Chapter 4

In Problems \(65-68\), find the measure of a central angle \(\theta\) in a circle of radius \(r\) that subtends an arc length s. Give \(\theta\) in (a) radians and (b) degrees. $$ r=5 \mathrm{ft}, s=7.5 \mathrm{ft} $$

Describe in words how you would obtain the graph of the given function by starting with the graph of \(y=\sin x\) (Problem 65 ) and the graph of \(y=\cos x(\) Problem 66\()\). $$ y=-6+\frac{1}{4} \cos \left(\frac{1}{2} x+\pi\right) $$

Show that the given expression is not an identity. $$ (\sin x+\cos x)^{2}=\sin ^{2} x+\cos ^{2} x $$

Using a inverse trigonometric function find the solutions of the given equation in the indicated interval. Round your answers to two decimal places. $$ 3 \sin ^{2} x-8 \sin x+4=0,[-\pi / 2, \pi / 2] $$

In Problems \(65-68\), find the measure of a central angle \(\theta\) in a circle of radius \(r\) that subtends an arc length s. Give \(\theta\) in (a) radians and (b) degrees. $$ r=10 \mathrm{in}, s=36 \mathrm{in} $$

Find the period of the given function. $$ f(x)=\sin \frac{1}{2} x \sin 2 x $$

Using a inverse trigonometric function find the solutions of the given equation in the indicated interval. Round your answers to two decimal places. $$ \tan ^{2} x+\tan x-1=0,(-\pi / 2, \pi / 2) $$

In Problems \(65-68\), find the measure of a central angle \(\theta\) in a circle of radius \(r\) that subtends an arc length s. Give \(\theta\) in (a) radians and (b) degrees. $$ r=9 \mathrm{~m}, s=15 \mathrm{~m} $$

Rewrite the given function as a single trigonometric function involving no products or squares. Give the amplitude and period of the function. $$ y=4 \cos ^{2} x-2 $$

Find the period of the given function. $$ f(x)=\sin \frac{3}{2} x+\cos \frac{5}{2} x $$

Using a inverse trigonometric function find the solutions of the given equation in the indicated interval. Round your answers to two decimal places. $$ 3 \sin 2 x+\cos x=0,[-\pi / 2, \pi / 2] $$

In Problems \(65-68\), find the measure of a central angle \(\theta\) in a circle of radius \(r\) that subtends an arc length s. Give \(\theta\) in (a) radians and (b) degrees. $$ r=20 \mathrm{~cm}, s=90 \mathrm{~cm} $$

Rewrite the given function as a single trigonometric function involving no products or squares. Give the amplitude and period of the function. $$ y=\sin (x / 2) \cos (x / 2) $$

Discuss and then sketch the graph of the given function. $$ f(x)=|\sin x| $$

Using a inverse trigonometric function find the solutions of the given equation in the indicated interval. Round your answers to two decimal places. $$ 5 \cos ^{3} x-3 \cos ^{2} x-\cos x=0,[0, \pi] $$

In Problems \(69-72\), find the arc length \(s\) subtended by a central angle \(\theta\) in a circle of radius \(r\). $$ \theta=3 \text { radians, } r=5 \text { in } $$

Rewrite the given function as a single trigonometric function involving no products or squares. Give the amplitude and period of the function. $$ y=2 \sin 2 x \cos 2 x $$

Discuss and then sketch the graph of the given function. $$ f(x)=|\cos x| $$

Using a inverse trigonometric function find the solutions of the given equation in the indicated interval. Round your answers to two decimal places. $$ \tan ^{4} x-3 \tan ^{2} x+1=0,(-\pi / 2, \pi / 2) $$

In Problems \(69-72\), find the arc length \(s\) subtended by a central angle \(\theta\) in a circle of radius \(r\). $$ \theta=1.5 \text { radians, } r=4 \mathrm{~cm} $$

Rewrite the given function as a single trigonometric function involving no products or squares. Give the amplitude and period of the function. $$ y=5 \cos ^{2} 4 x-5 \sin ^{2} 4 x $$

In Problems \(69-72\), find the arc length \(s\) subtended by a central angle \(\theta\) in a circle of radius \(r\). $$ \theta=30^{\circ}, r=2 \mathrm{~m} $$

If \(P\left(x_{1}\right)\) and \(P\left(x_{2}\right)\) are points in quadrant II on the terminal side of the angles \(x_{1}\) and \(x_{2}\), respectively, with \(\cos x_{1}=-\frac{1}{3}\) and \(\sin x_{2}=\frac{2}{3}\), find (a) \(\sin \left(x_{1}+x_{2}\right),(b) \cos \left(x_{1}+x_{2}\right),(\mathbf{c}) \sin \left(x_{1}-x_{2}\right)\), and (d) \(\cos \left(x_{1}-x_{2}\right)\).

An object travels in a circular path centered at the origin with constant angular speed. The \(y\) -coordinate of the object at any time \(t\) seconds is given by \(y=8 \cos (\pi t-\pi / 12) .\) At what time(s) does the object cross the \(x\) -axis?

In Problems \(69-72\), find the arc length \(s\) subtended by a central angle \(\theta\) in a circle of radius \(r\). $$ \theta=15^{\circ}, r=6 \mathrm{ft} $$

If \(x\) is a quadrant II angle, \(x_{2}\) is a quadrant III angle, \(\sin x_{1}=\frac{8}{17}\), and \(\tan x_{2}=\frac{3}{4}\), find (a) \(\sin \left(x_{1}+x_{2}\right),(b) \sin \left(x_{1}-x_{2}\right),(c) \cos \left(x_{1}+x_{2}\right)\), and (d) \(\cos \left(x_{1}-x_{2}\right)\).

