Coterminal Angles Explain how to find coterminal angles in degrees.

Amy J.

Numerade Educator

Degrees to Radians Explain how to convert from degrees to radians.

Noah C.

Numerade Educator

Trigonometric Functions

Find $\sin \theta, \cos \theta,$ and $\tan \theta$

Amy J.

Numerade Educator

Characteristics of a Graph In your own words, describe the meaning of amplitude and period.

Noah C.

Numerade Educator

Coterminal Angles in Degrees In Exercises 5 and 6 , determine two coterminal angles in degree measure (one positive and one negative) for each angle.

Amy J.

Numerade Educator

Noah C.

Numerade Educator

Coterminal Angles in Radians In Exercises 7 and 8 , determine two coterminal angles in radian measure (one positive and one negative) for each angle.

Amy J.

Numerade Educator

Noah C.

Numerade Educator

Degrees to Radians In Exercises 9 and 10, convert the degree measure to radian measure as a multiple of $\pi$ and as a decimal accurate to three decimal places.

$\begin{array}{lll}{\text { (a) }-20^{\circ}} & {\text { (b) }-240^{\circ}} & {\text { (c) }-270^{\circ}} & {\text { (d) } 144^{\circ}}\end{array}$

Amy J.

Numerade Educator

$\begin{array}{lll}{\text { (a) }-20^{\circ}} & {\text { (b) }-240^{\circ}} & {\text { (c) }-270^{\circ}} & {\text { (d) } 144^{\circ}}\end{array}$

Noah C.

Numerade Educator

Radians to Degrees In Exercises 11 and $12,$ convert the radian measure to degree measure.

$\begin{array}{lll}{\text { (a) } \frac{3 \pi}{2}} & {\text { (b) } \frac{7 \pi}{6}} & {\text { (c) }-\frac{7 \pi}{12}}\end{array} \quad$ (d) $-2.367$

Amy J.

Numerade Educator

Radians to Degrees In Exercises 11 and $12$ convert the radian measure to degree measure.

(a) $\frac{7 \pi}{3} \quad$ (b) $-\frac{11 \pi}{30}$ (c) $\frac{11 \pi}{6} \quad$ (d) 0.438

Noah C.

Numerade Educator

Completing a Table Let $r$ represent the radius of a circle, $\theta$ the central angle (measured in radians), and s the length of the arc subtended by the angle. Use the relationship $s=r \theta$ to complete the table.

Amy J.

Numerade Educator

Angular Speed A car is moving at the rate of 50 miles per hour, and the diameter of its wheels is 2.5 feet.

(a) Find the number of revolutions per minute that the wheels are rotating.

(b) Find the angular speed of the wheels in radians per minute.

Noah C.

Numerade Educator

Evaluating Trigonometric Functions In Exercises 15 and $16,$ evaluate the six trigonometric functions of the angle $\theta$ .

Amy J.

Numerade Educator

Noah C.

Numerade Educator

Evaluating Trigonometric Functions In Exercises $17-20$ , sketch a right triangle corresponding to the trigonometric function of the acute angle $\theta$ . Then evaluate the other five trigonometric functions of $\theta$ .

$$\sin \theta=\frac{1}{2}$$

Amy J.

Numerade Educator

Evaluating Trigonometric Functions In Exercises $17-20$ , sketch a right triangle corresponding to the trigonometric function of the acute angle $\theta$ . Then evaluate the other five trigonometric functions of $\theta$.

$$\sin \theta=\frac{1}{3}$$

Noah C.

Numerade Educator

Evaluating Trigonometric Functions In Exercises $17-20$ , sketch a right triangle corresponding to the trigonometric function of the acute angle $\theta$ . Then evaluate the other five trigonometric functions of $\theta$.

$$\cos \theta=\frac{4}{5}$$

Amy J.

Numerade Educator

Evaluating Trigonometric Functions In Exercises $17-20$ , sketch a right triangle corresponding to the trigonometric function of the acute angle $\theta$ . Then evaluate the other five trigonometric functions of $\theta$.

$$\sec \theta=\frac{13}{5}$$

Noah C.

Numerade Educator

Evaluating Trigonometric Functions In Exerrises $21-24$ , evaluate the sine, cosine, and tangent of each angle. Do not use a calculator.

$$\begin{array}{lll}{\text { (a) } 60^{\circ}} & {\text { (b) } 120^{\circ}} & {\text { (c) } \frac{\pi}{4}} & {\text { (d) } \frac{5 \pi}{4}}\end{array}$$

Amy J.

Numerade Educator

Evaluating Trigonometric Functions In Exerrises $21-24$ , evaluate the sine, cosine, and tangent of each angle. Do not use a calculator.

$$\begin{array}{lll}{(a)-30^{\circ}} & {\text { (b) } 150^{\circ}} & {\text { (c) }-\frac{\pi}{6}} & {\text { (d) } \frac{\pi}{2}}\end{array}$$

Noah C.

