Chapter 7

Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is at its maximum at time \(t=0\) . amplitude 60 ft, period 0.5 min

Graph the function. $$ h(x)=|\cos x| $$

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

\(13-18=\) The point \(P\) is on the unit circle. Find \(P(x, y)\) from the given information. The \(x\) -coordinate of \(P\) is \(\frac{4}{5}\) and the \(y\) -coordinate is positive.

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{llll}{\text { (a) } \cos \frac{7 \pi}{6}} & {\text { (b) } \sec \frac{7 \pi}{6}} & {\text { (c) } \csc \frac{7 \pi}{6}}\end{array} $$

Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is at its maximum at time \(t=0\) . amplitude 35 cm, period 8 s

Graph the function. $$ h(x)=|\sin x| $$

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

\(13-18=\) The point \(P\) is on the unit circle. Find \(P(x, y)\) from the given information. The \(y\) -coordinate of \(P\) is \(-\frac{1}{3}\) and the \(x\) -coordinate is positive.

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{llll}{\text { (a) } \tan \frac{5 \pi}{6}} & {\text { (b) } \tan \frac{7 \pi}{6}} & {\text { (c) } \tan \frac{11 \pi}{6}}\end{array} $$

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

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

\(13-18=\) The point \(P\) is on the unit circle. Find \(P(x, y)\) from the given information. The \(y\) -coordinate of \(P\) is \(\frac{2}{3}\) and the \(x\) -coordinate is negative.

Find the exact value of the trigonometric function at the given real number. $$ \text { (a) }\cot \left(-\frac{\pi}{3}\right) \quad \text { (b) } \cot \frac{2 \pi}{3} \quad \text { (c) } \cot \frac{5 \pi}{3} $$

Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is at its maximum at time \(t=0\) . amplitude 6.25 in., frequency 60 Hz

Find the amplitude and period of the function, and sketch its graph. $$ y=-\sin 2 x $$

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

Find the exact value of the trigonometric function at the given real number. $$ \text { (a) }\cos \left(-\frac{\pi}{4}\right) \quad \text { (b) } \csc \left(-\frac{\pi}{4}\right) \quad \text { (c) } \cot \left(-\frac{\pi}{4}\right) $$

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. $$k=2, \quad c=1.5, \quad f=3$$

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

\(13-18=\) The point \(P\) is on the unit circle. Find \(P(x, y)\) from the given information. The \(x\) -coordinate of \(P\) is \(-\sqrt{2} / 3\) and \(P\) lies below the \(x\) -axis.

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

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{llll}{\text { (a) } \sin \frac{5 \pi}{4}} & {\text { (b) } \sec \frac{5 \pi}{4}} & {\text { (c) } \tan \frac{5 \pi}{4}}\end{array} $$

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. $$k=15, \quad c=0.25, \quad f=0.6$$

Find the amplitude and period of the function, and sketch its graph. $$ y=\frac{1}{2} \cos 4 x $$

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

\(13-18=\) The point \(P\) is on the unit circle. Find \(P(x, y)\) from the given information. The \(x\) -coordinate of \(P\) is \(-\frac{2}{5}\) and \(P\) lies above the \(x\) -axis.

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{lll}{\text { (a) } \csc \left(-\frac{\pi}{2}\right)} & {\text { (b) } \csc \frac{\pi}{2}} & {\text { (c) } \csc \frac{3 \pi}{2}}\end{array} $$

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. $$k=100, \quad c=0.05, \quad p=4$$

Find the amplitude and period of the function, and sketch its graph. $$ y=10 \sin \frac{1}{2} x $$

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

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{llll}{\text { (a) } \sec (-\pi)} & {\text { (b) } \sec \pi} & {\text { (c) } \sec 4 \pi}\end{array} $$

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. $$k=0.75, \quad c=3, \quad p=3 \pi$$

Find the amplitude and period of the function, and sketch its graph. $$ y=5 \cos \frac{1}{4} x $$

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

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{llll}{\text { (a) } \sin 13 \pi} & {\text { (b) } \cos 14 \pi} & {\text { (c) } \tan 15 \pi}\end{array} $$

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. $$k=7, \quad c=10, \quad p=\pi / 6$$

Find the amplitude and period of the function, and sketch its graph. $$ y=-\frac{1}{3} \cos \frac{1}{3} x $$

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

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

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{llll}{\text { (a) } \sin \frac{25 \pi}{2}} & {\text { (b) } \cos \frac{25 \pi}{2}} & {\text { (c) } \cot \frac{25 \pi}{2}}\end{array} $$

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. $$k=1, \quad c=1, \quad p=1$$

Find the amplitude and period of the function, and sketch its graph. $$ y=4 \sin (-2 x) $$

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

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

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. $$k=0.3, \quad c=0.2, \quad f=20$$

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

7–52 Find the period and graph the function. $$y=\frac{1}{2} \sec \left(x-\frac{\pi}{6}\right)$$

\(21-30=\) Find the terminal point \(P(x, y)\) on the unit circle determined by the given value of \(t .\) $$ t=\frac{5 \pi}{6} $$

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. $$ t=\frac{\pi}{2} $$ table can't copy