Chapter 7

An initial amplitude \(k,\) damping constant \(c,\) and frequency \(f\) or period \(p\) are given. (Recall that frequency and period are related by the equation \(f=1 / p . )\) (a) Find a function that models the damped harmonic motion. Use a function of the form \(y=k e^{-c t} \cos \omega t\) in Exercises \(17-20,\) and of the form \(y=k e^{-c t}\) sin \(\omega t\) in Exercises \(21-24\). (b) Graph the function. $$k=12, \quad c=0.01, \quad f=8$$

Find the amplitude and period of the function, and sketch its graph. $$ y=-3 \sin \pi x $$

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

\(21-30=\) Find the terminal point \(P(x, y)\) on the unit circle determined by the given value of \(t .\) $$ t=\frac{7 \pi}{6} $$

Find the value of each of the six trigonometric functions (if it is defined) at the given real number \(t .\) Use your answers to complete the table. $$ t=\pi $$ table can't copy

A cork floating in a lake is bobbing in simple harmonic motion. Its displacement above the bottom of the lake is modeled by $$y=0.2 \cos 20 \pi t+8$$ where \(y\) is measured in meters and \(t\) is measured in minutes. (a) Find the frequency of the motion of the cork. (b) Sketch a graph of \(y\) (c) Find the maximum displacement of the cork above the lake bottom.

Find the amplitude and period of the function, and sketch its graph. $$ y=1+\frac{1}{2} \cos \pi x $$

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

\(21-30=\) Find the terminal point \(P(x, y)\) on the unit circle determined by the given value of \(t .\) $$ t=-\frac{\pi}{3} $$

Find the value of each of the six trigonometric functions (if it is defined) at the given real number \(t .\) Use your answers to complete the table. $$ t=\frac{3 \pi}{2} $$ table can't copy

The carrier wave for an FM radio signal is modeled by the function $$y=a \sin \left(2 \pi\left(9.15 \times 10^{7}\right) t\right)$$ where \(t\) is measured in seconds. Find the period and frequency of the carrier wave.

Find the amplitude and period of the function, and sketch its graph. $$ y=-2+\cos 4 \pi x $$

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

\(21-30=\) Find the terminal point \(P(x, y)\) on the unit circle determined by the given value of \(t .\) $$ t=\frac{5 \pi}{3} $$

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\) $$ \left(\frac{3}{5}, \frac{4}{5}\right) $$

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

7–52 Find the period and graph the function. $$y=\tan \frac{\pi}{4} x$$

\(21-30=\) Find the terminal point \(P(x, y)\) on the unit circle determined by the given value of \(t .\) $$ t=\frac{2 \pi}{3} $$

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\) $$ \left(-\frac{3}{5}, \frac{4}{5}\right) $$

Each time your heart beats, your blood pressure increases, then decreases as the heart rests between beats. A certain person's blood pressure is modeled by the function $$p(t)=115+25 \sin (160 \pi t)$$ where \(p(t)\) is the pressure in \(\mathrm{mmHg}\) at time \(t,\) measured in minutes. (a) Find the amplitude, period, and frequency of \(p .\) (b) Sketch a graph of \(p\) . (c) If a person is exercising, his heart beats faster. How does this affect the period and frequency of \(p ?\)

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

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

\(21-30=\) Find the terminal point \(P(x, y)\) on the unit circle determined by the given value of \(t .\) $$ t=-\frac{\pi}{2} $$

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$ y=-2 \sin \left(x-\frac{\pi}{6}\right) $$

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

\(21-30=\) Find the terminal point \(P(x, y)\) on the unit circle determined by the given value of \(t .\) $$ t=-\frac{3 \pi}{4} $$

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\) $$ \left(-\frac{1}{3},-\frac{2 \sqrt{2}}{3}\right) $$

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$ y=3 \cos \left(x+\frac{\pi}{4}\right) $$