Numerade Educator

Evaluating Trigonometric Functions In Exerrises $21-24$ , evaluate the sine, cosine, and tangent of each angle. Do not use a calculator.

$$\begin{array}{ll}{\text { (a) } 225^{\circ}} & {\text { (b) }-225^{\circ}} & {\text { (c) } \frac{5 \pi}{3}} & {\text { (d) } \frac{11 \pi}{6}}\end{array}$$

Amy J.

Numerade Educator

Evaluating Trigonometric Functions In Exerrises $21-24$ , evaluate the sine, cosine, and tangent of each angle. Do not use a calculator.

$$\begin{array}{ll}{\text { (a) } 750^{\circ}} & {\text { (b) } 510^{\circ}} & {\text { (c) } \frac{10 \pi}{3}} & {\text { (d) } \frac{17 \pi}{3}}\end{array}$$

Noah C.

Numerade Educator

Evaluating Trigonometric Functions Using Technology In Exercises $25-28$ , use a calculator to evaluate each trigonometric function. Round your answers to four decimal places.

$$\begin{array}{l}{\text { (a) } \sin 10^{\circ}} \\ {\text { (b) } \csc 10^{\circ}}\end{array}$$

Amy J.

Numerade Educator

Evaluating Trigonometric Functions Using Technology

In Exercises $25-28$ , use a calculator to evaluate each trigonometric function. Round your answers to four decimal places.

$$\begin{array}{l}{\text { (a) } \sec 225^{\circ}} \\ {\text { (b) } \sec 135^{\circ}}\end{array}$$

Noah C.

Numerade Educator

Evaluating Trigonometric Functions Using Technology

In Exercises $25-28$ , use a calculator to evaluate each trigonometric function. Round your answers to four decimal places.

$$\begin{array}{l}{\text { (a) } \tan \frac{\pi}{9}} \\ {\text { (b) } \tan \frac{10 \pi}{9}}\end{array}$$

Amy J.

Numerade Educator

Evaluating Trigonometric Functions Using Technology

In Exercises $25-28$ , use a calculator to evaluate each trigonometric function. Round your answers to four decimal places.

$$\begin{array}{l}{\text { (a) } \cot (1.35)} \\ {\text { (b) } \tan (1.35)}\end{array}$$

Noah C.

Numerade Educator

Determining a Quadrant In Exercises 29 and 30 , determine the quadrant in which $\theta$ lies.

$$\begin{array}{l}{\text { (a) } \sin \theta<0 \text { and } \cos \theta<0} \\ {\text { (b) } \sec \theta>0 \text { and } \cot \theta<0}\end{array}$$

Amy J.

Numerade Educator

Determining a Quadrant In Exercises 29 and 30 , determine the quadrant in which $\theta$ lies.

$$\begin{array}{l}{\text { (a) } \sin \theta>0 \text { and } \cos \theta<0} \\ {\text { (b) } \csc \theta<0 \text { and } \tan \theta>0}\end{array}$$

Noah C.

Numerade Educator

Solving a Trigonometric Equation In Exercises $31-34$ , find two solutions of each equation. Give your answers in radians $(0 \leq \theta \leq 2 \pi) .$ Do not use a calculator.

$$\begin{array}{l}{\text { (a) } \cos \theta=\frac{\sqrt{2}}{2}} \\ {\text { (b) } \cos \theta=-\frac{\sqrt{2}}{2}}\end{array}$$

Amy J.

Numerade Educator

Solving a Trigonometric Equation In Exercises $31-34$ , find two solutions of each equation. Give your answers in radians $(0 \leq \theta \leq 2 \pi) .$ Do not use a calculator.

$$\begin{array}{l}{\text { (a) } \sec \theta=2} \\ {\text { (b) } \sec \theta=-2}\end{array}$$

Noah C.

Numerade Educator

Solving a Trigonometric Equation In Exercises $31-34$ , find two solutions of each equation. Give your answers in radians $(0 \leq \theta \leq 2 \pi) .$ Do not use a calculator.

$$\begin{array}{l}{\text { (a) } \tan \theta=1} \\ {\text { (b) } \cot \theta=-\sqrt{3}}\end{array}$$

Amy J.

Numerade Educator

Solving a Trigonometric Equation In Exercises $31-34$ , find two solutions of each equation. Give your answers in radians $(0 \leq \theta \leq 2 \pi) .$ Do not use a calculator.

$$\begin{array}{l}{\text { (a) } \sin \theta=\frac{\sqrt{3}}{2}} \\ {\text { (b) } \sin \theta=-\frac{\sqrt{3}}{2}}\end{array}$$

Noah C.

