Chapter 7

Exer. 53-64: Use inverse trigonometric functions to find the solutions of the equation that are in the given interval, and approximate the solutions to four decimal places. \(6 \sin 2 x-8 \cos x+9 \sin x-6=0 ; \quad\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ \sec ^{5} \theta=4 \sec \theta $$

Exer. 51-60: Show that the equation is not an identity. (Hint: Find one number for which the equation is false.) $$ \cot (\tan \theta)=1 $$

Exer. 57-62: Use an addition or subtraction formula to find the solutions of the equation that are in the interval \([0, \pi\) ). $$ \sin 3 t \cos t+\cos 3 t \sin t=-\frac{1}{2} $$

Exer. 53-64: Use inverse trigonometric functions to find the solutions of the equation that are in the given interval, and approximate the solutions to four decimal places. \((\cos x)(15 \cos x+4)=3\) \([0,2 \pi)\)

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ 2 \tan t \csc t+2 \csc t+\tan t+1=0 $$

Exer. 61-64: Either show that the equation is an identity or show that the equation is not an identity. $$ (\sec x+\tan x)^{2}=2 \tan x(\tan x+\sec x) $$

Exer. 57-62: Use an addition or subtraction formula to find the solutions of the equation that are in the interval \([0, \pi\) ). $$ \tan 2 t+\tan t=1-\tan 2 t \tan t $$

Exer. 53-64: Use inverse trigonometric functions to find the solutions of the equation that are in the given interval, and approximate the solutions to four decimal places. \(6 \sin ^{2} x=\sin x+2\) \([0,2 \pi)\)

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ 2 \sin v \csc v-\csc v=4 \sin v-2 $$

Exer. 61-64: Either show that the equation is an identity or show that the equation is not an identity. $$ \frac{\tan ^{2} x}{\sec x-1}=\sec x $$

Exer. 57-62: Use an addition or subtraction formula to find the solutions of the equation that are in the interval \([0, \pi\) ). $$ \tan t-\tan 4 t=1+\tan 4 t \tan t $$

Exer. 53-64: Use inverse trigonometric functions to find the solutions of the equation that are in the given interval, and approximate the solutions to four decimal places. \(3 \cos 2 x-7 \cos x+5=0\) \([0,2 \pi)\)

Exer. 63-68: Approximate, to the nearest 10', the solutions of the equation in the interval \(\left[0^{\circ}, 360^{\circ}\right)\). $$ \sin ^{2} t-4 \sin t+1=0 $$

Exer. 61-64: Either show that the equation is an identity or show that the equation is not an identity. $$ \cos x(\tan x+\cot x)=\csc x $$

Exer. 53-64: Use inverse trigonometric functions to find the solutions of the equation that are in the given interval, and approximate the solutions to four decimal places. \(\sin 2 x=-1.5 \cos x\) \([0,2 \pi)\)

Exer. 63-68: Approximate, to the nearest 10', the solutions of the equation in the interval \(\left[0^{\circ}, 360^{\circ}\right)\). $$ \cos ^{2} t-4 \cos t+2=0 $$

Exer. 61-64: Either show that the equation is an identity or show that the equation is not an identity. $$ \csc ^{2} x+\sec ^{2} x=\csc ^{2} x \sec ^{2} x $$

Exer. 65-66: If an earthquake has a total horizontal displacement of \(S\) meters along its fault line, then the horizontal movement \(M\) of a point on the surface of Earth \(d\) kilometers from the fault line can be estimated using the formula $$ M=\frac{S}{2}\left(1-\frac{2}{\pi} \tan ^{-1} \frac{d}{D}\right) $$ where \(D\) is the depth (in kilometers) below the surface of the focal point of the earthquake. For the San Francisco earthquake of \(1906, S\) was 4 meters and \(D\) was \(3.5\) kilometers. Approximate \(M\) for the stated values of \(d\). (a) 1 kilometer (b) 4 kilometers (c) 10 kilometers

Exer. 63-68: Approximate, to the nearest 10', the solutions of the equation in the interval \(\left[0^{\circ}, 360^{\circ}\right)\). $$ \tan ^{2} \theta+3 \tan \theta+2=0 $$

Exer. 65-66: If an earthquake has a total horizontal displacement of \(S\) meters along its fault line, then the horizontal movement \(M\) of a point on the surface of Earth \(d\) kilometers from the fault line can be estimated using the formula $$ M=\frac{S}{2}\left(1-\frac{2}{\pi} \tan ^{-1} \frac{d}{D}\right) $$ where \(D\) is the depth (in kilometers) below the surface of the focal point of the earthquake. Approximate the depth \(D\) of the focal point of an earthquake with \(S=3 \mathrm{~m}\) if a point on the surface of Earth 5 kilometers from the fault line moved \(0.6\) meter horizontally.

