Chapter 6

Rotation of compact discs (CDs) The drive motor of a particular CD player is controlled to rotate at a speed of \(200 \mathrm{rpm}\) when reading a track \(5.7\) centimeters from the center of the CD. The speed of the drive motor must vary so that the reading of the data occurs at a constant rate. (a) Find the angular speed (in radians per minute) of the drive motor when it is reading a track \(5.7\) centimeters from the center of the CD. (b) Find the linear speed (in \(\mathrm{cm} / \mathrm{sec}\) ) of a point on the CD that is \(5.7\) centimeters from the center of the CD. (c) Find the angular speed (in rpm) of the drive motor when it is reading a track 3 centimeters from the center of the CD. (d) Find a function \(S\) that gives the drive motor speed in rpm for any radius \(r\) in centimeters, where \(2.3 \leq r \leq 5.9\). What type of variation exists between the drive motor speed and the radius of the track being read? Check your answer by graphing \(S\) and finding the speeds for \(r=3\) and \(r=5.7\).

From a point \(A\) that is \(8.20\) meters above level ground, the angle of elevation of the top of a building is \(31^{\circ} 20^{\prime}\) and the angle of depression of the base of the building is \(12^{\circ} 50^{\prime}\). Approximate the height of the building.

Exer. 47-50: Refer to the graph of \(y=\tan x\) to find the exact values of \(x\) in the interval \((-\pi / 2,3 \pi / 2)\) that satisfy the equation. $$ \tan x=\sqrt{3} $$

The popular biorhythm theory uses the graphs of three simple sine functions to make predictions about an individual's physical, emotional, and intellectual potential for a particular day. The graphs are given by \(y=a \sin b t\) for \(t\) in days, with \(t=0\) corresponding to birth and \(a=1\) denoting \(100 \%\) potential. (a) Find the value of \(b\) for the physical cycle, which has a period of 23 days; for the emotional cycle (period 28 days); and for the intellectual cycle (period 33 days). (b) Evaluate the biorhythm cycles for a person who has just become 21 years of age and is exactly 7670 days old.

Find the period and sketch the graph of the equation. Show the asymptotes. $$ y=4 \csc \left(\frac{1}{2} x-\frac{\pi}{4}\right) $$

Use fundamental identities to write the first expression in terms of the second, for any acute angle \(\theta\). $$\cos \theta, \cot \theta$$

Tire revolutions A typical tire for a compact car is 22 inches in diameter. If the car is traveling at a speed of \(60 \mathrm{mi} / \mathrm{hr}\), find the number of revolutions the tire makes per minute.

Exer. 47-50: Refer to the graph of \(y=\tan x\) to find the exact values of \(x\) in the interval \((-\pi / 2,3 \pi / 2)\) that satisfy the equation. $$ \tan x=0 $$

The height of the tide at a particular point on shore can be predicted by using seven trigonometric functions (called tidal components) of the form $$ f(t)=a \cos (b t+c) . $$ The principal lunar component may be approximated by $$ f(t)=a \cos \left(\frac{\pi}{6} t-\frac{11 \pi}{12}\right), $$ where \(t\) is in hours and \(t=0\) corresponds to midnight. Sketch the graph of \(f\) if \(a=0.5 \mathrm{~m}\).

Verify the identity by transforming the lefthand side into the right-hand side. $$\cos \theta \sec \theta=1$$

Cargo winch A large winch of diameter 3 feet is used to hoist cargo, as shown in the figure. (a) Find the distance the cargo is lifted if the winch rotates through an angle of radian measure \(7 \pi / 4\). (b) Find the angle (in radians) through which the winch must rotate in order to lift the cargo \(d\) feet.

Refer to Exercise 49. The principal solar diurnal component may be approximated by $$ f(t)=a \cos \left(\frac{\pi}{12} t-\frac{7 \pi}{12}\right) $$ Sketch the graph of \(f\) if \(a=0.2 \mathrm{~m}\).

Find the period and sketch the graph of the equation. Show the asymptotes. $$ y=\cot \pi x $$

Verify the identity by transforming the lefthand side into the right-hand side. $$\tan \theta \cot \theta=1$$

Pendulum's swing A pendulum in a grandfather clock is 4 feet long and swings back and forth along a 6-inch arc. Approximate the angle (in degrees) through which the pendulum passes during one swing.

