Chapter 7

For an object in simple harmonic motion with amplitude \(a\) and period \(2 \pi / \omega,\) find an equation that models the displacement \(y\) at time \(t\) if (a) \(y=0\) at time \(t=0 : y=\) _________ (b) \(y=a\) at time \(t=0 : y=\) _________

(a) To define the inverse sine function, we restrict the domain of sine to the interval______On this interval the sine function is one-to-one, and its inverse function \(\sin ^{-1}\) is defined by \(\sin ^{-1} x=y \Leftrightarrow \sin\) ______=_________For example, \(\sin ^{-1} \frac{1}{2}=\) ______because sin________=_______ (b) To define the inverse cosine function we restrict the domain of cosine to the interval______On this interval the cosine function is one-to-one and its inverse function \(\cos ^{-1}\) is defined by \(\cos ^{-1} x=y \Leftrightarrow\) cos_______=_______For example, \(\cos ^{-1} \frac{1}{2}=\)_______because cos _________=________

The trigonometric function \(y=\tan x\) has period __________ and asymptotes \(x=\) __________ Sketch a graph of this function on the interval \((-\pi / 2, \pi / 2)\)

The trigonometric functions \(y=\sin x\) and \(y=\cos x\) have amplitude _____ and period _____ Sketch a graph of each function on the interval \([0,2 \pi]\)

For an object in damped harmonic motion with initial amplitude \(k\) , period \(2 \pi / \omega,\) and damping constant \(c\) , find an equation that models the displacement \(y\) at time \(t\) if (a) \(y=0\) at time \(t=0 : y=\)________ (b) \(y=a\) at time \(t=0 : y=\) _______

The cancellation property \(\sin ^{-1}(\sin x)=x\) is valid for \(x\) in the interval ___________ Which of the following is not true? (a) \(\sin ^{-1}\left(\sin \frac{\pi}{3}\right)=\frac{\pi}{3}\) (b) \(\sin ^{-1}\left(\sin \frac{10 \pi}{3}\right)=\frac{10 \pi}{3}\)

The trigonometric function \(y=3 \sin 2 x\) has amplitude _____ and period _____.

The trigonometric function \(y=\csc x\) has period ________ and asymptotes \(x=\) __________ Sketch a graph of this function on the interval \((-\pi, \pi)\)

If \(P(x, y)\) is on the unit circle, then \(x^{2}+y^{2}=\) _________ . So for all \(t\) we have \(\sin ^{2} t+\cos ^{2} t=\) _______.

(a) If we mark off a distance \(t\) along the unit circle, starting at \((1,0)\) and moving in a counterclockwise direction, we arrive at the (b) The terminal points determined by \(\pi / 2, \pi,-\pi / 2,2 \pi\) are respectively.

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. $$ y=2 \sin 3 t $$

\(3-10=\) Find the exact value of each expression, if it is defined. $$ \begin{array}{lll}{\text { (a) } \sin ^{-1} 1} & {\text { (b) } \sin ^{-1} \frac{\sqrt{3}}{2}} & {\text { (c) } \sin ^{-1} 2}\end{array} $$

\(3-16\) Graph the function. $$ f(x)=1+\cos x $$

\(3-8 \approx\) 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. $$ y=3 \cos \frac{1}{2} t $$

\(3-16\) Graph the function. $$ f(x)=3+\sin x $$

\(3-8 \approx\) Show that the point is on the unit circle. $$ \left(-\frac{5}{13}, \frac{12}{13}\right) $$

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. $$ y=-\cos 0.3 t $$

\(3-10=\) Find the exact value of each expression, if it is defined. $$ \begin{array}{llll}{\text { (a) } \cos ^{-1}(-1)} & {\text { (b) } \cos ^{-1} \frac{1}{2}} & {\text { (c) } \cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)}\end{array} $$

\(3-16\) Graph the function. $$ f(x)=-\sin x $$

Find the exact value of the trigonometric function at the given real number. (a) \(\sin \frac{2 \pi}{3} \quad\) (b) \(\cos \frac{2 \pi}{3} \quad\) (c) \(\tan \frac{2 \pi}{3}\)

\(3-8 \approx\) Show that the point is on the unit circle. $$ \left(\frac{7}{25}, \frac{24}{25}\right) $$

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. $$ y=2.4 \sin 3.6 t $$

\(3-10=\) Find the exact value of each expression, if it is defined. $$ \text { (a) } \cos ^{-1}\left(\frac{\sqrt{2}}{2}\right) \quad \text { (b) } \cos ^{-1} 1 \quad \text { (c) } \cos ^{-1}\left(-\frac{\sqrt{2}}{2}\right) $$

