Stewart Precalc 6e Section 5.2: Trigonometric Functions of Real Numbers

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Stewart Precalc 6e Section 5.2: Trigonometric Functions of Real Numbers 0/91
1 Concept - Unit circle definitions · Level 1
Let \(P(x, y)\) be the terminal point on the unit circle determined by \(t\). Then \(\sin t = \) ___, \(\cos t = \) ___, and \(\tan t = \) ___.
2 Concept - Pythagorean identity · Level 1
If \(P(x, y)\) is on the unit circle, then \(x^2 + y^2 = \) ___. So for all \(t\) we have \(\sin^2 t + \cos^2 t = \) ___.
3 Skill - Reading trig values from unit circle · Level 1
Find \(\sin t\) and \(\cos t\) for \(t = \dfrac{\pi}{4}\), the terminal point shown on the unit circle in the figure (with \(t\) in increments of \(\dfrac{\pi}{4}\)). (See Exercises 21 and 22 in Section 5.1.)
4 Skill - Reading trig values from unit circle · Level 1
Find \(\sin t\) and \(\cos t\) for the values of \(t\) whose terminal points are shown on the unit circle in the figure with increments of \(\dfrac{\pi}{6}\). (See Exercises 21 and 22 in Section 5.1.)
5 Skill - Exact trigonometric values · Level 2
Find the exact value: (a) \(\sin \dfrac{2 \pi}{3}\) (b) \(\cos \dfrac{2 \pi}{3}\) (c) \(\tan \dfrac{2 \pi}{3}\)
6 Skill - Exact trigonometric values · Level 2
Find the exact value: (a) \(\sin \dfrac{5 \pi}{6}\) (b) \(\cos \dfrac{5 \pi}{6}\) (c) \(\tan \dfrac{5 \pi}{6}\)
7 Skill - Exact trigonometric values · Level 2
Find the exact value: (a) \(\sin \dfrac{7 \pi}{6}\) (b) \(\sin\left(-\dfrac{\pi}{6}\right)\) (c) \(\sin \dfrac{11 \pi}{6}\)
8 Skill - Exact trigonometric values · Level 2
Find the exact value: (a) \(\cos \dfrac{5 \pi}{2}\) (b) \(\cos\left(-\dfrac{5 \pi}{2}\right)\) (c) \(\cos \dfrac{7 \pi}{2}\)
9 Skill - Exact trigonometric values · Level 2
Find the exact value: (a) \(\cos \dfrac{3 \pi}{4}\) (b) \(\cos \dfrac{5 \pi}{4}\) (c) \(\cos \dfrac{7 \pi}{4}\)
10 Skill - Exact trigonometric values · Level 2
Find the exact value: (a) \(\sin \dfrac{3 \pi}{4}\) (b) \(\sin \dfrac{5 \pi}{4}\) (c) \(\sin \dfrac{7 \pi}{4}\)
11 Skill - Exact trigonometric values · Level 2
Find the exact value: (a) \(\sin \dfrac{7 \pi}{3}\) (b) \(\csc \dfrac{7 \pi}{3}\) (c) \(\cot \dfrac{7 \pi}{3}\)
12 Skill - Exact trigonometric values · Level 2
Find the exact value: (a) \(\cos\left(-\dfrac{\pi}{3}\right)\) (b) \(\sec\left(-\dfrac{\pi}{3}\right)\) (c) \(\tan\left(-\dfrac{\pi}{3}\right)\)
13 Skill - Exact trigonometric values · Level 2
Find the exact value: (a) \(\sin\left(-\dfrac{\pi}{2}\right)\) (b) \(\cos\left(-\dfrac{\pi}{2}\right)\) (c) \(\cot\left(-\dfrac{\pi}{2}\right)\)
14 Skill - Exact trigonometric values · Level 2
Find the exact value: (a) \(\sin\left(-\dfrac{3 \pi}{2}\right)\) (b) \(\cos\left(-\dfrac{3 \pi}{2}\right)\) (c) \(\cot\left(-\dfrac{3 \pi}{2}\right)\)
15 Skill - Exact trigonometric values · Level 2
Find the exact value: (a) \(\sec \dfrac{11 \pi}{3}\) (b) \(\csc \dfrac{11 \pi}{3}\) (c) \(\sec\left(-\dfrac{\pi}{3}\right)\)
16 Skill - Exact trigonometric values · Level 2
Find the exact value: (a) \(\cos \dfrac{7 \pi}{6}\) (b) \(\sec \dfrac{7 \pi}{6}\) (c) \(\csc \dfrac{7 \pi}{6}\)
17 Skill - Exact trigonometric values · Level 2
Find the exact value: (a) \(\tan \dfrac{5 \pi}{6}\) (b) \(\tan \dfrac{7 \pi}{6}\) (c) \(\tan \dfrac{11 \pi}{6}\)
18 Skill - Exact trigonometric values · Level 2
Find the exact value: (a) \(\cot\left(-\dfrac{\pi}{2}\right)\) (b) \(\cot \dfrac{2 \pi}{2}\) (c) \(\cot \dfrac{5 \pi}{2}\)