An airplane flying at an altitude of 10,000 feet passes directly over a fixed object on the ground. One minute later, the angle of depression of the object is \(42^{\circ}\). Approximate the speed of the airplane to the nearest mile per hour.

Exer. 51-54: Refer to the graph of the equation on the specified interval. Find all values of \(x\) such that for the real number \(a\), (a) \(y=a\), (b) \(y>a\), and (c) \(y

Find the period and sketch the graph of the equation. Show the asymptotes. $$ y=\csc 2 \pi x $$

Verify the identity by transforming the lefthand side into the right-hand side. $$\sin \theta \sec \theta=\tan \theta$$

Pizza values A vender sells two sizes of pizza by the slice. The small slice is \(\frac{1}{6}\) of a circular 18 -inch-diameter pizza, and it sells for \(\$ 2.00\). The large slice is \(\frac{1}{8}\) of a circular 26 -inchdiameter pizza, and it sells for \(\$ 3.00\). Which slice provides more pizza per dollar?

A motorist, traveling along a level highway at a speed of \(60 \mathrm{~km} / \mathrm{hr}\) directly toward a mountain, observes that between 1:00 P.M. and 1:10 P.M. the angle of elevation of the top of the mountain changes from \(10^{\circ}\) to \(70^{\circ}\). Approximate the height of the mountain.

Exer. 51-54: Refer to the graph of the equation on the specified interval. Find all values of \(x\) such that for the real number \(a\), (a) \(y=a\), (b) \(y>a\), and (c) \(y

Based on years of weather data, the expected low temperature \(T\) (in \({ }^{\circ} \mathrm{F}\) ) in Fairbanks, Alaska, can be approximated by $$ T=36 \sin \left[\frac{2 \pi}{365}(t-101)\right]+14, $$ where \(t\) is in days and \(t=0\) corresponds to January 1 . (a) Sketch the graph of \(T\) for \(0 \leq t \leq 365\). (b) Predict when the coldest day of the year will occur.

Find the period and sketch the graph of the equation. Show the asymptotes. $$ y=\sec \frac{\pi}{8} x $$

Verify the identity by transforming the lefthand side into the right-hand side. $$\sin \theta \cot \theta=\cos \theta$$

Verify the identity by transforming the lefthand side into the right-hand side. $$\frac{\csc \theta}{\sec \theta}=\cot \theta$$

Exer. 53-56: Scientists sometimes use the formula $$ f(t)=a \sin (b t+c)+d $$ to simulate temperature variations during the day, with time \(t\) in hours, temperature \(f(t)\) in \({ }^{\circ} \mathrm{C}\), and \(t=0\) corresponding to midnight. Assume that \(f(t)\) is decreasing at midnight. (a) Determine values of \(a, b, c\), and \(d\) that fit the information. (b) Sketch the graph of \(f\) for \(0 \leq t \leq 24\). The temperature at midnight is \(15^{\circ} \mathrm{C}\), and the high and low temperatures are \(20^{\circ} \mathrm{C}\) and \(10^{\circ} \mathrm{C}\).

Verify the identity by transforming the lefthand side into the right-hand side. $$\cot \theta \sec \theta=\csc \theta$$

Magnetic pole drift The geographic and magnetic north poles have different locations. Currently, the magnetic north pole is drifting westward through \(0.0017\) radian per year, where the angle of drift has its vertex at the center of Earth. If this movement continues, approximately how many years will it take for the magnetic north pole to drift a total of \(5^{\circ}\) ?

Exer. 53-56: Scientists sometimes use the formula $$ f(t)=a \sin (b t+c)+d $$ to simulate temperature variations during the day, with time \(t\) in hours, temperature \(f(t)\) in \({ }^{\circ} \mathrm{C}\), and \(t=0\) corresponding to midnight. Assume that \(f(t)\) is decreasing at midnight. (a) Determine values of \(a, b, c\), and \(d\) that fit the information. (b) Sketch the graph of \(f\) for \(0 \leq t \leq 24\). The temperature varies between \(10^{\circ} \mathrm{C}\) and \(30^{\circ} \mathrm{C}\), and the average temperature of \(20^{\circ} \mathrm{C}\) first occurs at 9 A.M.

