Stewart Precalc 6e Chapter 5 Review: Concept Check

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Stewart Precalc 6e Chapter 5 Review: Concept Check 0/73
1 Concept Check - Unit Circle · Level 1
(a) What is the unit circle? (b) Use a diagram to explain what is meant by the terminal point determined by a real number \(t\). (c) What is the reference number \(\overline{t}\) associated with \(t\)? (d) If \(t\) is a real number and \(P(x, y)\) is the terminal point determined by \(t\), write equations that define \(\sin t\), \(\cos t\), \(\tan t\), \(\cot t\), \(\sec t\), and \(\csc t\). (e) What are the domains of the six functions that you defined in part (d)? (f) Which trigonometric functions are positive in Quadrants I, II, III, and IV?
2 Concept Check - Even and Odd Functions · Level 1
(a) What is an even function? (b) Which trigonometric functions are even? (c) What is an odd function? (d) Which trigonometric functions are odd?
3 Concept Check - Identities · Level 1
(a) State the reciprocal identities. (b) State the Pythagorean identities.
4 Concept Check - Periodic Functions · Level 1
(a) What is a periodic function? (b) What are the periods of the six trigonometric functions?
5 Concept Check - Graphs of Sine and Cosine · Level 1
Graph the sine and cosine functions. How is the graph of cosine related to the graph of sine?
6 Concept Check - Amplitude, Period, Phase Shift · Level 1
Write expressions for the amplitude, period, and phase shift of the sine curve \(y = a \sin(k(x - b))\) and the cosine curve \(y = a \cos(k(x - b))\).
7 Concept Check - Tangent and Cotangent · Level 1
(a) Graph the tangent and cotangent functions. (b) State the periods of the tangent curve \(y = a \tan(k x)\) and the cotangent curve \(y = a \cot(k x)\).
8 Concept Check - Secant and Cosecant · Level 1
(a) Graph the secant and cosecant functions. (b) State the periods of the secant curve \(y = a \sec(k x)\) and the cosecant curve \(y = a \csc(k x)\).
9 Concept Check - Inverse Sine · Level 2
(a) Define the inverse sine function \(\sin^{-1}\). What are its domain and range? (b) For what values of \(x\) is the equation \(\sin(\sin^{-1} x) = x\) true? (c) For what values of \(x\) is the equation \(\sin^{-1}(\sin x) = x\) true?
10 Trigonometric Function Values · Level 2
Find the value of the trigonometric function. Use a calculator to find an approximate value correct to five decimal places.
(a) \(\cos \dfrac{\pi}{5}\)
(b) \(\cos\left(-\dfrac{\pi}{5}\right)\)

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11 Trigonometric Function Values · Level 3
Find the value of the trigonometric function.
(a) \(\cos \dfrac{9 \pi}{2}\)
(b) \(\sec \dfrac{9 \pi}{2}\)

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12 Trigonometric Function Values · Level 2
Find the value of the trigonometric function. Use a calculator to find an approximate value correct to five decimal places.
(a) \(\sin \dfrac{\pi}{7}\)
(b) \(\csc \dfrac{\pi}{7}\)

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13 Trigonometric Function Values · Level 3
Find the value of the trigonometric function.
(a) \(\tan \dfrac{5 \pi}{2}\)
(b) \(\cot \dfrac{5 \pi}{2}\)

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14 Trigonometric Function Values · Level 2
Find the value of the trigonometric function.
(a) \(\sin 2 \pi\)
(b) \(\csc 2 \pi\)

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15 Trigonometric Function Values · Level 2
Find the value of the trigonometric function.
(a) \(\tan \dfrac{5 \pi}{6}\)
(b) \(\cot \dfrac{5 \pi}{6}\)

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16 Trigonometric Function Values · Level 1
Find the value of the trigonometric function.
(a) \(\cos \dfrac{\pi}{3}\)
(b) \(\sin \dfrac{\pi}{6}\)