19 Skill - Exact trigonometric values · Level 2
Find the exact value: (a) \(\cos\left(-\dfrac{\pi}{4}\right)\) (b) \(\csc\left(-\dfrac{\pi}{4}\right)\) (c) \(\cot\left(-\dfrac{\pi}{4}\right)\)
20 Skill - Exact trigonometric values · Level 2
Find the exact value: (a) \(\sin \dfrac{5 \pi}{4}\) (b) \(\sec \dfrac{5 \pi}{4}\) (c) \(\tan \dfrac{5 \pi}{4}\)
21 Skill - Exact trigonometric values · Level 2
Find the exact value: (a) \(\csc\left(-\dfrac{\pi}{2}\right)\) (b) \(\csc \dfrac{\pi}{2}\) (c) \(\csc \dfrac{3 \pi}{2}\)
22 Skill - Exact trigonometric values · Level 2
Find the exact value: (a) \(\sec(-\pi)\) (b) \(\sec \pi\) (c) \(\sec 4 \pi\)
23 Skill - Exact trigonometric values · Level 2
Find the exact value: (a) \(\sin(13 \pi)\) (b) \(\cos(14 \pi)\) (c) \(\tan(15 \pi)\)
24 Skill - Exact trigonometric values · Level 2
Find the exact value: (a) \(\sin \dfrac{25 \pi}{2}\) (b) \(\cos \dfrac{25 \pi}{2}\) (c) \(\cot \dfrac{25 \pi}{2}\)
25 Skill - Trig values at quadrantal angles · Level 1
Find the value of each of the six trigonometric functions (if it is defined) at \(t = 0\).
26 Skill - Trig values at quadrantal angles · Level 1
Find the value of each of the six trigonometric functions (if it is defined) at \(t = \dfrac{\pi}{2}\).
27 Skill - Trig values at quadrantal angles · Level 1
Find the value of each of the six trigonometric functions (if it is defined) at \(t = \pi\).
28 Skill - Trig values at quadrantal angles · Level 1
Find the value of each of the six trigonometric functions (if it is defined) at \(t = \dfrac{3 \pi}{2}\).
29 Skill - Trig from terminal point · Level 2
The terminal point \(P(x, y)\) on the unit circle determined by \(t\) is \(\left(\dfrac{3}{5}, \dfrac{4}{5}\right)\). Find \(\sin t\), \(\cos t\), and \(\tan t\).
30 Skill - Trig from terminal point · Level 2
The terminal point \(P(x, y)\) on the unit circle determined by \(t\) is \(\left(-\dfrac{3}{5}, \dfrac{4}{5}\right)\). Find \(\sin t\), \(\cos t\), and \(\tan t\).
31 Skill - Trig from terminal point · Level 2
The terminal point \(P(x, y)\) on the unit circle determined by \(t\) is \(\left(\dfrac{\sqrt{5}}{4}, -\dfrac{\sqrt{11}}{4}\right)\). Find \(\sin t\), \(\cos t\), and \(\tan t\).
32 Skill - Trig from terminal point · Level 2
The terminal point \(P(x, y)\) on the unit circle determined by \(t\) is \(\left(-\dfrac{1}{3}, -\dfrac{2 \sqrt{2}}{3}\right)\). Find \(\sin t\), \(\cos t\), and \(\tan t\).
33 Skill - Trig from terminal point · Level 2
The terminal point \(P(x, y)\) on the unit circle determined by \(t\) is \(\left(-\dfrac{6}{7}, \dfrac{\sqrt{13}}{7}\right)\). Find \(\sin t\), \(\cos t\), and \(\tan t\).
34 Skill - Trig from terminal point · Level 2
The terminal point \(P(x, y)\) on the unit circle determined by \(t\) is \(\left(\dfrac{40}{41}, \dfrac{9}{41}\right)\). Find \(\sin t\), \(\cos t\), and \(\tan t\).
35 Skill - Trig from terminal point · Level 2
The terminal point \(P(x, y)\) on the unit circle determined by \(t\) is \(\left(-\dfrac{5}{13}, -\dfrac{12}{13}\right)\). Find \(\sin t\), \(\cos t\), and \(\tan t\).
36 Skill - Trig from terminal point · Level 2
The terminal point \(P(x, y)\) on the unit circle determined by \(t\) is \(\left(\dfrac{\sqrt{5}}{5}, \dfrac{2 \sqrt{5}}{5}\right)\). Find \(\sin t\), \(\cos t\), and \(\tan t\).
37 Skill - Trig from terminal point · Level 2
