4.2 Graphs of Trigonometric Functions

A Periodic Function of Angle

Imagine that you are riding on a Ferris wheel. As the wheel turns, your height above the ground increases and then decreases again, repeating the same pattern each time the Ferris wheel makes a complete rotation. This pattern is an example of a periodic function. We use periodic functions to model phenomena that exhibit cyclical behavior, such as the height of tides, seasonal patterns of growth in plants and animals, radio waves, and planetary motion.

Function graph showing the parametric curve (10*cos(t), 11 + 10*sin(t)) for t in [0, 6.28], the parametric curve (t*10*cos(a*pi/180), 11 + t*10*sin(a*pi/180)) for t in [0, 1], the parametric curve (10*cos(a*pi/180) + 0.7*cos(t), 11 + 10*sin(a*pi/180) + 0.7*sin(t)) for t in [0, 6.28] and the parametric curve (10*cos(a*pi/180), t*(11 + 10*sin(a*pi/180))) for t in [0, 1]. Adjustable parameter: Angle turned (a) = 30 °. Viewing window: x from -20 to 20, y from -1.37 to 23.37. — The Ferris wheel this paragraph imagines: a wheel of radius 10 meters whose hub sits 11 meters up. Drag the angle slider to ride: your height above the ground is 11 + 10 sin(a°), climbing to 21 meters at the top of the wheel (a = 90°) and dipping to 1 meter at the bottom (a = 270°). Keep dragging past 360° — the same heights return in the same order, once each full turn. That repetition, the same output every 360° of input, is what the section goes on to name a periodic function, and it is the reason the graph of sin θ you are about to meet is a wave.

We'll create a mathematical model for a ride on a Ferris wheel that has a radius of 100 feet and rotates counterclockwise. Our model will be a function that describes your height above the ground as you ride the wheel. In order to graph the Ferris wheel function, we must first specify the input and output variables, and then choose a coordinate system to display their values.

We'll place the origin at the center of the Ferris wheel. Then the line from the origin to your position on the wheel makes an angle with the horizontal, as shown at right. This angle, $θ$ , will be the input variable for the function. Your height, $h$ , is also a variable, and is related to the $y$ -coordinate of your position; in fact, we see that $h = y + 100$ , because the center of the wheel is 100 feet above the ground.

To simplify the model, we'll first graph $y$ as the output variable, instead of $h$ . As the angle $θ$ increases from $0 degrees$ to $90 degrees$ , your $y$ -coordinate increases from 0 to 100. You are then at the top of the wheel. Then, as $θ$ increases from $90 degrees$ to $180 degrees$ , your $y$ -coordinate decreases from 100 back to 0.

Finally, as $θ$ increases from $180 degrees$ to $360 degrees$ , your $y$ -coordinate decreases from 0 to $minus 100$ and then increases from $minus 100$ back to 0. You have made one complete rotation on the Ferris wheel. If you go around again, $θ$ increases from $360 degrees$ to $720 degrees$ , and the graph of your $y$ -coordinate will repeat the pattern of the first rotation. The figure above shows how your $y$ -coordinate is plotted as a function of the angle $θ$ .

Look back at the diagram of the Ferris wheel and notice that $\sin (θ) = \frac{y}{100}$ , so

$y equals 100 sin ⁡ open parenthesis θ close parenthesis$

For example, when $θ equals 30 degrees$ , the $y$ -coordinate is

$y = 100 \sin (30^{\circ}) = 100 (\frac{1}{2}) = 50$

and your height above the ground is

$h = y + 100 = 150 feet$

In general, then, $h$ is given as a function of $θ$ by

$h = y + 100 = 100 \sin (θ) + 100$

This is our model for your height on a Ferris wheel ride.

The Sine Function

Our Ferris wheel model used values of $\sin (θ)$ , so let us explore its properties. Remember that the trigonometric ratio $\sin (θ)$ is actually a function of the angle $θ$ . Thus, for each value of $θ$ , there is only one value of $\sin (θ)$ , and we may write $f (θ) = \sin (θ)$ .

Activity

Graph of the Sine Function

Graph the sine function $f (θ) = \sin (θ)$ .

First make a table of values. You can use your calculator's table feature to fill in the following values, rounded to two decimal places.