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17 Fundamental Identities · Level 3
Use the fundamental identities to write the first expression in terms of the second. \(\dfrac{\tan t}{\cos t}\), in terms of \(\sin t\)
18 Fundamental Identities · Level 3
Use the fundamental identities to write the first expression in terms of the second. \(\tan^2 t \sec t\), in terms of \(\cos t\)
19 Fundamental Identities · Level 3
Use the fundamental identities to write the first expression in terms of the second. \(\tan t\), in terms of \(\sin t\); \(t\) in Quadrant IV.
20 Fundamental Identities · Level 3
Use the fundamental identities to write the first expression in terms of the second. \(\sec t\), in terms of \(\sin t\); \(t\) in Quadrant II.
21 Finding Trigonometric Function Values · Level 2
Find the values of the remaining trigonometric functions at \(t\) from the given information. \(\sin t = \dfrac{5}{13}\), \(\cos t = -\dfrac{12}{13}\)
22 Finding Trigonometric Function Values · Level 3
Find the values of the remaining trigonometric functions at \(t\) from the given information. \(\sin t = -\dfrac{1}{2}\), \(\cos t > 0\)
23 Finding Trigonometric Function Values · Level 3
Find the values of the remaining trigonometric functions at \(t\) from the given information. \(\cot t = -\dfrac{1}{2}\), \(\csc t = \dfrac{\sqrt{5}}{2}\)
24 Finding Trigonometric Function Values · Level 3
Find the values of the remaining trigonometric functions at \(t\) from the given information. \(\cos t = -\dfrac{3}{5}\), \(\tan t < 0\)
25 Finding Trigonometric Function Values · Level 3
If \(\tan t = \dfrac{1}{4}\) and the terminal point for \(t\) is in Quadrant III, find \(\sin t\) and \(\cos t\).
26 Finding Trigonometric Function Values · Level 3
If \(\sin t = -\dfrac{8}{17}\) and the terminal point for \(t\) is in Quadrant IV, find \(\csc t + \sec t\).
27 Finding Trigonometric Function Values · Level 2
If \(\cos t = \dfrac{3}{5}\) and the terminal point for \(t\) is in Quadrant I, find \(\tan t + \sec t\).
28 Pythagorean Identity · Level 1
If \(\sec t = -5\) and the terminal point for \(t\) is in Quadrant II, find \(\sin^2 t + \cos^2 t\).
29 Amplitude, Period, Phase Shift · Level 3
A trigonometric function is given.
(a) Find the amplitude, period, and phase shift of the function.
(b) Sketch the graph. \(y = 10 \cos\left(\dfrac{1}{2} x\right)\)

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30 Amplitude, Period, Phase Shift · Level 3
A trigonometric function is given.
(a) Find the amplitude, period, and phase shift of the function.
(b) Sketch the graph. \(y = 4 \sin(2 \pi x)\)

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31 Amplitude, Period, Phase Shift · Level 3
A trigonometric function is given.
(a) Find the amplitude, period, and phase shift of the function.
(b) Sketch the graph. \(y = -\sin\left(\dfrac{1}{2} x\right)\)

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32 Amplitude, Period, Phase Shift · Level 3
A trigonometric function is given.
(a) Find the amplitude, period, and phase shift of the function.
(b) Sketch the graph. \(y = 2 \sin\left(x - \dfrac{\pi}{4}\right)\)

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33 Amplitude, Period, Phase Shift · Level 3
A trigonometric function is given.
(a) Find the amplitude, period, and phase shift of the function.
(b) Sketch the graph. \(y = 3 \sin(2 x - 2)\)

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34 Amplitude, Period, Phase Shift · Level 3
A trigonometric function is given.
(a) Find the amplitude, period, and phase shift of the function.
(b) Sketch the graph. \(y = \cos(2\left(x - \dfrac{\pi}{2}\right))\)

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35 Amplitude, Period, Phase Shift · Level 3
A trigonometric function is given.
(a) Find the amplitude, period, and phase shift of the function.
(b) Sketch the graph. \(y = -\cos\left(\dfrac{\pi}{2} x + \dfrac{\pi}{6}\right)\)

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36 Amplitude, Period, Phase Shift · Level 3
A trigonometric function is given.
(a) Find the amplitude, period, and phase shift of the function.
(b) Sketch the graph. \(y = 10 \sin\left(2 x - \dfrac{\pi}{2}\right)\)