The terminal point \(P(x, y)\) on the unit circle determined by \(t\) is \(\left(-\dfrac{20}{29}, \dfrac{21}{29}\right)\). Find \(\sin t\), \(\cos t\), and \(\tan t\).
38 Skill - Trig from terminal point · Level 2
The terminal point \(P(x, y)\) on the unit circle determined by \(t\) is \(\left(\dfrac{24}{25}, -\dfrac{7}{25}\right)\). Find \(\sin t\), \(\cos t\), and \(\tan t\).
39 Skill - Approximate trig values · Level 2
Find an approximate value of \(\sin 1\) by using (a) the figure and (b) a calculator. Compare the two values.
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40 Skill - Approximate trig values · Level 2
Find an approximate value of \(\cos 0.8\) by using (a) the figure and (b) a calculator. Compare the two values.
41 Skill - Approximate trig values · Level 2
Find an approximate value of \(\sin 1.2\) by using (a) the figure and (b) a calculator. Compare the two values.
42 Skill - Approximate trig values · Level 2
Find an approximate value of \(\cos 5\) by using (a) the figure and (b) a calculator. Compare the two values.
43 Skill - Approximate trig values · Level 2
Find an approximate value of \(\tan 0.8\) by using (a) the figure and (b) a calculator. Compare the two values.
44 Skill - Approximate trig values · Level 2
Find an approximate value of \(\tan(-1.3)\) by using (a) the figure and (b) a calculator. Compare the two values.
45 Skill - Approximate trig values · Level 2
Find an approximate value of \(\cos 4.1\) by using (a) the figure and (b) a calculator. Compare the two values.
46 Skill - Approximate trig values · Level 2
Find an approximate value of \(\sin(-5.2)\) by using (a) the figure and (b) a calculator. Compare the two values.
47 Skill - Sign by quadrant · Level 2
Find the sign of the expression \(\sin t \cos t\) if the terminal point determined by \(t\) is in Quadrant II.
48 Skill - Sign by quadrant · Level 2
Find the sign of the expression \(\tan t \sec t\) if the terminal point determined by \(t\) is in Quadrant IV.
49 Skill - Sign by quadrant · Level 2
Find the sign of the expression \(\dfrac{\tan t \sin t}{\cot t}\) if the terminal point determined by \(t\) is in Quadrant III.
50 Skill - Sign by quadrant · Level 2
Find the sign of the expression \(\cos t \sec t\), valid in any quadrant where it is defined.
51 Skill - Determine quadrant · Level 2
From the information \(\sin t > 0\) and \(\cos t < 0\), find the quadrant in which the terminal point determined by \(t\) lies.
52 Skill - Determine quadrant · Level 2
From the information \(\tan t > 0\) and \(\sin t < 0\), find the quadrant in which the terminal point determined by \(t\) lies.
53 Skill - Determine quadrant · Level 2
From the information \(\csc t > 0\) and \(\sec t < 0\), find the quadrant in which the terminal point determined by \(t\) lies.
54 Skill - Determine quadrant · Level 2
From the information \(\cos t < 0\) and \(\cot t < 0\), find the quadrant in which the terminal point determined by \(t\) lies.
55 Skill - Express trig functions in terms of one another · Level 3
Write \(\sin t\) in terms of \(\cos t\) if the terminal point determined by \(t\) is in Quadrant II.
56 Skill - Express trig functions in terms of one another · Level 3
Write \(\cos t\) in terms of \(\sin t\) if the terminal point determined by \(t\) is in Quadrant IV.
57 Skill - Express trig functions in terms of one another · Level 3
Write \(\tan t\) in terms of \(\sin t\) if the terminal point determined by \(t\) is in Quadrant IV.
58 Skill - Express trig functions in terms of one another · Level 3
