Chapter 7

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$ y=\frac{1}{2}-\frac{1}{2} \cos \left(2 x-\frac{\pi}{3}\right) $$

7–52 Find the period and graph the function. $$y=5 \csc \frac{3 \pi}{2} x$$

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\) $$ \left(\frac{24}{25},-\frac{7}{25}\right) $$

The pendulum in a grandfather clock makes one complete swing every 2 s. The maximum angle that the pendulum makes with respect to its rest position is \(10^{\circ} .\) We know from physical principles that the angle \(\theta\) between the pendulum and its rest position changes in simple harmonic fashion. Find an equation that describes the size of the angle \(\theta\) as a function of time. (Take \(t=0\) to be a time when the pendulum is vertical.)

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$ y=1+\cos \left(3 x+\frac{\pi}{2}\right) $$

7–52 Find the period and graph the function. $$y=5 \sec 2 \pi x$$

The variable star Zeta Gemini has a period of 10 days. The average brightness of the star is 3.8 magnitudes, and the maximum variation from the average is 0.2 magnitude. Assuming that the variation in brightness is simple harmonic, find an equation that gives the brightness of the star as a function of time.

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$ y=3 \cos \pi\left(x+\frac{1}{2}\right) $$

7–52 Find the period and graph the function. $$y=\tan 2\left(x+\frac{\pi}{2}\right)$$

Astronomers believe that the radius of a variable star increases and decreases with the brightness of the star. The variable star Delta Cephei (Example 4) has an average radius of 20 million miles and changes by a maximum of 1.5 million miles from this average during a single pulsation. Find an equation that describes the radius of this star as a function of time.

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$ y=3+2 \sin 3(x+1) $$

Find (a) the reference number for each value of t, and (b) the terminal point determined by t. $$ t=\frac{4 \pi}{3} $$

7–52 Find the period and graph the function. $$y=\csc 2\left(x+\frac{\pi}{2}\right)$$

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$ y=\sin (\pi+3 x) $$

7–52 Find the period and graph the function. $$y=\tan 2(x-\pi)$$

Circadian rhythms are biological processes that oscillate with a period of approximately 24 hours. That is, a circadian rhythm is an internal daily biological clock. Blood pressure appears to follow such a rhythm. For a certain individual the average resting blood pressure varies from a maximum of 100 mmHg at 2:00 P.M. to a minimum of 80 mmHg at 2:00 A.M. Find a sine function of the form $$f(t)=a \sin (\omega(t-c))+b$$ that models the blood pressure at time t, measured in hours from midnight.

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$ y=\cos \left(\frac{\pi}{2}-x\right) $$

Find (a) the reference number for each value of t, and (b) the terminal point determined by t. $$ t=\frac{7 \pi}{3} $$

7–52 Find the period and graph the function. $$y=\sec 2\left(x-\frac{\pi}{2}\right)$$

Find (a) the reference number for each value of t, and (b) the terminal point determined by t. $$ t=-\frac{2 \pi}{3} $$

7–52 Find the period and graph the function. $$y=\cot \left(2 x-\frac{\pi}{2}\right)$$

When a car with its horn blowing drives by an observer, the pitch of the horn seems higher as it approaches and lower as it recedes (see the figure). This phenomenon is called the Doppler effect. If the sound source is moving at speed \(v\) relative to the observer and if the speed of sound is \(v_{0},\) then the perceived frequency \(f\) is related to the actual frequency \(f_{0}\) as follows: $$f=f_{0}\left(\frac{v_{0}}{v_{0} \pm v}\right)$$ We choose the minus sign if the source is moving toward the observer and the plus sign if it is moving away. Suppose that a car drives at 110 ft/s past a woman standing on the shoulder of a highway, blowing its horn, which has a frequency of 500 Hz. Assume that the speed of sound is 1130 ft/s. (This is the speed in dry air at \(70^{\circ} \mathrm{F}\) . (a) What are the frequencies of the sounds that the woman hears as the car approaches her and as it moves away from her? (b) Let A be the amplitude of the sound. Find functions of the form $$y=A \sin \omega t$$ that model the perceived sound as the car approaches the woman and as it recedes.