Numerade Educator

Solving a Trigonometric Equation In Exercises $35-42,$ solve the equation for $\theta,$ where $0 \leq \theta \leq 2 \pi .$

$$2 \sin ^{2} \theta=1$$

Amy J.

Numerade Educator

Solving a Trigonometric

Equation In Exercises $35-42,$ solve the equation for $\theta,$ where $0 \leq \theta \leq 2 \pi .$

$$\tan ^{2} \theta=3$$

Noah C.

Numerade Educator

Solving a Trigonometric

Equation In Exercises $35-42,$ solve the equation for $\theta,$ where $0 \leq \theta \leq 2 \pi .$

$$\tan ^{2} \theta-\tan \theta=0$$

Amy J.

Numerade Educator

Solving a Trigonometric

Equation In Exercises $35-42,$ solve the equation for $\theta,$ where $0 \leq \theta \leq 2 \pi .$

$$2 \cos ^{2} \theta-\cos \theta=1$$

Noah C.

Numerade Educator

Solving a Trigonometric Equation

In Exercises $35-42$ , solve the equation for $\theta,$ where $0 \leq \theta \leq 2 \pi$

$$\sec \theta \csc \theta=2 \csc \theta$$

Amy J.

Numerade Educator

Solving a Trigonometric Equation

In Exercises $35-42$ , solve the equation for $\theta,$ where $0 \leq \theta \leq 2 \pi$

$$\sin \theta=\cos \theta$$

Noah C.

Numerade Educator

Solving a Trigonometric Equation

In Exercises $35-42$ , solve the equation for $\theta,$ where $0 \leq \theta \leq 2 \pi$

$$\cos ^{2} \theta+\sin \theta=1$$

Amy J.

Numerade Educator

Solving a Trigonometric Equation

In Exercises $35-42$ , solve the equation for $\theta,$ where $0 \leq \theta \leq 2 \pi$

$$\cos \frac{\theta}{2}-\cos \theta=1$$

Check back soon!

Airplane Ascent An airplane leaves the runway climbing at an angle of $18^{\circ}$ with a speed of 275 feet per second (see figure). Find the altitude $a$ of the plane after 1 minute.

Amy J.

Numerade Educator

Height of a Mountain While traveling across flat land, you notice a mountain directly in front of you. Its angle of elevation (to the peak) is $3.5^{\circ}$ . After you drive 13 miles closer to the mountain, the angle of elevation is $9^{\circ}$ . Approximate the height of the mountain.

Noah C.

Numerade Educator

Period and Amplitude In Exercises $45-48,$ determine the period and amplitude of each function.

$$y=2 \sin 2 x$$

Amy J.

Numerade Educator

Period and Amplitude In Exercises $45-48,$ determine the period and amplitude of each function.

$$y=\frac{3}{2} \cos \frac{x}{2}$$

Noah C.

Numerade Educator

Period and Amplitude In Exercises $45-48,$ determine the period and amplitude of each function.

$$y=-3 \sin 4 \pi x$$

Amy J.

Numerade Educator

Period and Amplitude In Exercises $45-48,$ determine the period and amplitude of each function.

$$y=\frac{2}{3} \cos \frac{\pi x}{10}$$

Noah C.

Numerade Educator

Period In Exercises $49-52,$ find the period of the function.

$$y=7 \tan 2 \pi x$$

Amy J.

Numerade Educator

Period In Exercises $49-52,$ find the period of the function.

$$y=7 \tan 2 \pi x$$

Noah C.

Numerade Educator

Period In Exercises $49-52,$ find the period of the function.

$$y=\sec 5 x$$

Amy J.

Numerade Educator

Period In Exercises $49-52,$ find the period of the function.

$$y=\csc 4 x$$

Noah C.

Numerade Educator

Writing In Exercises 53 and $54,$ use a graphing utility to graph each function $f$ in the same viewing window for $c=-2$ , $c=-1, c=1,$ and $c=2 .$ Give a written description of the change in the graph caused by changing $c .$

$$\begin{array}{l}{\text { (a) } f(x)=c \sin x} \\ {\text { (b) } f(x)=\cos (c x)} \\ {\text { (c) } f(x)=\cos (\pi x-c)}\end{array}$$

Amy J.

Numerade Educator

Writing In Exercises 53 and $54,$ use a graphing utility to graph each function $f$ in the same viewing window for $c=-2,$ $c=-1, c=1,$ and $c=2 .$ Give a written description of the change in the graph caused by changing $c .$

$$\begin{array}{l}{\text { (a) } f(x)=\sin x+c} \\ {\text { (b) } f(x)=-\sin (2 \pi x-c)} \\ {\text { (c) } f(x)=c \cos x}\end{array}$$

Noah C.

