Chapter 9

Use the unit circle to evaluate the six trigonometric functions of \(\theta\). \(\theta=-2 \pi\)

ERROR ANALYSIS Describe and correct the error in describing the transformation of \(f(x)=\tan x\) represented by \(g(x)=2 \tan 5 x\).

Identify the amplitude and period of the function. Then graph the function and describe the graph of \(g\) as a transformation of the graph of its parent function. \(g(x)=\cos 3 x\)

\(\tan \theta=\frac{7}{6}\)

Evaluate the expression given that \(\cos a=\frac{4}{5}\) with \(0

In Exercises 15-22, sketch the angle. Then find its reference angle. \(-100^{\circ}\)

Identify the amplitude and period of the function. Then graph the function and describe the graph of \(g\) as a transformation of the graph of its parent function. \(g(x)=\cos 4 x\)

\(\csc \theta=\frac{15}{8}\)

Sketch the angle. Then find its reference angle. \(150^{\circ}\)

USING EQUATIONS Which of the following are asymptotes of the graph of \(y=3 \tan 4 x\) ? (A) \(x=\frac{\pi}{8}\) (B) \(x=\frac{\pi}{4}\) (C) \(x=0\) (D) \(x=-\frac{5 \pi}{8}\)

Identify the amplitude and period of the function. Then graph the function and describe the graph of \(g\) as a transformation of the graph of its parent function. \(g(x)=\sin 2 \pi x\)

\(\sec \theta=\frac{14}{9}\)

In Exercises 17-22, simplify the expression. \(\tan (x+\pi)\)

Sketch the angle. Then find its reference angle. \(320^{\circ}\)

\(g(x)=3 \csc x\)

Identify the amplitude and period of the function. Then graph the function and describe the graph of \(g\) as a transformation of the graph of its parent function. \(g(x)=3 \sin 2 x\)

ERROR ANALYSIS Describe and correct the error in fi nding the vertical shift of a sinusoid with a maximum point at (3, ?2) and a minimum point at (7, ?8).

\(\cot \theta=\frac{16}{11}\)

Simplify the expression. \(\cos \left(x-\frac{\pi}{2}\right)\)

Sketch the angle. Then find its reference angle. \(-370^{\circ}\)

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

Identify the amplitude and period of the function. Then graph the function and describe the graph of \(g\) as a transformation of the graph of its parent function. \(g(x)=\frac{1}{3} \cos 4 x\)

Simplify the expression. \(\cos (x+2 \pi)\)

Sketch the angle. Then find its reference angle. \(\frac{15 \pi}{4}\)

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

Identify the amplitude and period of the function. Then graph the function and describe the graph of \(g\) as a transformation of the graph of its parent function. \(g(x)=\frac{1}{2} \cos 4 \pi x\)

ERROR ANALYSIS Describe and correct the error in fi nding csc ?, given that ? is an acute angle of a right triangle and cos ? = 7 —11 .

Simplify the expression. \(\tan (x-2 \pi)\)

Sketch the angle. Then find its reference angle. \(\frac{8 \pi}{3}\)

\(g(x)=\sec 3 x\)

Which functions have an amplitude of 4 and a period of 2 ? (A) \(y=4 \cos 2 x\) (B) \(y=-4 \sin \pi x\) (C) \(y=2 \sin 4 x\) (D) \(y=4 \cos \pi x\)

Simplify the expression. \(\sin \left(x-\frac{3 \pi}{2}\right)\)

Sketch the angle. Then find its reference angle. \(-\frac{5 \pi}{6}\)

\(g(x)=\frac{1}{2} \sec \pi x\)

Simplify the expression. \(\tan \left(x+\frac{\pi}{2}\right)\)

Sketch the angle. Then find its reference angle. \(-\frac{13 \pi}{6}\)

\(g(x)=\frac{1}{4} \sec 2 \pi x\)

The motion of a pendulum can be modeled by the function \(d=4 \cos 8 \pi t\), where \(d\) is the horizontal displacement (in inches) of the pendulum relative to its position at rest and \(t\) is the time (in seconds). Find and interpret the period and amplitude in the context of this situation. Then graph the function.

MODELING WITH MATHEMATICS A circuit has an alternating voltage of 100 volts that peaks every \(0.5\) second. Write a sinusoidal model for the voltage \(V\) as a function of the time \(t\) (in seconds).

Let \((-3,2)\) be a point on the terminal side of an angle \(\theta\) in standard position. Describe and correct the error in finding \(\tan \theta\).

\(g(x)=\csc \frac{\pi}{2} x\)

A buoy bobs up and down as waves go past. The vertical displacement \(y\) (in feet) of the buoy with respect to sea level can be modeled by \(y=1.75 \cos \frac{\pi}{3} t\), where \(t\) is the time (in seconds). Find and interpret the period and amplitude in the context of the problem. Then graph the function.

Describe and correct the error in simplifying the expression. $$ \begin{aligned} \sin \left(x-\frac{\pi}{4}\right) &=\sin \frac{\pi}{4} \cos x-\cos \frac{\pi}{4} \sin x \\ &=\frac{\sqrt{2}}{2} \cos x-\frac{\sqrt{2}}{2} \sin x \\ &=\frac{\sqrt{2}}{2}(\cos x-\sin x) \end{aligned} $$

Describe and correct the error in finding a reference angle \(\theta^{\prime}\) for \(\theta=650^{\circ}\).

\(g(x)=\csc \frac{\pi}{4} x\)

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

What are the solutions of the equation \(2 \sin x-1=0\) for \(0 \leq x<2 \pi\) ? (A) \(\frac{\pi}{3}\) (B) \(\frac{\pi}{6}\) (C) \(\frac{2 \pi}{3}\) (D) \(\frac{5 \pi}{6}\)

In Exercises 25-32, evaluate the function without using a calculator. \(\sec 135^{\circ}\)

Graph the function. \(g(x)=\cos x-4\)

What are the solutions of the equation \(\tan x+1=0\) for \(0 \leq x<2 \pi\) ? (A) \(\frac{\pi}{4}\) (B) \(\frac{3 \pi}{4}\) (C) \(\frac{5 \pi}{4}\) (D) \(\frac{7 \pi}{4}\)