In Exercises 9396, the graphs of the sine and cosine

In Exercises 1–3, state the sign (positive or negative) of the function in each quadrant. sin x

In Exercises 1–3, state the sign (positive or negative) of the function in each quadrant. cos x

In Exercises 1–3, state the sign (positive or negative) of the function in each quadrant. tan x

In Exercises 4–6, give the radian measure of the angle. \(135^{\circ}\)

In Exercises 4–6, give the radian measure of the angle. \(-150^{\circ}\)

In Exercises 4–6, give the radian measure of the angle. \(450^{\circ}\)

In Exercises 7–10, find a transformation that will transform the graph of \(y_{1}\) to the graph of \(y_{2}\). \(y_{1}=\sqrt{x}\) and \(y_{2}=3 \sqrt{x}\)

In Exercises 7–10, find a transformation that will transform the graph of \(y_{1}\) to the graph of \(y_{2}\). \(y_{1}=e^{x}\) and \(y_{2}=e^{-x}\)

In Exercises 7–10, find a transformation that will transform the graph of \(y_{1}\) to the graph of \(y_{2}\). \(y_{1}=\ln x\) and \(y_{2}=0.5 \ln x\)

In Exercises 7–10, find a transformation that will transform the graph of \(y_{1}\) to the graph of \(y_{2}\). \(y_{1}=x^{3}\) and \(y_{2}=x^{3}-2\)

In Exercises 1–6, find the amplitude of the function and use the language of transformations to describe how the graph of the function is related to the graph of y = sin x. y = 2 sin x

In Exercises 1–6, find the amplitude of the function and use the language of transformations to describe how the graph of the function is related to the graph of y = sin x. \(y=\frac{2}{3} \sin x\)

In Exercises 1–6, find the amplitude of the function and use the language of transformations to describe how the graph of the function is related to the graph of y = sin x. \(y=-4 \sin x\)

In Exercises 1–6, find the amplitude of the function and use the language of transformations to describe how the graph of the function is related to the graph of y = sin x. \(y=-\frac{7}{4} \sin x\)

In Exercises 1–6, find the amplitude of the function and use the language of transformations to describe how the graph of the function is related to the graph of y = sin x. y = 0.73 sin x

In Exercises 1–6, find the amplitude of the function and use the language of transformations to describe how the graph of the function is related to the graph of y = sin x. \(y=-2.34 \sin x\)

In Exercises 7–12, find the period of the function and use the language of transformations to describe how the graph of the function is related to the graph of y = cos x. y = cos 3x

In Exercises 7–12, find the period of the function and use the language of transformations to describe how the graph of the function is related to the graph of y = cos x. y = cos x/5

In Exercises 7–12, find the period of the function and use the language of transformations to describe how the graph of the function is related to the graph of y = cos x. y = cos (-7x)

In Exercises 7–12, find the period of the function and use the language of transformations to describe how the graph of the function is related to the graph of y = cos x. y = cos (-0.4x)

In Exercises 7–12, find the period of the function and use the language of transformations to describe how the graph of the function is related to the graph of y = cos x. y = 3 cos 2x

In Exercises 7–12, find the period of the function and use the language of transformations to describe how the graph of the function is related to the graph of y = cos x. \(y=\frac{1}{4} \cos \frac{2 x}{3}\)

In Exercises 13–16, find the amplitude, period, and frequency of the function and use this information (not your calculator) to sketch a graph of the function in the window \([-3\pi,\ 3\pi]\) by \([-4,\ 4]\) \(y=3 \sin \frac{x}{2}\)

In Exercises 13–16, find the amplitude, period, and frequency of the function and use this information (not your calculator) to sketch a graph of the function in the window \([-3\pi,\ 3\pi]\) by \([-4,\ 4]\) \(y=2 \cos \frac{x}{3}\)

In Exercises 13–16, find the amplitude, period, and frequency of the function and use this information (not your calculator) to sketch a graph of the function in the window \([-3\pi,\ 3\pi]\) by \([-4,\ 4]\) \(y=-\frac{3}{2} \sin 2 x\)