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37 Determining Function from Graph · Level 4
The graph of one period of a function of the form \(y = a \sin(k(x - b))\) or \(y = a \cos(k(x - b))\) is shown. Determine the function.
문제 이미지
38 Determining Function from Graph · Level 4
The graph of one period of a function of the form \(y = a \sin(k(x - b))\) or \(y = a \cos(k(x - b))\) is shown. Determine the function.
문제 이미지
39 Determining Function from Graph · Level 4
The graph of one period of a function of the form \(y = a \sin(k(x - b))\) or \(y = a \cos(k(x - b))\) is shown. Determine the function.
문제 이미지
40 Period and Graph - Tangent · Level 3
Find the period, and sketch the graph. \(y = 3 \tan x\)
41 Period and Graph - Tangent · Level 3
Find the period, and sketch the graph. \(y = \tan(\pi x)\)
42 Period and Graph - Cotangent · Level 3
Find the period, and sketch the graph. \(y = 2 \cot\left(x - \dfrac{\pi}{2}\right)\)
43 Period and Graph - Secant · Level 3
Find the period, and sketch the graph. \(y = \sec\left(\dfrac{1}{2} x - \dfrac{\pi}{2}\right)\)
44 Period and Graph - Cosecant · Level 3
Find the period, and sketch the graph. \(y = 4 \csc(2 x + \pi)\)
45 Period and Graph - Tangent · Level 3
Find the period, and sketch the graph. \(y = \tan\left(x + \dfrac{\pi}{6}\right)\)
46 Period and Graph - Tangent · Level 3
Find the period, and sketch the graph. \(y = \tan\left(\dfrac{1}{2} x - \dfrac{\pi}{8}\right)\)
47 Period and Graph - Secant · Level 3
Find the period, and sketch the graph. \(y = -4 \sec(4 \pi x)\)
48 Inverse Trigonometric Functions · Level 2
Find the exact value of the expression, if it is defined. \(\cos^{-1}\left(-\dfrac{1}{2}\right)\)
49 Inverse Trigonometric Functions · Level 3
Find the exact value of the expression, if it is defined. \(\sin^{-1}\left(\sin \dfrac{13 \pi}{6}\right)\)
50 Inverse Trigonometric Functions · Level 2
Find the exact value of the expression, if it is defined. \(\tan(\cos^{-1}\left(\dfrac{1}{2}\right))\)
51 Graph Analysis - Periodicity and Symmetry · Level 3
A function is given.
(a) Use a graphing device to graph the function.
(b) Determine from the graph whether the function is periodic and, if so, determine the period.
(c) Determine from the graph whether the function is odd, even, or neither. \(y = |\cos x|\)

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52 Graph Analysis - Periodicity and Symmetry · Level 3
A function is given.
(a) Use a graphing device to graph the function.
(b) Determine from the graph whether the function is periodic and, if so, determine the period.
(c) Determine from the graph whether the function is odd, even, or neither. \(y = \sin(\cos x)\)

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53 Graph Analysis - Periodicity and Symmetry · Level 3
A function is given.
(a) Use a graphing device to graph the function.
(b) Determine from the graph whether the function is periodic and, if so, determine the period.
(c) Determine from the graph whether the function is odd, even, or neither. \(y = \cos(2^{0.1 x})\)

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54 Graph Analysis - Periodicity and Symmetry · Level 3
A function is given.
(a) Use a graphing device to graph the function.
(b) Determine from the graph whether the function is periodic and, if so, determine the period.
(c) Determine from the graph whether the function is odd, even, or neither. \(y = 1 + 2^{\cos x}\)

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55 Graph Analysis - Periodicity and Symmetry · Level 4
A function is given.
(a) Use a graphing device to graph the function.
(b) Determine from the graph whether the function is periodic and, if so, determine the period.
(c) Determine from the graph whether the function is odd, even, or neither. \(y = |x| \cos(3 x)\)

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56 Graph Analysis - Periodicity and Symmetry · Level 4
A function is given.
(a) Use a graphing device to graph the function.
(b) Determine from the graph whether the function is periodic and, if so, determine the period.
(c) Determine from the graph whether the function is odd, even, or neither. \(y = \sqrt{x} \sin(3 x)\) (where \(x > 0\))