$θ$	$0 degrees$	$10 degrees$	$20 degrees$	$30 degrees$	$40 degrees$	$50 degrees$	$60 degrees$	$70 degrees$	$80 degrees$	$90 degrees$
$\sin (θ)$

Plot the points in the table and connect them with a smooth curve to graph the sine function in the first quadrant, from $0 degrees$ to $90 degrees$ .

Recall that the values of

\sin (θ)

in the second quadrant can be found using reference angles:

\sin (180^{\circ} - θ) = \sin (θ)

. This fact gives us values of

\sin θ

from

90 degrees

180 degrees

$θ$	$100 degrees$	$110 degrees$	$120 degrees$	$130 degrees$	$140 degrees$	$150 degrees$	$160 degrees$	$170 degrees$	$180 degrees$
$\sin (θ)$

Continue your graph from part (2) for second quadrant angles.

The values of

\sin (θ)

in the third and fourth quadrants are the negatives of their values in the second and first quadrants.

$θ$	$190 degrees$	$200 degrees$	$210 degrees$	$220 degrees$	$230 degrees$	$240 degrees$	$250 degrees$	$260 degrees$	$270 degrees$
$\sin (θ)$

$θ$	$280 degrees$	$290 degrees$	$300 degrees$	$310 degrees$	$320 degrees$	$330 degrees$	$340 degrees$	$350 degrees$	$360 degrees$
$\sin (θ)$

The graph of $f (θ) = \sin (θ)$ from $0 degrees$ to $360 degrees$ is shown below.

If we continue the graph for angles larger than $360 degrees$ or smaller than $0 degrees$ , we find that the same pattern repeats, as shown below. This should not be surprising, because we know that coterminal angles have the same trigonometric ratios.

The sine is an example of a periodic function. The smallest interval on which the graph repeats is called the period of the graph. From the graph in the previous example, we make the following observations:

You can use your calculator to graph the sine function, by entering

$Y sub 1 equals$ SIN $X, T, θ, n$

and pressing ZOOM $7$ for the trig window. The graph shows two periods of the sine function, from $θ = - 360^{\circ}$ to $θ equals 360 degrees$ .

Your height $h$ on the Ferris wheel is a function of $θ$ ,

$h = 100 + y = 100 + 100 \sin (θ)$

Complete the table of values and graph the Ferris wheel function,

h = F (θ)

$θ$	$0 degrees$	$30 degrees$	$60 degrees$	$90 degrees$	$120 degrees$	$150 degrees$	$180 degrees$
$\sin (θ)$
$h = F (θ)$

$θ$	$210 degrees$	$240 degrees$	$270 degrees$	$300 degrees$	$330 degrees$	$360 degrees$
$\sin (θ)$
$h = F (θ)$

Give the period, amplitude, and midline of the graph.

$θ$	$0 degrees$	$30 degrees$	$60 degrees$	$90 degrees$	$120 degrees$	$150 degrees$	$180 degrees$
$\sin (θ)$	$0$	$0.5$	$0.866$	$1.0$	$0.866$	$0.5$	$0$
$h = F (θ)$	$100$	$150$	$186.6$	$200$	$186.6$	$150$	$100$

$θ$	$210 degrees$	$240 degrees$	$270 degrees$	$300 degrees$	$330 degrees$	$360 degrees$
$\sin (θ)$	$minus 0.5$	$minus 0.866$	$minus 1.0$	$minus 0.866$	$minus 0.5$	$0$
$h = F (θ)$	$50$	$13.4$	$0$	$13.4$	$50$	$100$

Period: $360 degrees$ , amplitude: 100, midline: $h equals 100$

The Cosine Function

In the previous exercise you graphed the height of a person riding on a Ferris wheel. Your graph involved $\sin (θ)$ , because the sine function tells us the $y$ -coordinate of a point that travels around a circle. The cosine function tells us the $x$ -coordinate of a point that travels around a circle.

Example 1

Small adjustments to the fit of a bicycle can affect both the cyclist's efficiency and the stress on his or her joints. The KOPS rule (Knee Over Pedal Stem) aligns the cyclist's knee directly over the pedal at the point of maximum force, as shown at right. As the cyclist's foot rotates away from this KOPS line, stress on the knee increases.

Suppose the pedal crank is 18 centimeters long. When the crank makes an angle $θ$ with the horizontal, how far is the cyclist's foot displaced horizontally from the KOPS line?
Graph the horizontal displacement, $d$ , as a function of $θ$ .