In Exercises 13–16, find the amplitude, period, and frequency of the function and use this information (not your calculator) to sketch a graph of the function in the window \([-3\pi,\ 3\pi]\) by \([-4,\ 4]\) \(y=-4 \sin \frac{2x}{3}\)

In Exercises 17–22, graph one period of the function. Use your understanding of transformations, not your graphing calculators. Be sure to show the scale on both axes. y = 2 sin x

In Exercises 17–22, graph one period of the function. Use your understanding of transformations, not your graphing calculators. Be sure to show the scale on both axes. y = 2.5 sin x

In Exercises 17–22, graph one period of the function. Use your understanding of transformations, not your graphing calculators. Be sure to show the scale on both axes. y = 3 cos x

In Exercises 17–22, graph one period of the function. Use your understanding of transformations, not your graphing calculators. Be sure to show the scale on both axes. y = -2 cos x

In Exercises 17–22, graph one period of the function. Use your understanding of transformations, not your graphing calculators. Be sure to show the scale on both axes. y = -0.5 sin x

In Exercises 17–22, graph one period of the function. Use your understanding of transformations, not your graphing calculators. Be sure to show the scale on both axes. y = 4 cos x

In Exercises 23–28, graph three periods of the function. Use your understanding of transformations, not your graphing calculators. Be sure to show the scale on both axes. y = 5 sin 2x

In Exercises 23–28, graph three periods of the function. Use your understanding of transformations, not your graphing calculators. Be sure to show the scale on both axes. \(y=3 \cos \frac{x}{2}\)

In Exercises 23–28, graph three periods of the function. Use your understanding of transformations, not your graphing calculators. Be sure to show the scale on both axes. y = 0.5 cos 3x

In Exercises 23–28, graph three periods of the function. Use your understanding of transformations, not your graphing calculators. Be sure to show the scale on both axes. y = 20 sin 4x

In Exercises 23–28, graph three periods of the function. Use your understanding of transformations, not your graphing calculators. Be sure to show the scale on both axes. \(y=4 \sin \frac{x}{4}\)

In Exercises 23–28, graph three periods of the function. Use your understanding of transformations, not your graphing calculators. Be sure to show the scale on both axes. y = 8 cos 5x

In Exercises 29–34, specify the period and amplitude of each function. Then give the viewing window in which the graph is shown. Use your understanding of transformations, not your graphing calculators. y = 1.5 sin 2x

In Exercises 29–34, specify the period and amplitude of each function. Then give the viewing window in which the graph is shown. Use your understanding of transformations, not your graphing calculators. y = 2 cos 3x

In Exercises 29–34, specify the period and amplitude of each function. Then give the viewing window in which the graph is shown. Use your understanding of transformations, not your graphing calculators. y = -3 cos 2x

In Exercises 29–34, specify the period and amplitude of each function. Then give the viewing window in which the graph is shown. Use your understanding of transformations, not your graphing calculators. \(y=5 \sin \frac{x}{2}\)

In Exercises 29–34, specify the period and amplitude of each function. Then give the viewing window in which the graph is shown. Use your understanding of transformations, not your graphing calculators. \(y=-4 \sin \frac{\pi}{3} x\)

In Exercises 29–34, specify the period and amplitude of each function. Then give the viewing window in which the graph is shown. Use your understanding of transformations, not your graphing calculators. \(y=3 \cos \pi x\)

In Exercises 35–40, identify the maximum and minimum values and the zeros of the function in the interval \([-2\pi,\ 2\pi]\). Use your understanding of transformations, not your graphing calculators. y = 2 sin x

In Exercises 35–40, identify the maximum and minimum values and the zeros of the function in the interval \([-2\pi,\ 2\pi]\). Use your understanding of transformations, not your graphing calculators. \(y=3 \cos \frac{x}{2}\)

In Exercises 35–40, identify the maximum and minimum values and the zeros of the function in the interval \([-2\pi,\ 2\pi]\). Use your understanding of transformations, not your graphing calculators. y = cos 2x