\(3-16\) Graph the function. $$ f(x)=2-\cos x $$

Find the exact value of the trigonometric function at the given real number. (a) \(\sin \frac{5 \pi}{6}\) \(\quad\) (b) \(\cos \frac{5 \pi}{6}\) \(\quad\) (c) \(\tan \frac{5 \pi}{6}\)

\(3-8 \approx\) Show that the point is on the unit circle. $$ \left(-\frac{5}{7},-\frac{2 \sqrt{6}}{7}\right) $$

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. $$ y=-0.25 \cos \left(1.5 t-\frac{\pi}{3}\right) $$

\(3-10=\) Find the exact value of each expression, if it is defined. $$ \begin{array}{llll}{\text { (a) } \tan ^{-1}(-1)} & {\text { (b) } \tan ^{-1} \sqrt{3}} & {} & {\text { (c) } \tan ^{-1} \frac{\sqrt{3}}{3}}\end{array} $$

\(3-16\) Graph the function. $$ f(x)=-2+\sin x $$

Find the exact value of the trigonometric function at the given real number. (a) \(\sin \frac{7 \pi}{6}\) \(\quad\) (b) \(\sin \left(-\frac{\pi}{6}\right)\) \(\quad\) (c) \(\sin \frac{11 \pi}{6}\)

\(3-8 \approx\) Show that the point is on the unit circle. $$ \left(-\frac{\sqrt{5}}{3}, \frac{2}{3}\right) $$

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. $$ y=-\frac{3}{2} \sin (0.2 t+1.4) $$

\(3-10=\) Find the exact value of each expression, if it is defined. $$ \begin{array}{lll}{\text { (a) } \tan ^{-1} 0} & {\text { (b) } \tan ^{-1}(-\sqrt{3})} & {\text { (c) } \tan ^{-1}\left(-\frac{\sqrt{3}}{3}\right)}\end{array} $$

Find the exact value of the trigonometric function at the given real number. (a) \(\cos \frac{5 \pi}{3}\) \(\quad\) (b) \(\cos \left(-\frac{5 \pi}{3}\right)\) \(\quad\) (c) \(\cos \frac{7 \pi}{3}\)

\(3-8 \approx\) Show that the point is on the unit circle. $$ \left(\frac{\sqrt{11}}{6}, \frac{5}{6}\right) $$

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. $$ y=5 \cos \left(\frac{2}{3} t+\frac{3}{4}\right) $$

\(3-10=\) Find the exact value of each expression, if it is defined. $$ \begin{array}{lll}{\text { (a) } \cos ^{-1}\left(-\frac{1}{2}\right)} & {\text { (b) } \sin ^{-1}\left(-\frac{\sqrt{2}}{2}\right)} & {\text { (c) } \tan ^{-1} 1}\end{array} $$

\(3-16\) Graph the function. $$ g(x)=3 \cos x $$

Find the exact value of the trigonometric function at the given real number. (a) \(\cos \frac{3 \pi}{4}\) \(\quad\) (b) \(\cos \frac{5 \pi}{4}\) \(\quad\) (c) \(\cos \frac{7 \pi}{4}\)

Find the period and graph the function. $$ y=4 \tan x $$

\(9-14\) . Find the missing coordinate of \(P\) , using the fact that \(P\) lies on the unit circle in the given quadrant. $$ \frac{\text { Coordinates }}{P\left(-\frac{3}{5},\right.} ) \frac{\text { Quadrant }}{\text { III }} $$

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. $$ y=1.6 \sin (t-1.8) $$

\(3-10=\) Find the exact value of each expression, if it is defined. $$ \begin{array}{llll}{\text { (a) } \cos ^{-1} 0} & {} & {\text { (b) } \sin ^{-1} 0} & {} & {\text { (e) } \sin ^{-1}\left(-\frac{1}{2}\right)}\end{array} $$

Find the exact value of the trigonometric function at the given real number. (a) \(\sin \frac{3 \pi}{4} \quad\) (b) \(\sin \frac{5 \pi}{4} \quad\) (c) \(\sin \frac{7 \pi}{4}\)

\(3-16\) Graph the function. $$ g(x)=2 \sin x $$

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

\(9-14\) . Find the missing coordinate of \(P\) , using the fact that \(P\) lies on the unit circle in the given quadrant. $$ \frac{\text { Coordinates }}{P\left(-\frac{7}{25},\right.} ) \frac{\text { Quadrant }}{\text { IV }} $$

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 \(\mathrm{s}\)

\(11-22\) . Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. \(\sin ^{-1} \frac{2}{3}\)