In Problems \(73-76\), find the area of the circular sector having the given radius \(r\) and central angle \(\theta\). $$ r=3 \mathrm{ft}, \theta=7.2 \text { radians } $$

An electric generator produces a 6o-cycle alternating current given by \(I(t)=30 \sin 120 \pi\left(t-\frac{7}{36}\right)\), where \(I(t)\) is the current in amperes at \(t\) seconds. Find the smallest positive value of \(t\) for which the current is 15 amperes.

In Problems \(73-76\), find the area of the circular sector having the given radius \(r\) and central angle \(\theta\). $$ r=18 \text { in, } \theta=\frac{2 \pi}{3} \text { radians } $$

If the voltage given by \(V=\) \(V_{\mathrm{o}} \sin (\omega t+\alpha)\) is impressed on a series circuit, an alternating current is produced. If \(V_{\mathrm{o}}=110\) volts, \(\omega=120 \pi\) radians per second, and \(\alpha=-\pi / 6,\) when is the voltage equal to zero?

In Problems \(73-76\), find the area of the circular sector having the given radius \(r\) and central angle \(\theta\). $$ r=6 \mathrm{~m}, \theta=30^{\circ} $$

We saw in Problem 56 of Exercises \(4.2\) that if a projectile, such as a shot put, is released from a height \(h\), upward at an angle \(\theta\) with velocity \(v_{\mathrm{o}}\), the range \(R\) at which it strikes the ground is given by $$ R=\frac{v_{0} \cos \theta}{g}\left(v_{0} \sin \theta+\sqrt{v_{0}^{2} \sin ^{2} \theta+2 g h}\right), $$ where \(g\) is the acceleration due to gravity. (a) Show that when \(h=0\) the range of the projectile is $$ R=\frac{v_{0}^{2} \sin 2 \theta}{g} $$ (b) It can be shown that the maximum range \(R_{\max }\) is achieved when the angle \(\theta\) satisfies the equation $$ \cos 2 \theta=\frac{g h}{v_{0}^{2}+g h} $$ Show that maximum range is $$ R_{\max }=\frac{v_{0} \sqrt{v_{0}^{2}+2 g h}}{g} $$ by using the expressions for \(R\) and \(\cos 2 \theta\) and the half-angle formulas for the sine and the cosine with \(t=2 \theta\).

In Problems \(73-76\), find the area of the circular sector having the given radius \(r\) and central angle \(\theta\). $$ r=12 \mathrm{~cm}, \theta=75^{\circ} $$

Why would you expect to get an error message from your calculator when you try to evaluate $$ \frac{\tan 35^{\circ}+\tan 55^{\circ}}{1-\tan 35^{\circ} \tan 55^{\circ}} ? $$

On the basis of data collected from 1966 to \(1980,\) the extent of snow cover \(S\) in the northern hemisphere, measured in millions of square kilometers, can be modeled by the function $$ S(w)=25+21 \cos \frac{\pi}{26}(w-5) $$ where \(w\) is the number of weeks past January \(1 .\) (a) How much snow cover does this formula predict for April Fool's Day? (Round \(w\) to the nearest integer.) (b) In which week does the formula predict the least amount of snow cover? (c) What month does this fall in?

How would you find a formula that expresses \(\sin 3 \theta\) in terms of \(\sin \theta\) ? Carry out your ideas.

In Problems 79 and \(80,\) use a graphing utility to obtain the graph of the given function. Find all solutions of the equation \(f(x)=0\) and an interval that contains all these solutions $$ f(x)=\frac{\sin x}{x} $$

Use a graphing utility to obtain the graph of the given function. Find all solutions of the equation \(f(x)=0\) and an interval that contains all these solutions. $$ f(x)=\cos \left(\frac{\pi}{2 x}\right) $$

What are degree and the radian measures of the angle through which the minute hand on the analog clock in Figure 4.1 .13 rotates in (a) \(\frac{3}{4}\) hour, and (b) 3.5 hours?

Planet Earth The Earth rotates on its axis once every 24 hours. How long does it take the Earth to rotate through an angle of (a) \(240^{\circ}\) and (b) \(\pi / 6\) radian?

Planet Mercury The planet Mercury completes one rotation on its axis every 59 days. Through what angle (measured in degrees) does it rotate in (a) 1 day \(,\) (b) 1 hour, and (c) 1 minute?

Pendulum Clock A clock pendulum is \(1.3 \mathrm{~m}\) long and swings back and forth along a \(15-\mathrm{cm}\) arc. Find (a) the central angle and (b) the area of the sector through which the pendulum sweeps in one swing.

Sailing at Sea A nautical mile is defined as the arc length subtended on the surface of the Earth by an angle of measure 1 minute. If the diameter of the Earth is 7927 miles, find how many statute (land) miles there are in a nautical mile.

Circular Motion of a Yo-Yo A yo-yo is whirled around in a circle at the end of its \(100-\mathrm{cm}\) string. (a) If it makes six revolutions in 4 seconds, find its rate of turning, or angular speed, in radians per second. (b) Find the speed at which the yo-yo travels in centimeters per second; that is its linear speed.

Circular Motion of a Car Tire An automobile with 26 -in. diameter tires is traveling at a rate of \(55 \mathrm{mi} / \mathrm{h}\) (a) Find the number of revolutions per minute that its tires are making. (b) Find the angular speed of its tires in radians per minute.