In Exercises 35–40, identify the maximum and minimum values and the zeros of the function in the interval \([-2\pi,\ 2\pi]\). Use your understanding of transformations, not your graphing calculators. \(y=\frac{1}{2} \sin x\)

In Exercises 35–40, identify the maximum and minimum values and the zeros of the function in the interval \([-2\pi,\ 2\pi]\). Use your understanding of transformations, not your graphing calculators. y = -cos 2x

In Exercises 35–40, identify the maximum and minimum values and the zeros of the function in the interval \([-2\pi,\ 2\pi]\). Use your understanding of transformations, not your graphing calculators. y = -2 sin x

Write the function y = - sin x as a phase shift of y = sin x.

Write the function y = - cos x as a phase shift of y = sin x.

In Exercises 43–48, describe the transformations required to obtain the graph of the given function from a basic trigonometric graph. y = 0.5 sin 3x

In Exercises 43–48, describe the transformations required to obtain the graph of the given function from a basic trigonometric graph. y = 1.5 cos 4x

In Exercises 43–48, describe the transformations required to obtain the graph of the given function from a basic trigonometric graph. \(y=-\frac{2}{3} \cos \frac{x}{3}\)

In Exercises 43–48, describe the transformations required to obtain the graph of the given function from a basic trigonometric graph. \(y=\frac{3}{4} \sin \frac{x}{5}\)

In Exercises 43–48, describe the transformations required to obtain the graph of the given function from a basic trigonometric graph. \(y=3 \cos \frac{2 \pi x}{3}\)

In Exercises 43–48, describe the transformations required to obtain the graph of the given function from a basic trigonometric graph. \(y=-2 \sin \frac{\pi x}{4}\)

In Exercises 49–52, describe the transformations required to obtain the graph of \(y_{2}\) from the graph of \(y_{1}\). \(y_{1}=\cos 2 x\) and \(y_{2}=\frac{5}{3} \cos 2 x\)

In Exercises 49–52, describe the transformations required to obtain the graph of \(y_{2}\) from the graph of \(y_{1}\). \(y_{1}=2 \cos \left(x+\frac{\pi}{3}\right)\) and \(y_{2}=\cos \left(x+\frac{\pi}{4}\right)\)

In Exercises 49–52, describe the transformations required to obtain the graph of \(y_{2}\) from the graph of \(y_{1}\). \(y_{1}=2 \cos \pi x\) and \(y_{2}=2 \cos 2 \pi x\)

In Exercises 49–52, describe the transformations required to obtain the graph of \(y_{2}\) from the graph of \(y_{1}\). \(y_{1}=3 \sin \frac{2 \pi x}{3}\) and \(y_{2}=2 \sin \frac{\pi x}{3}\)

In Exercises 53–56, select the pair of functions that have identical graphs. (a) y = cos x (b) \(y=\sin \left(x+\frac{\pi}{2}\right)\) (c) \(y=\cos \left(x+\frac{\pi}{2}\right)\)

In Exercises 53–56, select the pair of functions that have identical graphs. (a) y = sin x (b) \(y=\cos \left(x-\frac{\pi}{2}\right)\) (c) y = cos x

In Exercises 53–56, select the pair of functions that have identical graphs. (a) \(y=\sin \left(x+\frac{\pi}{2}\right)\) (b) \(y=-\cos (x-\pi)\) (c) \(y=\cos \left(x-\frac{\pi}{2}\right)\)

In Exercises 53–56, select the pair of functions that have identical graphs. (a) \(y=\sin \left(2x+\frac{\pi}{4}\right)\) (b) \(y=-\cos \left(2x-\frac{\pi}{2}\right)\) (c) \(y=\cos \left(2x-\frac{\pi}{4}\right)\)

In Exercises 57–60, construct a sinusoid with the given amplitude and period that goes through the given point. Amplitude 3, period \(\pi\), point (0, 0)

In Exercises 57–60, construct a sinusoid with the given amplitude and period that goes through the given point. Amplitude 2, period \(3\pi\), point (0, 0)