Exer. 63-68: Approximate, to the nearest 10', the solutions of the equation in the interval \(\left[0^{\circ}, 360^{\circ}\right)\). $$ 2 \tan ^{2} x-3 \tan x-1=0 $$

Exer. 63-66: (a) Use the formula from Example 6 to express \(f\) in terms of the cosine function. (b) Determine the amplitude, period, and phase shift of \(f\). (c) Sketch the graph of \(f\). $$ f(x)=5 \cos 10 x-5 \sin 10 x $$

A golfer, centered in a 30-yard-wide straight fairway, hits a ball 280 yards. Approximate the largest angle the drive can have from the center of the fairway if the ball is to stay in the fairway (see the figure).

Exer. 63-68: Approximate, to the nearest 10', the solutions of the equation in the interval \(\left[0^{\circ}, 360^{\circ}\right)\). $$ 12 \sin ^{2} u-5 \sin u-2=0 $$

Exer. 67-68: For certain applications in electrical engineering, the sum of several voltage signals or radio waves of the same frequency is expressed in the compact form \(y=A \cos (B t-C)\). Express the given signal in this form. $$ y=50 \sin 60 \pi t+40 \cos 60 \pi t $$

Exer. 63-68: Approximate, to the nearest 10', the solutions of the equation in the interval \(\left[0^{\circ}, 360^{\circ}\right)\). $$ 5 \cos ^{2} \alpha+3 \cos \alpha-2=0 $$

Exer. 67-68: For certain applications in electrical engineering, the sum of several voltage signals or radio waves of the same frequency is expressed in the compact form \(y=A \cos (B t-C)\). Express the given signal in this form. $$ y=10 \sin \left(120 \pi t-\frac{\pi}{2}\right)+5 \sin 120 \pi t $$

A tidal wave of height 50 feet and period 30 minutes is approaching a sea wall that is \(12.5\) feet above sea level (see the figure). From a particular point on shore, the distance \(y\) from sea level to the top of the wave is given by $$ y=25 \cos \frac{\pi}{15} t $$ with \(t\) in minutes. For approximately how many minutes of each 30 -minute period is the top of the wave above the level of the top of the sea wall?

Exer. 69-72: Make the trigonometric substitution $$ x=a \tan \theta \text { for }-\pi / 2<\theta<\pi / 2 \text { and } a>0 \text {. } $$ Simplify the resulting expression. $$ \sqrt{a^{2}+x^{2}} $$

If a mass that is attached to a spring is raised \(y_{0}\) feet and released with an initial vertical velocity of \(v_{0} \mathrm{ft} / \mathrm{sec}\), then the subsequent position \(y\) of the mass is given by $$ y=y_{0} \cos \omega t+\frac{v_{0}}{\omega} \sin \omega t, $$ where \(t\) is time in seconds and \(\omega\) is a positive constant. (a) If \(\omega=1, y_{0}=2 \mathrm{ft}\), and \(v_{0}=3 \mathrm{ft} / \mathrm{sec}\), express \(y\) in the form \(A \cos (B t-C)\), and find the amplitude and period of the resulting motion. (b) Determine the times when \(y=0\)-that is, the times when the mass passes through the equilibrium position.

The expected low temperature \(T\) (in \({ }^{\circ} \mathrm{F}\) ) in Fairbanks, Alaska, may be approximated by $$ T=36 \sin \left[\frac{2 \pi}{365}(t-101)\right]+14 $$ where \(t\) is in days, with \(t=0\) corresponding to January 1 . For how many days during the year is the low temperature expected to be below \(-4^{\circ} \mathrm{F}\) ?

