Chapter 5

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

Find the missing coordinate of \(P,\) using the fact that \(P\) lies on the unit circle in the given quadrant. Coordinates $$P\left(\quad, \frac{1}{3}\right)$$ Quadrant II

\(11-14\) a Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is zero at time \(t=0\) amplitude \(10 \mathrm{cm}, \quad\) period \(3 \mathrm{s}\)

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

Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. $$\sin ^{-1}\left(-\frac{8}{9}\right)$$

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

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

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

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

Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. $$\cos ^{-1}\left(-\frac{3}{7}\right)$$

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

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

Find the missing coordinate of \(P,\) using the fact that \(P\) lies on the unit circle in the given quadrant. Coordinates $$P\left(\quad,-\frac{2}{7}\right)$$ Quadrant 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 6 in., frequency \(5 / \pi \mathrm{Hz}\)

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

Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. $$\cos ^{-1}\left(\frac{4}{9}\right)$$

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

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

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

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

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

Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. $$\cos ^{-1}(-0.92761)$$

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

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

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 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 \(60 \mathrm{ft}\), period \(0.5 \mathrm{min}\)

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

Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. $$\sin ^{-1}(0.13844)$$

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

Find the exact value of the trigonometric function at the given real number. (a) \(\cos \frac{7 \pi}{6}\) (b) \(\sec \frac{7 \pi}{6}\) (c) \(\csc \frac{7 \pi}{6}\)

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 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 \(35 \mathrm{cm}, \quad\) period \(8 \mathrm{s}\)

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

Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. $$\tan ^{-1} 10$$

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

Find the exact value of the trigonometric function at the given real number. (a) \(\tan \frac{5 \pi}{6}\) (b) \(\tan \frac{7 \pi}{6}\) (c) \(\tan \frac{11 \pi}{6}\)

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 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 \(2.4 \mathrm{m}, \quad\) frequency \(750 \mathrm{Hz}\)

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

Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. $$\tan ^{-1}(-26)$$

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

The point \(P\) is on the unit circle. Find \(P(x, y)\) from the given information. The \(x\) -coordinate of \(P\) is positive, and the \(y\) -coordinate of \(P\) is \(-\sqrt{5} / 5\).

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 \(60 \mathrm{Hz}\)

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

Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. $$\tan ^{-1}(1.23456)$$

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. (a) \(\cos \left(-\frac{\pi}{4}\right)\) (b) \(\csc \left(-\frac{\pi}{4}\right)\) (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 \(19-22,\) and of the form \(y=k e^{-c t}\) sin \(\omega t\) in Exercises \(23-26\) (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. $$y=-3 \sin 3 x$$