Write \(\tan t\) in terms of \(\cos t\) if the terminal point determined by \(t\) is in Quadrant III.
59 Skill - Express trig functions in terms of one another · Level 3
Write \(\sec t\) in terms of \(\tan t\) if the terminal point determined by \(t\) is in Quadrant II.
60 Skill - Express trig functions in terms of one another · Level 3
Write \(\csc t\) in terms of \(\cot t\) if the terminal point determined by \(t\) is in Quadrant III.
61 Skill - Express trig functions in terms of one another · Level 3
Write \(\tan t\) in terms of \(\sec t\) if the terminal point determined by \(t\) is in Quadrant III.
62 Skill - Express trig functions in terms of one another · Level 3
Write \(\sin t\) in terms of \(\sec t\) if the terminal point determined by \(t\) is in Quadrant IV.
63 Skill - Express trig functions in terms of one another · Level 3
Write \(\tan^2 t\) in terms of \(\sin t\), valid in any quadrant.
64 Skill - Express trig functions in terms of one another · Level 3
Write \(\sec^2 t \sin^2 t\) in terms of \(\cos t\), valid in any quadrant.
65 Skill - Find all trig values from given info · Level 3
Find the values of the trigonometric functions of \(t\) given \(\sin t = \dfrac{3}{5}\) and the terminal point of \(t\) is in Quadrant II.
66 Skill - Find all trig values from given info · Level 3
Find the values of the trigonometric functions of \(t\) given \(\cos t = -\dfrac{4}{5}\) and the terminal point of \(t\) is in Quadrant III.
67 Skill - Find all trig values from given info · Level 3
Find the values of the trigonometric functions of \(t\) given \(\sec t = 3\) and the terminal point of \(t\) is in Quadrant IV.
68 Skill - Find all trig values from given info · Level 3
Find the values of the trigonometric functions of \(t\) given \(\tan t = \dfrac{1}{4}\) and the terminal point of \(t\) is in Quadrant III.
69 Skill - Find all trig values from given info · Level 3
Find the values of the trigonometric functions of \(t\) given \(\tan t = -\dfrac{3}{4}\) and \(\cos t > 0\).
70 Skill - Find all trig values from given info · Level 3
Find the values of the trigonometric functions of \(t\) given \(\sec t = 2\) and \(\sin t < 0\).
71 Skill - Find all trig values from given info · Level 3
Find the values of the trigonometric functions of \(t\) given \(\sin t = -\dfrac{1}{4}\) and \(\sec t < 0\).
72 Skill - Find all trig values from given info · Level 3
Find the values of the trigonometric functions of \(t\) given \(\tan t = -4\) and \(\csc t > 0\).
73 Skill - Even/odd functions · Level 2
Determine whether \(f(x) = x^2 \sin x\) is even, odd, or neither.
74 Skill - Even/odd functions · Level 2
Determine whether \(f(x) = x^2 \cos 2 x\) is even, odd, or neither.
75 Skill - Even/odd functions · Level 2
Determine whether \(f(x) = \sin x \cos x\) is even, odd, or neither.
76 Skill - Even/odd functions · Level 2
Determine whether \(f(x) = \sin x + \cos x\) is even, odd, or neither.
77 Skill - Even/odd functions · Level 2
Determine whether \(f(x) = |x| \cos x\) is even, odd, or neither.
78 Skill - Even/odd functions · Level 2
Determine whether \(f(x) = x \sin^3 x\) is even, odd, or neither.
79 Skill - Even/odd functions · Level 2
Determine whether \(f(x) = x^3 + \cos x\) is even, odd, or neither.
80 Skill - Even/odd functions · Level 2
Determine whether \(f(x) = \cos(\sin x)\) is even, odd, or neither.
81 Application - Harmonic motion · Level 3