Numerade Educator

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.

$$y=\sin \frac{x}{2}$$

Amy J.

Numerade Educator

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.

$$y=2 \cos 2 x$$

Noah C.

Numerade Educator

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.

$$y=-\sin \frac{2 \pi x}{3}$$

Amy J.

Numerade Educator

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.

$$y=2 \tan x$$

Noah C.

Numerade Educator

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.

$$y=\csc \frac{x}{2}$$

Amy J.

Numerade Educator

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.

$$y=\tan 2 x$$

Noah C.

Numerade Educator

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.

$$y=2 \sec 2 x$$

Amy J.

Numerade Educator

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.

$$y=\csc 2 \pi x$$

Noah C.

Numerade Educator

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.

$$y=\sin (x+\pi)$$

Amy J.

Numerade Educator

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.

$$y=\cos \left(x-\frac{\pi}{3}\right)$$

Noah C.

Numerade Educator

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.

$$y=1+\cos \left(x-\frac{\pi}{2}\right)$$

Amy J.

Numerade Educator

Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.

$$y=1+\sin \left(x+\frac{\pi}{2}\right)$$

Noah C.

Numerade Educator

Graphical Reasoning In Exercises 67 and $68,$ find $a, b,$ and $c$ such that the graph of the function matches the graph in the figure.

$$y=a \cos (b x-c)$$

Amy J.

Numerade Educator

Graphical Reasoning In Exercises 67 and $68,$ find $a, b,$ and $c$ such that the graph of the function matches the graph in the figure.

$$y=a \sin (b x-c)$$

Noah C.

Numerade Educator

Think About It You are given the value of tan $\theta$ . Is it possible to find the value of sec $\theta$ without finding the measure of $\theta ?$ Explain.

Amy J.

Numerade Educator

Restricted Domain Explain how to restrict the domain of the sine function so that it becomes a one-to-one function.

Noah C.

Numerade Educator

Think About It How do the ranges of the cosine function and the secant function compare?

Amy J.

Numerade Educator

HOW DO YOU SEE IT? Consider an angle in standard position with $r=12$ centimeters, as shown in the figure. Describe the changes in the values of $x, y, \sin \theta, \cos \theta,$ and $\tan \theta$ as $\theta$ increases continually from $0^{\circ}$ to $90^{\circ} .$

Noah C.

Numerade Educator

Think About It Sketch the graphs of

$$f(x)=\sin x, \quad g(x)=|\sin x|, \quad \text { and } \quad h(x)=\sin (|x|)$$

In general, how are the graphs of $|f(x)|$ and $f(|x|)$ related to the graph of $f ?$

Check back soon!

Ferris wheel. The model for the height $h$ of a Ferris wheel car is $h=51+50 \sin 8 \pi t$

where $t$ is measured in minutes. (The Ferris wheel has a radius of 50 feet.) This model yields a height of 51 feet when $t=0$ . Alter the model so that the height of the car is 1 foot when $t=0$ .

Noah C.

Numerade Educator

Sales The monthly sales $S$ (in thousands of units) of a seasonal product are modeled by

$$S=58.3+32.5 \cos \frac{\pi t}{6}$$

where $t$ is the time (in months), with $t=1$ corresponding to January. Use a graphing utility to graph the model for $S$ and determine the months when sales exceed $75,000$ units.

Check back soon!

Pattern Recognition Use a graphing utility to compare the graph of

$$f(x)=\frac{4}{\pi}\left(\sin \pi x+\frac{1}{3} \sin 3 \pi x\right)$$

with the given graph. Try to improve the approximation by

adding a term to $f(x) .$ Use a graphing utility to verify that

your new approximation is better than the original. Can you

find other terms to add to make the approximation even better?

What is the pattern? (Hint: Use sine terms.)

Noah C.

Numerade Educator

True or False? In Exercises $77-80$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$$\begin{array}{l}{\text { A measurement of } 4 \text { radians corresponds to two complete }} \\ {\text { revolutions from the initial side to the terminal side of an angle. }}\end{array}$$

Check back soon!

True or False? In Exercises $77-80$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

Amplitude is always positive

Noah C.

Numerade Educator

True or False? In Exercises $77-80$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$$\begin{array}{l}{\text { The function } y=\frac{1}{2} \sin 2 x \text { has an amplitude that is twice that of }} \\ {\text { the function } y=\sin x .}\end{array}$$

Amy J.

Numerade Educator

True or False? In Exercises $77-80$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$$\begin{array}{l}{\text { The function } y=3 \cos (x / 3) \text { has a period that is three times }} \\ {\text { that of the function } y=\cos x .}\end{array}$$

Noah C.

Numerade Educator