In Exercises 57–60, construct a sinusoid with the given amplitude and period that goes through the given point. Amplitude 1.5, period \(\pi / 6\), point (1, 0)

In Exercises 57–60, construct a sinusoid with the given amplitude and period that goes through the given point. Amplitude 3.2, period \(\pi / 7\), point (5, 0)

In Exercises 61–68, state the amplitude and period of the sinusoid, and (relative to the basic function) the phase shift and vertical translation. \(y=-2 \sin \left(x-\frac{\pi}{4}\right)+1\)

In Exercises 61–68, state the amplitude and period of the sinusoid, and (relative to the basic function) the phase shift and vertical translation. \(y=-3.5 \sin \left(2 x-\frac{\pi}{2}\right)-1\)

In Exercises 61–68, state the amplitude and period of the sinusoid, and (relative to the basic function) the phase shift and vertical translation. \(y=5 \cos \left(3 x-\frac{\pi}{6}\right)+0.5\)

In Exercises 61–68, state the amplitude and period of the sinusoid, and (relative to the basic function) the phase shift and vertical translation. y = 3 cos (x + 3) - 2

In Exercises 61–68, state the amplitude and period of the sinusoid, and (relative to the basic function) the phase shift and vertical translation. \(y=2 \cos 2 \pi x+1\)

In Exercises 61–68, state the amplitude and period of the sinusoid, and (relative to the basic function) the phase shift and vertical translation. \(y=4 \cos 3 \pi x-2\)

In Exercises 61–68, state the amplitude and period of the sinusoid, and (relative to the basic function) the phase shift and vertical translation. \(y=\frac{7}{3} \sin \left(x+\frac{5}{2}\right)-1\)

In Exercises 61–68, state the amplitude and period of the sinusoid, and (relative to the basic function) the phase shift and vertical translation. \(y=\frac{2}{3} \cos \left(\frac{x-3}{4}\right)+1\)

In Exercises 69 and 70, find values a, b, h, and k so that the graph of the function y = a sin (b(x + h)) + k is the curve shown [0, 6.28] by [–4, 4]

In Exercises 69 and 70, find values a, b, h, and k so that the graph of the function y = a sin (b(x + h)) + k is the curve shown [–0.5, 5.78] by [–4, 4]

Points of Intersection Graph \(y=1.3^{-x}\) and \(y=1.3^{-x} \cos x\) for x in the interval [-1, 8]. (a) How many points of intersection do there appear to be? (b) Find the coordinates of each point of intersection.

Motion of a Buoy A signal buoy in the Chesapeake Bay bobs up and down with the height h of its transmitter (in feet) above sea level modeled by h = a sin bt + 5. During a small squall its height varies from 1 ft to 9 ft and there are 3.5 sec from one 9-ft height to the next. What are the values of the constants a and b?

Ferris Wheel A Ferris wheel 50 ft in diameter makes one revolution every 40 sec. If the center of the wheel is 30 ft above the ground, how long after reaching the low point is a rider 50 ft above the ground?

Tsunami Wave An earthquake occurred at 9:40 A.M. on Nov. 1, 1755, at Lisbon, Portugal, and started a tsunami (often called a tidal wave) in the ocean. It produced waves that traveled more than 540 ft/sec (370 mph) and reached a height of 60 ft. If the period of the waves was 30 min or 1800 sec, estimate the length L between the crests.

Ebb and Flow On a particular Labor Day, the high tide in Southern California occurs at 7:12 A.M. At that time you measure the water at the end of the Santa Monica Pier to be 11 ft deep. At 1:24 P.M. it is low tide, and you measure the water to be only 7 ft deep. Assume the depth of the water is a sinusoidal function of time with a period of 1/2 a lunar day, which is about 12 hr 24 min. (a) At what time on that Labor Day does the first low tide occur? (b) What was the approximate depth of the water at 4:00 A.M. and at 9:00 P.M.? (c) What is the first time on that Labor Day that the water is 9 ft deep?

