Chapter 5

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

Potation of compact disces (COs) 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. Find the angular speed (in rpm) of the drive motor when it is reading a track 3 centimeters from the center of the CD. Find a function \(S\) that gives the drive motor speed in \(\mathrm{rpm}\) 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 \(r=5.7\)

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. $$\tan x=1$$

Use fundamental Identities to write the first expression in terms of the second, for any acute angle \(\boldsymbol{\theta}\). $$\sin \theta, \sec \theta$$

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

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.

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.

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.

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

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

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 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. $$\cos \theta \sec \theta=1$$

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=\tan \frac{\pi}{2} x$$

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.

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 \(a=0.2 \mathrm{m}\)

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. $$\tan x=-1 / \sqrt{3}$$

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

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

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?

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.

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

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

Low temperature in Fairbanics Based on years of weather data, the expected low temperature \(T\) (in '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 I. 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. $$\sin \theta \cot \theta=\cos \theta$$

Graph the equation \(y=f(t)\) on the inter val \([0,24] .\) Let \(y\) represent the outdoor temperature \(\left(\text { in }^{\circ} \mathrm{F}\right)\) at time \(t\) (in hours), where \(t=0\) corresponds to 9 A.M. Describe the temperature during the 24 -hour interval. \(y=20+15 \sin \frac{\pi}{12} t\)

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

Find an equation using the cotangent function that has the same graph as \(y=\tan x\)

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

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 \(5^{\circ} 7\)

Graph the equation \(y=f(t)\) on the inter val \([0,24] .\) Let \(y\) represent the outdoor temperature \(\left(\text { in }^{\circ} \mathrm{F}\right)\) at time \(t\) (in hours), where \(t=0\) corresponds to 9 A.M. Describe the temperature during the 24 -hour interval. \(y=80+22 \cos \left[\frac{\pi}{12}(t-3)\right]\)

Find an equation using the cosecant function that has the same graph as \(y=\sec x\)

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

Use the graph of a trigonometric function to ald 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. $$(1+\cos 2 \theta)(1-\cos 2 \theta)=\sin ^{2} 2 \theta$$

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

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 \(^{\circ} \mathbf{C},\) and \(t=\mathbf{0}\) corresponding to midnight. Assume that \(f(t)\) is decreasing at midnight. A. Determine values of \(a, b, c,\) and \(d\) that int 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 \(9 \mathrm{A} . \mathrm{M}\).

Use the graph of a trigonometric function to ald 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. $$\cos ^{2} \theta\left(\sec ^{2} \theta-1\right)=\sin ^{2} \theta$$

Use the graph of a trigonometric function to ald in sketching the graph of the equation without plotting points. $$y=|\cos x|-3$$

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. $$1-2 \sin ^{2}(\theta / 2)=2 \cos ^{2}(\theta / 2)-1$$

The number of daylight hours \(D\) at a particular location varies with both the month and the latitude. The table lists the number of daylight hours on the first day of each month at \(60^{\circ} \mathrm{N}\) latitude. $$\begin{array}{|lc|c|c|cc|} \hline \text { Month } & \boldsymbol{D} & \text { Month } & \boldsymbol{D} & \text { Month } & \boldsymbol{D} \\ \hline \text { Jan. } & 6.03 & \text { May } & 15.97 & \text { Sept. } & 14.18 \\ \hline \text { Feb. } & 7.97 & \text { June } & 18.28 & \text { Oct. } & 11.50 \\ \hline \text { March } & 10.43 & \text { July } & 18.72 & \text { Nov. } & 8.73 \\ \hline \text { April } & 13.27 & \text { Aug. } & 16.88 & \text { Dec. } & 5.88 \\ \hline \end{array}$$ A. Let \(t\) be time in months, with \(t=1\) corresponding to January, \(t=2\) to February, \(\ldots, t=12\) to December, \(t=13\) to January, and so on. Plot the data for a twoyear period. B. Find a function \(D(t)=a \sin (b t+c)+d\) that ap proximates the number of daylight hours. Graph the function \(D\) with the data.

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

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

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

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