Chapter 9

Evaluate the function without using a calculator. \(\tan 240^{\circ}\)

Graph the function. \(g(x)=\cos \left(x-\frac{\pi}{2}\right)\)

. \(\cos 14^{\circ}\)

In Exercises 27-32, solve the equation for \(0 \leq x<2 \pi\). \(\sin \left(x+\frac{\pi}{2}\right)=\frac{1}{2}\)

Evaluate the function without using a calculator. \(\sin \left(-150^{\circ}\right)\)

Graph the function. \(g(x)=\sin \left(x+\frac{\pi}{4}\right)\)

\(\tan 31^{\circ}\)

Solve the equation for \(0 \leq x<2 \pi\). \(\tan \left(x-\frac{\pi}{4}\right)=0\)

Evaluate the function without using a calculator. \(\csc \left(-420^{\circ}\right)\)

Graph the function. \(g(x)=2 \cos x-1\)

\(\csc 59^{\circ}\)

Solve the equation for \(0 \leq x<2 \pi\). \(\cos \left(x+\frac{\pi}{6}\right)-\cos \left(x-\frac{\pi}{6}\right)=1\)

Evaluate the function without using a calculator. \(\tan \left(-\frac{3 \pi}{4}\right)\)

\(g(x)=4 \tan x\)

Graph the function. \(g(x)=3 \sin x+1\)

\(\sin 23^{\circ}\)

Solve the equation for \(0 \leq x<2 \pi\). \(\sin \left(x+\frac{\pi}{4}\right)+\sin \left(x-\frac{\pi}{4}\right)=0\)

Evaluate the function without using a calculator. \(\cot \left(\frac{-8 \pi}{3}\right)\)

\(g(x)=4 \cot x\)

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

\(\cot 6^{\circ}\)

Solve the equation for \(0 \leq x<2 \pi\). \(\tan (x+\pi)-\tan (\pi-x)=0\)

Evaluate the function without using a calculator. \(\cos \frac{7 \pi}{4}\)

\(g(x)=4 \csc \pi x\)

Graph the function. \(g(x)=\cos 2(x-\pi)\)

\(\sec 11^{\circ}\)

Solve the equation for \(0 \leq x<2 \pi\). \(\sin (x+\pi)+\cos (x+\pi)=0\)

Evaluate the function without using a calculator. \(\sec \frac{11 \pi}{6}\)

\(g(x)=4 \sec \pi x\)

Graph the function. \(g(x)=\sin \frac{1}{2}(x+2 \pi)+3\)

Derive the cofunction identity \(\sin \left(\frac{\pi}{2}-\theta\right)=\cos \theta\) using the difference formula for sine.

\(g(x)=\sec 2 x\)

Graph the function. \(g(x)=\cos \frac{1}{2}(x-3 \pi)-5\)

Your friend claims it is possible to use the difference formula for tangent to derive the cofunction identity \(\tan \left(\frac{\pi}{2}-\theta\right)=\cot \theta\). Is your friend correct? Explain your reasoning.

\(g(x)=\csc 2 x\)

Describe and correct the error in finding the period of the function \(y=\sin \frac{2}{3} x\).

WRITING Explain why there is more than one tangent function whose graph passes through the origin and has asymptotes at \(x=-\pi\) and \(x=\pi\).

Describe and correct the error in determining the point where the maximum value of the function \(y=2 \sin \left(x-\frac{\pi}{2}\right)\) occurs.

When a wave travels through a taut string, the displacement \(y\) of each point on the string depends on the time \(t\) and the point's position \(x\). The equation of a standing wave can be obtained by adding the displacements of two waves traveling in opposite directions. Suppose a standing wave can be modeled by the formula \(y=A \cos \left(\frac{2 \pi t}{3}-\frac{2 \pi x}{5}\right)+A \cos \left(\frac{2 \pi t}{3}+\frac{2 \pi x}{5}\right)\). When \(t=1\), show that the formula can be rewritten as \(y=-A \cos \frac{2 \pi x}{5}\).

USING EQUATIONS Graph one period of each function. Describe the transformation of the graph of its parent function. a. \(g(x)=\sec x+3\) b. \(g(x)=\csc x-2\) c. \(g(x)=\cot (x-\pi)\) d. \(g(x)=-\tan x\)

Describe the transformation of the graph of \(f\) represented by the function \(g\). \(f(x)=\cos x, g(x)=2 \cos \left(x-\frac{\pi}{2}\right)+1\)

The busy signal on a touch-tone phone is a combination of two tones with frequencies of 480 hertz and 620 hertz. The individual tones can be modeled by the equations: 480 hertz: \(y_1=\cos 960 \pi t\) 620 hertz: \(y_2=\cos 1240 \pi t\) The sound of the busy signal can be modeled by \(y_1+y_2\). Show that \(y_1+y_2=2 \cos 1100 \pi t \cos 140 \pi t\).

A rock climber is using a rock climbing treadmill that is 10 feet long. The climber begins by lying horizontally on the treadmill, which is then rotated about its midpoint by \(110^{\circ}\) so that the rock climber is climbing toward the top. If the midpoint of the treadmill is 6 feet above the ground, how high above the ground is the top of the treadmill?

\(f(x)=\cot 2 x\); translation 3 units up and \(\frac{\pi}{2}\) units left

Describe the transformation of the graph of \(f\) represented by the function \(g\). \(f(x)=\sin x, g(x)=3 \sin \left(x+\frac{\pi}{4}\right)-2\)

Explain how to use the figure to solve the equation \(\sin \left(x+\frac{\pi}{4}\right)-\sin \left(\frac{\pi}{4}-x\right)=0\) for \(0 \leq x<2 \pi\).

A Ferris wheel has a radius of 75 feet. You board a car at the bottom of the Ferris wheel, which is 10 feet above the ground, and rotate \(255^{\circ}\) counterclockwise before the ride temporarily stops. How high above the ground are you when the ride stops? If the radius of the Ferris wheel is doubled, is your height above the ground doubled? Explain your reasoning.

\(f(x)=2 \tan x\), translation \(\pi\) units right, followed by a horizontal shrink by a factor of \(\frac{1}{3}\)

Describe the transformation of the graph of \(f\) represented by the function \(g\). \(f(x)=\sin x, g(x)=\sin 3(x+3 \pi)-5\)

\(f(x)=5 \sec (x-\pi)\); translation 2 units down, followed by a reflection in the \(x\)-axis