7–52 Find the period and graph the function. $$y=5 \csc 3 x$$

\(21-30=\) Find the terminal point \(P(x, y)\) on the unit circle determined by the given value of \(t .\) $$ t=\frac{11 \pi}{6} $$

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\) $$ \left(-\frac{6}{7}, \frac{\sqrt{13}}{7}\right) $$

The Bay of Fundy in Nova Scotia has the highest tides in the world. In one 12 -hour period the water starts at mean sea level, rises to 21 ft above, drops to 21 ft below, then returns to mean sea level. Assuming that the motion of the tides is simple harmonic, find an equation that describes the height of the tide in the Bay of Fundy above mean sea level. Sketch a graph that shows the level of the tides over a 12 -hour period.

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$ y=-4 \sin 2\left(x+\frac{\pi}{2}\right) $$

Suppose that the terminal point determined by \(t\) is the point \(\left(\frac{3}{5}, \frac{4}{5}\right)\) on the unit circle. Find the terminal point determined by each of the following. \(\begin{array}{ll}{\text { (a) } \pi-t} & {\text { (b) }-t} \\ {\text { (c) } \pi+t} & {\text { (d) } 2 \pi+t}\end{array}\)

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

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\) $$ \left(\frac{40}{41}, \frac{9}{41}\right) $$

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$ y=\sin \frac{1}{2}\left(x+\frac{\pi}{4}\right) $$

7–52 Find the period and graph the function. $$y=\csc \frac{1}{2} x$$

Suppose that the terminal point determined by \(t\) is the point \(\left(\frac{3}{4}, \sqrt{7} / 4\right)\) on the unit circle. Find the terminal point deter- mined by each of the following. \(\begin{array}{ll}{\text { (a) }-t} & {\text { (b) } 4 \pi+t} \\ {\text { (c) } \pi-t} & {\text { (d) } t-\pi}\end{array}\)

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\) $$ \left(-\frac{5}{13},-\frac{12}{13}\right) $$

A mass is suspended on a spring. The spring is compressed so that the mass is located 5 cm above its rest position. The mass is released at time \(t=0\) and allowed to oscillate. It is observed that the mass reaches its lowest point \(\frac{1}{2}\) s after it is released. Find an equation that describes the motion of the mass.

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$ y=5 \cos \left(3 x-\frac{\pi}{4}\right) $$

7–52 Find the period and graph the function. $$y=2 \tan 3 \pi x$$

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\) $$ \left(\frac{\sqrt{5}}{5}, \frac{2 \sqrt{5}}{5}\right) $$

The frequency of oscillation of an object suspended on a spring depends on the stiffness \(k\) of the spring (called the spring constant) and the mass \(m\) of the object. If the spring is compressed a distance \(a\) and then allowed to oscillate, its displacement is given by $$f(t)=a \cos \sqrt{k / m} t$$ (a) A 10 -g mass is suspended from a spring with stiffness \(k=3 .\) If the spring is compressed a distance 5 cm and then released, find the equation that describes the oscillation of the spring. (b) Find a general formula for the frequency (in terms of \(k\) and \(m ) .\) (c) How is the frequency affected if the mass is increased? Is the oscillation faster or slower? (d) How is the frequency affected if a stiffer spring is used (larger \(k\) )? Is the oscillation faster or slower?

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$ y=2 \sin \left(\frac{2}{3} x-\frac{\pi}{6}\right) $$

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

\(33-36=\) Find the reference number for each value of \(t\) $$ \begin{array}{ll}{\text { (a) } t=\frac{5 \pi}{6}} & {\text { (b) } t=\frac{7 \pi}{6}} \\ {\text { (c) } t=\frac{11 \pi}{3}} & {\text { (d) } t=-\frac{7 \pi}{4}}\end{array} $$

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\) $$ \left(-\frac{20}{29}, \frac{21}{29}\right) $$

A ferris wheel has a radius of 10 m, and the bottom of the wheel passes 1 m above the ground. If the ferris wheel makes one complete revolution every 20 s, find an equation that gives the height above the ground of a person on the ferris wheel as a function of time.