Blood Pressure The function \(P=120+30 \sin 2 \pi t\) models the blood pressure (in millimeters of mercury) for a person who has a (high) blood pressure of 150/90; t represents seconds. (a) What is the period of this function? (b) How many heartbeats are there each minute? (c) Graph this function to model a 10-sec time interval.

Bouncing Block A block mounted on a spring is set into motion directly above a motion detector, which registers the distance to the block at intervals of 0.1 second. When the block is released, it is 7.2 cm above the motion detector. The table below shows the data collected by the motion detector during the first two seconds, with distance d measured in centimeters: (a) Make a scatter plot of d as a function of t and estimate the maximum d visually. Use this number and the given minimum (7.2) to compute the amplitude of the block’s motion. (b) Estimate the period of the block’s motion visually from the scatter plot. (c) Model the motion of the block as a sinusoidal function d(t). (d) Graph your function with the scatter plot to support your model graphically.

LP Turntable A suction-cup-tipped arrow is secured vertically to the outer edge of a turntable designed for playing LP phonograph records (ask your parents). A motion detector is situated 60 cm away. The turntable is switched on and a mo-tion detector measures the distance to the arrow as it revolves around the turntable. The table below shows the distance d as a function of time during the first 4 seconds. (a) If the turntable is 25.4 cm in diameter, find the amplitude of the arrow’s motion. (b) Find the period of the arrow’s motion by analyzing the data. (c) Model the motion of the arrow as a sinusoidal function d(t). (d) Graph your function with a scatter plot to support your model graphically.

Temperature Data The normal monthly Fahrenheit temperatures in Albuquerque, NM, are shown in the table below (month 1 = Jan, month 2 = Feb, etc.): Model the temperature T as a sinusoidal function of time, using 36 as the minimum value and 79 as the maximum value. Support your answer graphically by graphing your function with a scatter plot.

Temperature Data The normal monthly Fahrenheit temperatures in Helena, MT, are shown in the table below (month 1 = Jan, month 2 = Feb, etc.): Model the temperature T as a sinusoidal function of time, using 20 as the minimum value and 68 as the maximum value. Support your answer graphically by graphing your function with a scatter plot.

True or False The graph of y = sin 2x has half the period of the graph of y = sin 4x. Justify your answer.

True or False Every sinusoid can be written as y = A cos (Bx + C) for some real numbers A, B, and C. Justify your answer.

You may use a graphing calculator when answering these questions. Multiple Choice A sinusoid with amplitude 4 has a minimum value of 5. Its maximum value is (A) 7. (B) 9. (C) 11. (D) 13. (E) 15.

You may use a graphing calculator when answering these questions. Multiple Choice The graph of y = f(x) is a sinusoid with period 45 passing through the point (6, 0). Which of the following can be determined from the given information? I. f(0) II. f(6) III. f(96) (A) I only (B) II only (C) I and III only (D) II and III only (E) I, II, and III only

You may use a graphing calculator when answering these questions. Multiple Choice The period of the function f(x) = 210 sin (420x + 840) is (A) \(\pi / 840\). (B) \(\pi / 420\). (C) \(\pi / 210\). (D) \(840 / \pi\). (E) \(420 / \pi\).

You may use a graphing calculator when answering these questions. Multiple Choice The number of solutions to the equation sin (2000x) = 3/7 in the interval \([0,\ 2\pi]\) is (A) 1000. (B) 2000. (C) 4000. (D) 6000. (E) 8000.

Approximating Cosine (a) Draw a scatter plot (x, cos, x) for the 17 special angles x, where \(-\pi \leq x \leq \pi\). (b) Find a quartic regression for the data. (c) Compare the approximation to the cosine function given by the quartic regression with the Taylor polynomial approximations given in Exercise 80 of Section 4.3.

Approximating Cosine (a) Draw a scatter plot (x, sin, x) for the 17 special angles x, where \(-\pi \leq x \leq \pi\). (b) Find a cubic regression for the data. (c) Compare the approximation to the sine function given by the cubic regression with the Taylor polynomial approximations given in Exercise 79 of Section 4.3.

