Chapter 13

Compare the period of \(y=\tan \theta\) with the period of \(y=\sin \theta .\) Use a graph of the two functions to support your statements.

Use a graphing calculator to graph each function in the interval from 0 to 2\(\pi .\) Then sketch each graph. $$ y=\sin (x+\cos x) $$

For sound waves, the period and the frequency of a pitch are reciprocals of each other: period \(=\frac{\text { seconds }}{\text { cycle }}\) and frequency \(=\frac{\text { cycles }}{\text { second }} .\) Write an equation for each pitch. Let \(\theta=\) time in seconds, Use \(a=1\). the lowest pitch easily heard by humans: 30 cycles per second

Sketch each angle in standard position. Use the unit circle and a right triangle to find exact values of the cosine and the sine of the angle. $$ -300^{\circ} $$

The given angle \(\theta\) is in standard position. Find the radian measure of the angle that results after the given number of revolutions from the terminal side of \(\theta\) . \(\theta=\frac{\pi}{3} ; 2\) clockwise revolutions

a. What are the domain, range, and period of \(y=\csc x ?\) b. What is the relative minimum in the interval \(0 \leq x \leq \pi ?\) c. What is the relative maximum in the interval \(\pi \leq x \leq 2 \pi ?\)

Solve each equation in the interval from 0 to 2\(\pi .\) Round your answer to the nearest hundredth. $$ \cos t=\frac{1}{4} $$

Write the explicit formula for each geometric sequence. List the first five terms. $$ a_{1}=10, r=3 $$

Use a graphing calculator to graph each function in the interval from 0 to 2\(\pi .\) Then sketch each graph. $$ y=\sin (x+2 \cos x) $$

Sketch each angle in standard position. Use the unit circle and a right triangle to find exact values of the cosine and the sine of the angle. $$ 120^{\circ} $$

The given angle \(\theta\) is in standard position. Find the radian measure of the angle that results after the given number of revolutions from the terminal side of \(\theta\) . \(\theta=-\frac{2 \pi}{3} ; 1\) counterclockwise revolution

Solve each equation in the interval from 0 to 2\(\pi .\) Round your answer to the nearest hundredth. $$ 10 \cos t=-2 $$

Write the explicit formula for each geometric sequence. List the first five terms. \(a_{1}=12, r=-0.3\)

$$ \begin{array}{l}{\text { Which function is a phase shift of } y=\sin \theta \text { by } 5 \text { units to the left? }} \\ {\begin{array}{llll}{\text { A. } y=5 \sin \theta} & {\text { B. } y=\sin \theta+5} & {\text { C. } y=\sin (\theta+5)} & {\text { D. } y=\sin 5 \theta}\end{array}}\end{array} $$

Sound For sound waves, the period and the frequency of a pitch are reciprocals of each other: period \(=\frac{\text { seconds }}{\text { cycle }}\) and frequency \(=\frac{\text { cycles }}{\text { second }} .\) Write an equation for each pitch. Let \(\theta=\) time in seconds. Use \(a=1.\) The highest pitch heard by bats: 120,000 cycles per second.

Sketch each angle in standard position. Use the unit circle and a right triangle to find exact values of the cosine and the sine of the angle. $$ 225^{\circ} $$

The given angle \(\theta\) is in standard position. Find the radian measure of the angle that results after the given number of revolutions from the terminal side of \(\theta\) . \(\theta=\frac{5 \pi}{6} ; 2 \frac{1}{2}\) counterclockwise revolutions

Writing Explain why each expression is undefined. $$ \csc 180^{\circ} $$

Solve each equation in the interval from 0 to 2\(\pi .\) Round your answer to the nearest hundredth. $$ -2 \cos \theta=0.7 $$

Write the explicit formula for each geometric sequence. List the first five terms. $$ a_{1}=900, r=-\frac{1}{3} $$

$$ \begin{array}{ll}{\text { Which function is a translation of } y=\cos \theta \text { by } 5 \text { units down } ?} \\ {\text { F. } y=-5 \cos \theta} & {\text { G. } y=\cos \theta-5} \\ {\text { H. } y=\cos (\theta-5)} & {\text { J. } y=\cos (-5 \theta)}\end{array} $$

Sketch each angle in standard position. Use the unit circle and a right triangle to find exact values of the cosine and the sine of the angle. $$ -780^{\circ} $$

Reasoning. Use the proportion \(\frac{\text { measure of central angle }}{\text { measure of one complete rotation }}=\frac{\text { length of arc }}{\text { circumference }}\) to derive the formula \(s=r \theta .\) Use \(\theta\) for the central angle measure and \(s\) for the arc length. Measure the rotation in radians.

Writing Explain why each expression is undefined. $$ \sec 90^{\circ} $$

Solve each equation in the interval from 0 to 2\(\pi .\) Round your answer to the nearest hundredth. $$ 3 \cos \frac{t}{5}=1 $$

Which function is a translation of \(y=\sin \theta\) that is \(\frac{\pi}{3}\) units up and \(\frac{\pi}{2}\) units to the left? \(\begin{array}{ll}{\text { A. } y=\sin \left(\theta+\frac{\pi}{3}\right)+\frac{\pi}{2}} & {\text { B. } y=\sin \left(\theta+\frac{\pi}{2}\right)+\frac{\pi}{3}} \\ {\text { C. } y=\sin \left(\theta-\frac{\pi}{2}\right)+\frac{\pi}{3}} & {\text { D. } y=\sin \left(\theta-\frac{\pi}{3}\right)-\frac{\pi}{2}}\end{array}\)

