exact value: \(\left(\dfrac{1}{2}\right) \cos\left(\dfrac{\pi}{12}\right) + \left(\dfrac{\sqrt{3}}{2}\right) \sin\left(\dfrac{\pi}{12}\right)\).
59 Exact Value · Level 3
Find the exact value: \(\cos 37.5^{\circ} \cos 7.5^{\circ}\).
60 Exact Value · Level 3
Find the exact value: \(\cos 67.5^{\circ} + \cos 22.5^{\circ}\).
61 Given Conditions · Level 3
Given \(\sec x = \dfrac{3}{2}\), \(\csc y = 3\), with \(x\) and \(y\) in Quadrant I, find the exact value of \(\sin(x + y)\).
62 Given Conditions · Level 3
Given \(\sec x = \dfrac{3}{2}\), \(\csc y = 3\), with \(x\) and \(y\) in Quadrant I, find the exact value of \(\cos(x - y)\).
63 Given Conditions · Level 3
Given \(\sec x = \dfrac{3}{2}\), \(\csc y = 3\), with \(x\) and \(y\) in Quadrant I, find the exact value of \(\tan(x + y)\).
64 Given Conditions · Level 2
Given \(\sec x = \dfrac{3}{2}\), \(\csc y = 3\), with \(x\) and \(y\) in Quadrant I, find the exact value of \(\sin 2 x\).
65 Given Conditions · Level 3
Given \(\sec x = \dfrac{3}{2}\), \(\csc y = 3\), with \(x\) and \(y\) in Quadrant I, find the exact value of \(\cos\left(\dfrac{y}{2}\right)\).
66 Given Conditions · Level 3
Given \(\sec x = \dfrac{3}{2}\), \(\csc y = 3\), with \(x\) and \(y\) in Quadrant I, find the exact value of \(\tan\left(\dfrac{y}{2}\right)\).
67 Inverse Trig Exact Value · Level 3
Find the exact value: \(\tan(2 \arccos\left(\dfrac{3}{7}\right))\).
68 Inverse Trig Exact Value · Level 3
Find the exact value: \(\sin(\arctan\left(\dfrac{3}{4}\right) + \arccos\left(\dfrac{5}{13}\right))\).
69 Algebraic Expression · Level 3
Write as an algebraic expression in \(x\): \(\tan(2 \arctan x)\).
70 Algebraic Expression · Level 3
Write as an algebraic expression in \(x\) and \(y\): \(\cos(\arcsin x + \arccos y)\).
71 Applications - Geometry · Level 4
A 10-ft-wide highway sign is adjacent to a roadway, as shown in the figure. As a driver approaches the sign, the viewing angle \(\theta\) changes. (a) Express viewing angle \(\theta\) as a function of the distance \(x\) between the driver and the sign. (b) The sign is legible when the viewing angle is \(2^{\circ}\) or greater. At what distance \(x\) does the sign first become legible?
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72 Applications - Geometry · Level 4
A 380-ft-tall building supports a 40-ft communications tower. As a driver approaches the building, the viewing angle \(\theta\) of the tower changes. (a) Express viewing angle \(\theta\) as a function of distance \(x\) between the driver and the building. (b) At what distance from the building is the viewing angle \(\theta\) as large as possible?
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73 Concept Check · Level 1
(a) State the reciprocal identities. (b) State the Pythagorean identities. (c) State the even-odd identities. (d) State the cofunction identities.
74 Concept Check · Level 1
Explain the difference between an equation and an identity.
75 Concept Check · Level 1
How do you prove a trigonometric identity?
76 Concept Check · Level 1
(a) State the Addition Formulas for Sine, Cosine, and Tangent. (b) State the Subtraction Formulas for Sine, Cosine, and Tangent.
77 Concept Check · Level 1
(a) State the Double-Angle Formulas for Sine, Cosine, and Tangent. (b) State the formulas for lowering powers. (c) State the Half-Angle Formulas.
78 Concept Check · Level 1
(a) State the Product-to-Sum Formulas. (b) State the Sum-to-Product Formulas.
79 Concept Check · Level 1
Explain how you solve a trigonometric equation by factoring.
80 Concept Check · Level 1
What identity would you use to solve the equation \(\cos x \sin 2 x = 0\)?

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