Exer. 69-72: Make the trigonometric substitution $$ x=a \tan \theta \text { for }-\pi / 2<\theta<\pi / 2 \text { and } a>0 \text {. } $$ Simplify the resulting expression. $$ \frac{1}{\sqrt{a^{2}+x^{2}}} $$

Exer. 71-76: Verify the identity. $$ \sin ^{-1} x=\tan ^{-1} \frac{x}{\sqrt{1-x^{2}}} $$

On a clear day with \(D\) hours of daylight, the intensity of sunlight \(I\) (in calories \(/ \mathrm{cm}^{2}\) ) may be approximated by $$ I=I_{\mathrm{M}} \sin ^{3} \frac{\pi t}{D} \text { for } 0 \leq t \leq D, $$ where \(t=0\) corresponds to sunrise and \(I_{\mathrm{M}}\) is the maximum intensity. If \(D=12\), approximately how many hours after sunrise is \(I=\frac{1}{2} I_{\mathrm{M}}\) ?

Exer. 69-72: Make the trigonometric substitution $$ x=a \tan \theta \text { for }-\pi / 2<\theta<\pi / 2 \text { and } a>0 \text {. } $$ Simplify the resulting expression. $$ \frac{1}{x^{2}+a^{2}} $$

Exer. 71-76: Verify the identity. $$ \arccos x+\arccos \sqrt{1-x^{2}}=\frac{\pi}{2}, 0 \leq x \leq 1 $$

Exer. 69-72: Make the trigonometric substitution $$ x=a \tan \theta \text { for }-\pi / 2<\theta<\pi / 2 \text { and } a>0 \text {. } $$ Simplify the resulting expression. $$ \left(x^{2}+a^{2}\right)^{3 / 2} $$

Exer. 71-76: Verify the identity. $$ \arcsin (-x)=-\arcsin x $$

Exer. 73-76: Make the trigonometric substitution $$ x=a \sec \theta \text { for } 0<\theta<\pi / 2 \text { and } a>0 \text {. } $$ Simplify the resulting expression. $$ \sqrt{x^{2}-a^{2}} $$

Exer. 71-76: Verify the identity. $$ \arccos (-x)=\pi-\arccos x $$

In the study of frost penetration problems in highway engineering, the temperature \(T\) at time \(t\) hours and depth \(x\) feet is given by $$ T=T_{0} e^{-\lambda x} \sin (\omega t-\lambda x), $$ where \(T_{0}, \omega\), and \(\lambda\) are constants and the period of \(T\) is 24 hours. (a) Find a formula for the temperature at the surface. (b) At what times is the surface temperature a minimum? (c) If \(\lambda=2.5\), find the times when the temperature is a minimum at a depth of 1 foot.

Exer. 73-76: Make the trigonometric substitution $$ x=a \sec \theta \text { for } 0<\theta<\pi / 2 \text { and } a>0 \text {. } $$ Simplify the resulting expression. $$ \frac{1}{x^{2} \sqrt{x^{2}-a^{2}}} $$

Many animal populations, such as that of rabbits, fluctuate over ten-year cycles. Suppose that the number of rabbits at time \(t\) (in years) is given by $$ N(t)=1000 \cos \frac{\pi}{5} t+4000 . $$ (a) Sketch the graph of \(N\) for \(0 \leq t \leq 10\). (b) For what values of \(t\) in part (a) does the rabbit population exceed 4500 ?

Exer. 73-76: Make the trigonometric substitution $$ x=a \sec \theta \text { for } 0<\theta<\pi / 2 \text { and } a>0 \text {. } $$ Simplify the resulting expression. $$ x^{3} \sqrt{x^{2}-a^{2}} $$

Exer. 71-76: Verify the identity. $$ 2 \cos ^{-1} x=\cos ^{-1}\left(2 x^{2}-1\right), 0 \leq x \leq 1 $$

The flow rate (or water discharge rate) at the mouth of the Orinoco River in South America may be approximated by $$ F(t)=26,000 \sin \left[\frac{\pi}{6}(t-5.5)\right]+34,000, $$ where \(t\) is the time in months and \(F(t)\) is the flow rate in \(\mathrm{m}^{3} / \mathrm{sec}\). For approximately how many months each year does the flow rate exceed \(55,000 \mathrm{~m}^{3} / \mathrm{sec}\) ?

Exer. 73-76: Make the trigonometric substitution $$ x=a \sec \theta \text { for } 0<\theta<\pi / 2 \text { and } a>0 \text {. } $$ Simplify the resulting expression. $$ \frac{\sqrt{x^{2}-a^{2}}}{x^{2}} $$