Use the graph of a trigonometric function to aid in sketching the graph of the equation without plotting points. $$ y=|\sin x| $$

Verify the identity by transforming the lefthand side into the right-hand side. $$(1+\cos 2 \theta)(1-\cos 2 \theta)=\sin ^{2} 2 \theta$$

Exer. 55-62: Use the graph of a trigonometric function to sketch the graph of the equation without plotting points. $$ y=3+\cos x $$

Exer. 53-56: Scientists sometimes use the formula $$ f(t)=a \sin (b t+c)+d $$ to simulate temperature variations during the day, with time \(t\) in hours, temperature \(f(t)\) in \({ }^{\circ} \mathrm{C}\), and \(t=0\) corresponding to midnight. Assume that \(f(t)\) is decreasing at midnight. (a) Determine values of \(a, b, c\), and \(d\) that fit the information. (b) Sketch the graph of \(f\) for \(0 \leq t \leq 24\). The high temperature of \(28^{\circ} \mathrm{C}\) occurs at 2 P.M., and the average temperature of \(20^{\circ} \mathrm{C}\) occurs 6 hours later.

Verify the identity by transforming the lefthand side into the right-hand side. $$\cos ^{2} 2 \theta-\sin ^{2} 2 \theta=2 \cos ^{2} 2 \theta-1$$

Use the graph of a trigonometric function to aid in sketching the graph of the equation without plotting points. $$ y=|\sin x|+2 $$

Verify the identity by transforming the lefthand side into the right-hand side. $$\cos ^{2} \theta\left(\sec ^{2} \theta-1\right)=\sin ^{2} \theta$$

Verify the identity by transforming the lefthand side into the right-hand side. $$(\tan \theta+\cot \theta) \tan \theta=\sec ^{2} \theta$$

Verify the identity by transforming the lefthand side into the right-hand side. $$\frac{\sin (\theta / 2)}{\csc (\theta / 2)}+\frac{\cos (\theta / 2)}{\sec (\theta / 2)}=1$$

Verify the identity by transforming the lefthand side into the right-hand side. $$1-2 \sin ^{2}(\theta / 2)=2 \cos ^{2}(\theta / 2)-1$$

Sketch the graph of the equation. $$ y=x+\cos x $$

Sketch the graph of the equation. $$ y=x-\sin x $$

Verify the identity by transforming the lefthand side into the right-hand side. $$\left(1-\sin ^{2} \theta\right)\left(1+\tan ^{2} \theta\right)=1$$

A ship leaves port at 1:00 P.M. and sails in the direction \(\mathrm{N} 34^{\circ} \mathrm{W}\) at a rate of \(24 \mathrm{mi} / \mathrm{hr}\). Another ship leaves port at 1:30 P.M. and sails in the direction \(\mathrm{N} 56^{\circ} \mathrm{E}\) at a rate of \(18 \mathrm{mi} / \mathrm{hr}\). (a) Approximately how far apart are the ships at 3:00 P.M.? (b) What is the bearing, to the nearest degree, from the first ship to the second?

Exer. 63-66: Find the intervals between \(-2 \pi\) and \(2 \pi\) on which the given function is (a) increasing or (b) decreasing. secant

Sketch the graph of the equation. $$ y=2^{-x} \cos x $$

Verify the identity by transforming the lefthand side into the right-hand side. $$\sec \theta-\cos \theta=\tan \theta \sin \theta$$

Sketch the graph of the equation. $$ y=e^{x} \sin x $$

Verify the identity by transforming the lefthand side into the right-hand side. $$\frac{\sin \theta+\cos \theta}{\cos \theta}=1+\tan \theta$$

An airplane flying at a speed of \(360 \mathrm{mi} / \mathrm{hr}\) flies from a point \(A\) in the direction \(137^{\circ}\) for 30 minutes and then flies in the direction \(227^{\circ}\) for 45 minutes. Approximate, to the nearest mile, the distance from the airplane to \(A\).