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57 Common Screen Graphs · Level 3
Graph the three functions on a common screen. How are the graphs related? \(y = x\), \(y = -x\), \(y = x \sin x\)
58 Common Screen Graphs · Level 3
Graph the three functions on a common screen. How are the graphs related? \(y = 2^{-x}\), \(y = -2^{-x}\), \(y = 2^{-x} \cos(4 \pi x)\)
59 Common Screen Graphs · Level 3
Graph the three functions on a common screen. How are the graphs related? \(y = x\), \(y = \sin(4 x)\), \(y = x + \sin(4 x)\)
60 Common Screen Graphs - Pythagorean Identity · Level 2
Graph the three functions on a common screen. How are the graphs related? \(y = \sin^2 x\), \(y = \cos^2 x\), \(y = \sin^2 x + \cos^2 x\)
61 Maximum and Minimum Values · Level 4
Find the maximum and minimum values of the function. \(y = \cos x + \sin(2 x)\)
62 Maximum and Minimum Values · Level 4
Find the maximum and minimum values of the function. \(y = \cos x + \sin^2 x\)
63 Solving Trigonometric Equations · Level 3
Find the solutions of \(\sin x = 0.3\) in the interval \([0, 2 \pi]\).
64 Solving Trigonometric Equations · Level 3
Find the solutions of \(\cos(3 x) = x\) in the interval \([0, \pi]\).
65 Function Analysis · Level 4
Let \(f(x) = \dfrac{\sin^2 x}{x}\).
(a) Is the function \(f\) even, odd, or neither?
(b) Find the \(x\)-intercepts of the graph of \(f\).
(c) Graph \(f\) in an appropriate viewing rectangle.
(d) Describe the behavior of the function as \(x\) becomes large.
(e) Notice that \(f(x)\) is not defined when \(x = 0\). What happens as \(x\) approaches \(0\)?

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66 Function Analysis · Level 4
Let \(y_1 = \cos(\sin x)\) and \(y_2 = \sin(\cos x)\).
(a) Graph \(y_1\) and \(y_2\) in the same viewing rectangle.
(b) Determine the period of each of these functions from its graph.
(c) Find an inequality between \(\sin(\cos x)\) and \(\cos(\sin x)\) that is valid for all \(x\).

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67 Simple Harmonic Motion · Level 4
A point \(P\) moving in simple harmonic motion completes 8 cycles every second. If the amplitude of the motion is 50 cm, find an equation that describes the motion of \(P\) as a function of time. Assume that the point \(P\) is at its maximum displacement when \(t = 0\).
68 Simple Harmonic Motion · Level 4
A mass suspended from a spring oscillates in simple harmonic motion at a frequency of 4 cycles per second. The distance from the highest to the lowest point of the oscillation is 100 cm. Find an equation that describes the distance of the mass from its rest position as a function of time. Assume that the mass is at its lowest point when \(t = 0\).
69 Simple Harmonic Motion - Application · Level 4
The graph shows the variation of the water level relative to mean sea level in the Long Beach harbor for a particular 24-hour period. Assuming that this variation is simple harmonic, find an equation of the form \(y = a \cos(\omega t)\) that describes the variation in water level as a function of the number of hours after midnight.
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70 Damped Harmonic Motion - Application · Level 4
The top floor of a building undergoes damped harmonic motion after a sudden brief earthquake. At time \(t = 0\) the displacement is at a maximum, 16 cm from the normal position. The damping constant is \(c = 0.72\) and the building vibrates at 1.4 cycles per second.
(a) Find a function of the form \(y = k e^{-c t} \cos(\omega t)\) to model the motion.
(b) Graph the function you found in part (a).
(c) What is the displacement at time \(t = 10\) s?

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71 Concept Check - Inverse Cosine Function · Level 2
(a) Define the inverse cosine function \(\cos^{-1}\). What are its domain and range?
(b) For what values of \(x\) is the equation \(\cos(\cos^{-1} x) = x\) true?
(c) For what values of \(x\) is the equation \(\cos^{-1}(\cos x) = x\) true?

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72 Concept Check - Inverse Tangent Function · Level 2
(a) Define the inverse tangent function \(\tan^{-1}\). What are its domain and range?
(b) For what values of \(x\) is the equation \(\tan(\tan^{-1} x) = x\) true?
(c) For what values of \(x\) is the equation \(\tan^{-1}(\tan x) = x\) true?

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73 Concept Check - Harmonic Motion · Level 2
(a) What is simple harmonic motion?
(b) What is damped harmonic motion?
(c) Give three real-life examples of simple harmonic motion and of damped harmonic motion.

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