Visualizing a Musical Note A piano tuner strikes a tuning fork for the note middle C and creates a sound wave that can be modeled by \(y=1.5 \sin 524 \pi t\) where t is the time in seconds. (a) What is the period p of this function? (b) What is the frequency f = 1/p of this note? (c) Graph the function.

Writing to Learn In a certain video game a cursor bounces back and forth horizontally across the screen at a constant rate. Its distance d from the center of the screen varies with time t and hence can be described as a function of t. Explain why this horizontal distance d from the center of the screen does not vary according to an equation d = a sin bt, where t represents seconds. You may find it helpful to include a graph in your explanation.

Group Activity Using only integer values of a and b between 1 and 9 inclusive, look at graphs of functions of the form y = sin (ax) cos (bx) - cos (ax) sin (bx) for various values of a and b. (A group can look at more graphs at a time than one person can.) (a) Some values of a and b result in the graph of y = sin x. Find a general rule for such values of a and b. (b) Some values of a and b result in the graph of y = sin 2x. Find a general rule for such values of a and b. (c) Can you guess which values of a and b will result in the graph of y = sin kx for an arbitrary integer k?

Group Activity Using only integer values of a and b between 1 and 9 inclusive, look at graphs of functions of the form y = cos (ax) cos (bx) + sin (ax) sin (bx) for various values of a and b. (A group can look at more graphs at a time than one person can.) (a) Some values of a and b result in the graph of y = cos x. Find a general rule for such values of a and b. (b) Some values of a and b result in the graph of y = cos 2x. Find a general rule for such values of a and b. (c) Can you guess which values of a and b will result in the graph of y = cos kx for an arbitrary integer k?

In Exercises 93–96, the graphs of the sine and cosine functions are waveforms like the figure below. By correctly labeling the coordinates of points A, B, and C, you will get the graph of the function given. y = 3 cos 2x and \(A=\left(-\frac{\pi}{4},\ 0\right)\). Find B and C.

In Exercises 93–96, the graphs of the sine and cosine functions are waveforms like the figure below. By correctly labeling the coordinates of points A, B, and C, you will get the graph of the function given. \(y=4.5 \sin \left(x-\frac{\pi}{4}\right)\) and \(A=\left(\frac{\pi}{4},\ 0\right)\). Find B and C.

In Exercises 93–96, the graphs of the sine and cosine functions are waveforms like the figure below. By correctly labeling the coordinates of points A, B, and C, you will get the graph of the function given. \(y=2.5 \sin \left(3x-\frac{\pi}{4}\right)\) and \(A=\left(\frac{\pi}{12},\ 0\right)\). Find B and C.

In Exercises 93–96, the graphs of the sine and cosine functions are waveforms like the figure below. By correctly labeling the coordinates of points A, B, and C, you will get the graph of the function given. \(y=3 \sin (2 x-\pi)\), and A is the first x-intercept on the right of the y-axis. Find A, B, and C.

The Ultimate Sinusoidal Equation It is an interesting fact that any sinusoid can be written in the form y = a sin [b(x - H)] + k, where both a and b are positive numbers. (a) Explain why you can assume b is positive. [Hint: Replace b by \(-B\) and simplify.] (b) Use one of the horizontal translation identities to prove that the equation y = a cos [b(x - h)] + k has the same graph as y = a sin [b(x - H)] + k for a correctly chosen value of H. Explain how to choose H. (c) Give a unit circle argument for the identity \(\sin (\theta+\pi)=-\sin \theta\). Support your unit circle argument graphically. (d) Use the identity from (c) to prove that \(y=-a \sin [b(x-h)]+k, a>0\), has the same graph as \(y=a \sin [b(x-H)]+k, a>0\) for a correctly chosen value of H. Explain how to choose H. (e) Combine your results from (a)–(d) to prove that any sinusoid can be represented by the equation y = a sin [b(x - H)] + k where a and b are both positive.