Find (a) the reference number for each value of t, and (b) the terminal point determined by t. $$ t=-\frac{7 \pi}{6} $$

7–52 Find the period and graph the function. $$y=\frac{1}{2} \tan (\pi x-\pi)$$

A strong gust of wind strikes a tall building, causing it to sway back and forth in damped harmonic motion. The frequency of the oscillation is 0.5 cycle per second and the damping constant is \(c=0.9 .\) Find an equation that describes the motion of the building. (Assume \(k=1\) and take \(t=0\) to be the instant when the gust of wind strikes the building.)

Find (a) the reference number for each value of t, and (b) the terminal point determined by t. $$ t=\frac{13 \pi}{4} $$

7–52 Find the period and graph the function. $$y=2 \csc \left(\pi x-\frac{\pi}{3}\right)$$

Find (a) the reference number for each value of t, and (b) the terminal point determined by t. $$ t=\frac{13 \pi}{6} $$

7–52 Find the period and graph the function. $$y=2 \sec \left(\frac{1}{2} x-\frac{\pi}{3}\right)$$

Find the sign of the expression if the terminal point determined by \(t\) is in the given quadrant. \(\sin t \cos t, \quad\) quadrant II

A tuning fork is struck and oscillates in damped harmonic motion. The amplitude of the motion is measured, and 3 s later it is found that the amplitude has dropped to \(\frac{1}{4}\) of this value. Find the damping constant \(c\) for this tuning fork.

7–52 Find the period and graph the function. $$y=5 \sec \left(3 x-\frac{\pi}{2}\right)$$

A guitar string is pulled at point \(P\) a distance of 3 cm above its rest position. It is then released and vibrates in damped harmonic motion with a frequency of 165 cycles per second. After 2 s, it is observed that the amplitude of the vibration at point \(P\) is 0.6 cm. (a) Find the damping constant \(c\) . (b) Find an equation that describes the position of point \(P\) above its rest position as a function of time. Take \(t=0\) to be the instant that the string is released.

Find the sign of the expression if the terminal point determined by \(t\) is in the given quadrant. \(\tan t \sec t, \quad\) quadrant IV

Find (a) the reference number for each value of t, and (b) the terminal point determined by t. $$ t=\frac{17 \pi}{4} $$

7–52 Find the period and graph the function. $$y=\frac{1}{2} \sec (2 \pi x-\pi)$$

Find the sign of the expression if the terminal point determined by \(t\) is in the given quadrant. \(\frac{\tan t \sin t}{\cot t}, \quad\) quadrant III

Find (a) the reference number for each value of t, and (b) the terminal point determined by t. $$ t=-\frac{11 \pi}{3} $$

7–52 Find the period and graph the function. $$y=\tan \left(\frac{2}{3} x-\frac{\pi}{6}\right)$$

Find (a) the reference number for each value of t, and (b) the terminal point determined by t. $$ t=\frac{31 \pi}{6} $$

7–52 Find the period and graph the function. $$y=\tan \frac{1}{2}\left(x+\frac{\pi}{4}\right)$$

From the information given, find the quadrant in which the terminal point determined by \(t\) lies. $$ \sin t>0 \text { and } \cos t<0 $$

7–52 Find the period and graph the function. $$y=3 \sec \pi\left(x+\frac{1}{2}\right)$$

From the information given, find the quadrant in which the terminal point determined by \(t\) lies. $$ \tan t>0 \text { and } \sin t<0 $$

Find (a) the reference number for each value of t, and (b) the terminal point determined by t. $$ t=-\frac{41 \pi}{4} $$

7–52 Find the period and graph the function. $$y=\sec \left(3 x+\frac{\pi}{2}\right)$$

From the information given, find the quadrant in which the terminal point determined by \(t\) lies. $$ \csc t>0 \text { and } \sec t<0 $$

7–52 Find the period and graph the function. $$y=-2 \tan \left(2 x-\frac{\pi}{3}\right)$$

From the information given, find the quadrant in which the terminal point determined by \(t\) lies. $$ \cos t<0 \text { and } \cot t<0 $$

Determine an appropriate viewing rectangle for each function, and use it to draw the graph. $$ f(x)=\cos (x / 80) $$