Chapter 7

The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. $$y=2 \sin 3 t$$

Graph the function. $$ f(x)=1+\cos x $$

\(1-6=\) Show that the point is on the unit circle. $$ \left(\frac{4}{5},-\frac{3}{5}\right) $$

The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. $$y=3 \cos \frac{1}{2} t$$

Graph the function. $$ f(x)=3+\sin x $$

\(1-6=\) Show that the point is on the unit circle. $$ \left(-\frac{5}{13}, \frac{12}{13}\right) $$

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{llll}{\text { (a) } \sin \frac{2 \pi}{3}} & {\text { (b) } \cos \frac{2 \pi}{3}} & {\text { (c) } \tan \frac{2 \pi}{3}}\end{array} $$

The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. $$y=-\cos 0.3 t$$

Graph the function. $$ f(x)=-\sin x $$

\(1-6=\) Show that the point is on the unit circle. $$ \left(\frac{7}{25}, \frac{24}{25}\right) $$

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

The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. $$y=2.4 \sin 3.6 t$$

Graph the function. $$ f(x)=2-\cos x $$

\(1-6=\) Show that the point is on the unit circle. $$ \left(-\frac{5}{7},-\frac{2 \sqrt{6}}{7}\right) $$

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

The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. $$y=-0.25 \cos \left(1.5 t-\frac{\pi}{3}\right)$$

Graph the function. $$ f(x)=-2+\sin x $$

\(1-6=\) Show that the point is on the unit circle. $$ \left(-\frac{\sqrt{5}}{3}, \frac{2}{3}\right) $$

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{llll}{\text { (a) } \cos \frac{5 \pi}{3}} & {\text { (b) } \cos \left(-\frac{5 \pi}{3}\right)} & {\text { (c) } \cos \frac{7 \pi}{3}}\end{array} $$

The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. $$y=-\frac{3}{2} \sin (0.2 t+1.4)$$

Graph the function. $$ f(x)=-1+\cos x $$

\(1-6=\) Show that the point is on the unit circle. $$ \left(\frac{\sqrt{11}}{6}, \frac{5}{6}\right) $$

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{llll}{\text { (a) } \cos \frac{3 \pi}{4}} & {\text { (b) } \cos \frac{5 \pi}{4}} & {\text { (c) } \cos \frac{7 \pi}{4}}\end{array} $$

The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. $$y=5 \cos \left(\frac{2}{3} t+\frac{3}{4}\right)$$

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{llll}{\text { (a) } \sin \frac{3 \pi}{4}} & {\text { (b) } \sin \frac{5 \pi}{4}} & {\text { (c) } \sin \frac{7 \pi}{4}}\end{array} $$

The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. $$y=1.6 \sin (t-1.8)$$

Graph the function. $$ g(x)=2 \sin x $$

7–52 Find the period and graph the function. $$y=-4 \tan x$$

\(7-12\) . Find the missing coordinate of \(P\) , using the fact that \(P\) lies on the unit circle in the given quadrant. \(P\left(\quad,-\frac{7}{25}\right) \quad \mathrm{IV}\)

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{llll}{\text { (a) } \sin \frac{7 \pi}{3}} & {\text { (b) } \csc \frac{7 \pi}{3}} & {\text { (c) } \cot \frac{7 \pi}{3}}\end{array} $$

Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is zero at time \(t=0\) . amplitude \(10 \mathrm{cm},\) period 3 \(\mathrm{s}\)

Graph the function. $$ g(x)=-\frac{1}{2} \sin x $$

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

\(7-12\) . Find the missing coordinate of \(P\) , using the fact that \(P\) lies on the unit circle in the given quadrant. \(P\left(\begin{array}{cc}{P( } & {, \frac{1}{3} )} & {\text { II }}\end{array}\right.\)

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

Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is zero at time \(t=0\) . amplitude 24 \(\mathrm{ft},\) period 2 \(\mathrm{min}\)

Graph the function. $$ g(x)=-\frac{2}{3} \cos x $$

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

\(7-12\) . Find the missing coordinate of \(P\) , using the fact that \(P\) lies on the unit circle in the given quadrant. \(P\left(\frac{2}{5}, \quad\right) \quad\) I

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

Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is zero at time \(t=0\) . amplitude 6 in., frequency 5\(/ \pi \mathrm{Hz}\)

Graph the function. $$ g(x)=3+3 \cos x $$

\(7-12\) . Find the missing coordinate of \(P\) , using the fact that \(P\) lies on the unit circle in the given quadrant. \(P\left(\begin{array}{cc}{P( } & {,-\frac{2}{7} )} & {\mathrm{IV}}\end{array}\right.\)

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

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

Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is zero at time \(t=0\) . amplitude 1.2 \(\mathrm{m},\) frequency 0.5 \(\mathrm{Hz}\)

Graph the function. $$ g(x)=4-2 \sin x $$

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

\(7-12\) . Find the missing coordinate of \(P\) , using the fact that \(P\) lies on the unit circle in the given quadrant. \(P\left(-\frac{2}{3}, \quad\right) \quad\) II

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{llll}{\text { (a) } \sec \frac{11 \pi}{3}} & {\text { (b) } \csc \frac{11 \pi}{3}} & {\text { (c) } \sec \left(-\frac{\pi}{3}\right)}\end{array} $$