The displacement from equilibrium of an oscillating mass attached to a spring is given by \(y(t) = 4 \cos(3 \pi t)\), where \(y\) is measured in inches and \(t\) in seconds. Find the displacement at \(t = 0, 0.25, 0.50, 0.75, 1.00, 1.25\) seconds.
82 Application - Circadian rhythms · Level 3
The resting diastolic blood pressure at time \(t\) is given by \(B(t) = 80 + 7 \sin\left(\pi \dfrac{t}{12}\right)\), where \(t\) is hours since midnight and \(B(t)\) is in mmHg. Find this person's diastolic blood pressure at (a) 6:00 A.M., (b) 10:30 A.M., (c) Noon, (d) 8:00 P.M.
83 Application - Electric circuit · Level 3
After the switch is closed in the circuit shown, the current \(t\) seconds later is \(I(t) = 0.8 e^{-3 t} \sin(10 t)\). Find the current at (a) \(t = 0.1\) s and (b) \(t = 0.5\) s.
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84 Application - Damped oscillation · Level 3
A bungee jumper plummets from a bridge to the river below and bounces back over and over. At time \(t\) seconds after her jump, her height \(H\) (in meters) above the river is \(H(t) = 100 + 75 e^{-\dfrac{t}{20}} \cos\left(\dfrac{\pi}{4} t\right)\). Find her height at the times indicated in the table.
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85 Discovery/Writing - Reduction formulas · Level 3
A reduction formula reduces the input of a trigonometric function. Explain how the figure shows that the following reduction formulas are valid: \(\sin(t + \pi) = -\sin t\), \(\cos(t + \pi) = -\cos t\), \(\tan(t + \pi) = \tan t\).
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86 Discovery/Writing - Reduction formulas · Level 3
By the Angle-Side-Angle theorem, triangles \(C D O\) and \(A O B\) in the figure are congruent. Explain how this proves that if \(B\) has coordinates \((x, y)\), then \(D\) has coordinates \((-y, x)\). Then explain how the figure shows the reduction formulas \(\sin\left(t + \dfrac{\pi}{2}\right) = \cos t\), \(\cos\left(t + \dfrac{\pi}{2}\right) = -\sin t\), \(\tan\left(t + \dfrac{\pi}{2}\right) = -\cot t\).
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87 Example - Evaluating Trigonometric Functions · Level 2
Find the six trigonometric functions of each given real number \(t\).
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(a) \(t = \dfrac{\pi}{3}\)
(b) \(t = \dfrac{\pi}{2}\)

Enter your answer directly below each part above.

88 Example - Evaluating Trigonometric Functions Using Reference Numbers · Level 2
Find each value.
(a) \(\cos \dfrac{2 \pi}{3}\)
(b) \(\tan\left(-\dfrac{\pi}{3}\right)\)
(c) \(\sin \dfrac{19 \pi}{4}\)

Enter your answer directly below each part above.

89 Example - Using a Calculator to Evaluate Trigonometric Functions · Level 1
Making sure the calculator is set to radian mode and rounding the results to six decimal places, evaluate:
(a) \(\sin 2.2\)
(b) \(\cos 1.1\)
(c) \(\cot 28\)
(d) \(\csc 0.98\)

Enter your answer directly below each part above.

90 Example - Even and Odd Trigonometric Functions · Level 2
Use the even-odd properties of the trigonometric functions to determine each value.
(a) \(\sin\left(-\dfrac{\pi}{6}\right)\)
(b) \(\cos\left(-\dfrac{\pi}{4}\right)\)

Enter your answer directly below each part above.

91 Example - Finding All Trigonometric Functions from the Value of One · Level 3
If \(\cos t = \dfrac{3}{5}\) and \(t\) is in Quadrant IV, find the values of all the trigonometric functions at \(t\).

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