Sketch each angle in standard position. Use the unit circle and a right triangle to find exact values of the cosine and the sine of the angle. $$ -405^{\circ} $$

Writing Explain why each expression is undefined. $$ \cot 0^{\circ} $$

Solve each equation in the interval from 0 to 2\(\pi .\) Round your answer to the nearest hundredth. $$ \cos \frac{3}{4} \theta=-0.6 $$

Find values of a and \(b\) such that the function \(y=\sin \theta\) can be expressed as \(y=a \cos (\theta+b)\)

Which pairs of measurements represent the same angle measures? \(I.240^{\circ}, \frac{7 \pi}{6}\) radians \(\quad\) II. \(135^{\circ}, \frac{3 \pi}{4}\) radians \(\quad\) III. \(150^{\circ}, \frac{5 \pi}{6}\) radians A. I and II only \(\quad\) B. \(I\) and III only C. II and III only \(\quad\) D. I,II, and III

Sketch each angle in standard position. Use the unit circle and a right triangle to find exact values of the cosine and the sine of the angle. $$ 1020^{\circ} $$

Solve each equation in the interval from 0 to 2\(\pi .\) Round your answer to the nearest hundredth. $$ 5 \cos \pi t=0.9 $$

Write a function that is a transformation of \(y=\sin \theta\) so that its amplitude is 4 and its minimum value is 1 . Show your work.

What is the exact value of \(\cos \left(\frac{5 \pi}{4} \text { radians }\right) ?\) F. \(-\frac{\sqrt{3}}{2} \quad\) G. \(-\frac{\sqrt{2}}{2} \quad\) H. \(-\frac{1}{2} \quad\) I. \(\frac{\sqrt{2}}{2}\)

a. Graph \(y-\tan x\) and \(y-\cot x\) on the same axes. b. State the domain, the range, and the asymptotes of each function. c. Writing Compare the two graphs. How are they alike? How are they different? d. Geometry The graph of the cotangent function can be reflected about a line to graph the tangent function. Name at least two lines that have this property.

Find the mean, median, and mode for each set of values. $$ \begin{array}{lllllllllll}{9} & {6} & {8} & {1} & {3} & {4} & {5} & {2} & {6} & {8} & {4} & {9} & {12} & {3} & {4} & {10} & {7} & {6}\end{array} $$

In a circle, an arc of length 8\(\pi \mathrm{cm}\) is intercepted by a central angle of \(\frac{2 \pi}{3}\) radians. What is the radius of the circle? $$ \begin{array}{lllll}{\text { A. } \frac{3 \pi}{16} \mathrm{cm}} & {\text { B. } \frac{16 \pi}{3} \mathrm{cm}} & {\text { C. } \frac{16 \pi^{2}}{3} \mathrm{cm}} & {\text { D. } 12 \mathrm{cm}}\end{array} $$

Open-Ended Find the measures of four angles in standard position that have a sine of \(0.5 .\) . (Hint: Use the unit circle and right triangles.)

Find all the values of \(\theta\) between \(-\pi\) and 2\(\pi\) for which \(\sin \theta=3 \sin \theta .\) Show your work.

Graph each function in the interval from 0 to 2\(\pi .\) Describe any phase shift and vertical shift in the graph. $$ y=\sec 2 \theta+3 $$

Find the mean, median, and mode for each set of values. $$ \begin{array}{llllllllllll}{45} & {42} & {39} & {35} & {41} & {45} & {49} & {42} & {43} & {48} & {32} & {51} & {42}\end{array} $$

Identify the period of each function. Then tell where two asymptotes occur for each function. $$ y=\tan 6 \theta $$

Two arcs have the same length. One arc is intercepted by an angle of \(\frac{3 \pi}{2}\) radians in a circle of radius 15 \(\mathrm{cm} .\) If the radius of the other circle is \(25 \mathrm{cm},\) what central angle intercepts the arc? \(\mathrm{F} \cdot \frac{3 \pi}{2}\) radians \(\quad\) G. \(\frac{9 \pi}{10}\) radians H. \(\frac{3 \pi}{2}\) radians \(\quad\) J. \(\frac{5 \pi}{3}\) radians

Critical Thinking Suppose \(\theta\) is an angle in standard position and \(\cos \theta=-\frac{1}{2}\) and \(\sin \theta=-\frac{\sqrt{3}}{2} .\) Can the value of \(\theta\) be \(60^{\circ} ?\) Can it be \(-120^{\circ} ?\) Draw a diagram and justify your reasoning.

Graph each function in the interval from 0 to 2\(\pi .\) Describe any phase shift and vertical shift in the graph. $$ y=\sec 2\left(\theta+\frac{\pi}{2}\right) $$

Find the mean, median, and mode for each set of values. $$ \begin{array}{lllllllllll}{7.1} & {8.5} & {7.0} & {7.6} & {8.5} & {8.1} & {7.9} & {8.2} & {7.3} & {9.1} & {8.7} & {7.9}\end{array} $$

Identify the period of each function. Then tell where two asymptotes occur for each function. $$ y=\tan \frac{\theta}{4} $$

Describe the relationship between a central angle of one radian and the radius of the circle.

Navigation When navigators locate an object, they measure in a clockwise direction from due north. The measure of the angle is called the bearing. Suppose a lighthouse's bearing is \(110^{\circ}\) from a ship. a. Sketch the diagram at the right on a coordinate plane. Place north along the positive \(y\) -axis. b. Express the location of the lighthouse in terms of an angle in standard position.