The cyclist's foot travels around a circle of radius 18 centimeters. If we place the origin at the center of the chain gear, the $x$ -coordinate of the foot is given by
$x = r \cos (θ) = 18 \cos (θ)$
(See the figure at right.) The KOPS line is the vertical line $x equals 18$ , so the horizontal distance between the cyclist's foot and the KOPS line is
$d = 18 - 18 \cos (θ)$

You can use your calculator to verify the table and graph for the function

$d = 18 - 18 \cos (θ)$

shown below.

$θ$	$0 degrees$	$30 degrees$	$60 degrees$	$90 degrees$	$120 degrees$	$150 degrees$	$180 degrees$	$210 degrees$	$240 degrees$	$270 degrees$	$300 degrees$	$330 degrees$	$360 degrees$
$d$	$0$	$2.4$	$9$	$18$	$27$	$33.6$	$36$	$33.6$	$27$	$18$	$9$	$2.4$	$0$

In the example above, the period is $360 degrees$ , the amplitude is 18 cm, and the midline is $d equals 18$ .

You can see that the cosine graph is similar to the sine graph, but they are not identical.

Complete the table below with values rounded to two decimal places. Use the table and your knowledge of reference angles to graph the cosine function,

f (θ) = \cos (θ),

from

minus 180 degrees

540 degrees

$θ$	$0 degrees$	$10 degrees$	$20 degrees$	$30 degrees$	$40 degrees$	$50 degrees$	$60 degrees$	$70 degrees$	$80 degrees$	$90 degrees$
$\cos (θ)$

Use your graph to find the period, amplitude, and midline of the cosine function. How does the graph of cosine differ from the graph of sine? (Hint: Consider the intercepts of the graph, and the location of the maximum and minimum values.)

$θ$	$0 degrees$	$10 degrees$	$20 degrees$	$30 degrees$	$40 degrees$	$50 degrees$	$60 degrees$	$70 degrees$	$80 degrees$	$90 degrees$
$\cos (θ)$	$1$	$0.98$	$0.94$	$0.87$	$0.77$	$0.64$	$0.50$	$0.34$	$0.17$	$0$

Period: $360 degrees$ , amplitude: $1$ , midline: $y equals 0$ . The cosine graph starts $open parenthesis θ equals 0 degrees close parenthesis$ at its high point, while the sine graph starts $open parenthesis θ equals 0 degrees close parenthesis$ at its midline.

Interlude: Review of Function Notation

Perhaps it is time to review our use of function notation. Recall that we use the notation $y = f (x)$ to indicate that $y$ is a function of $x$ , that is, $x$ is the input variable and $y$ is the output variable.

Example 2

Make a table of input and output values and a graph for the function

$y = f (x) = \sqrt{9 - x^{2}}$

We choose several values for the input variable, $x$ , and evaluate the function to find the corresponding values of the output variable, $y$ . For example,

$f (- 3) = \sqrt{9 - (- 3)^{2}} = 0$

We plot the points in the table and connect them to obtain the graph shown at right.

$x$	$minus 3$	$minus 2$	$minus 1$	$0$	$1$	$2$	$3$
$y$	$0$	$the square root of 5$	$2 the square root of 2$	$3$	$2 the square root of 2$	$the square root of 5$	$0$

Of course, we don't always use $x$ and $y$ for the input and output variables. In the previous example, we could write $w = f (t) = \sqrt{9 - t^{2}}$ for the function, so that $t$ is the input and $w$ is the output. The table of values and the graph are the same; only the names of the variables have changed.

Note

When we discuss trigonometric functions, there are several variables involved. Our definitions of the trig ratios involve four variables: $x$ , $y$ , $r$ , and $θ$ , as illustrated below.

If the value of $r$ is fixed for a given situation, such as the Ferris wheel or the bicycle wheel discussed above, then $x$ and $y$ are both functions of $θ$ . This means that the values of $x$ and $y$ depend only on the value of the angle $θ$ . If $r equals 1$ , we have

$\begin{aligned} x & = f (θ) = \cos (θ) \\ y & = g (θ) = \sin (θ) \end{aligned}$

The graphs of these functions are shown below. Note particularly that the horizontal axis displays values of the input variable, and the vertical axis displays the output variable.

If we use different variables for the input and output, the functions and their graphs are the same, but the axes should be labeled with the appropriate variables.

Sketch a graph of each function, and label the axes.

$d = F (ϕ) = \sin (ϕ)$
$t = G (β) = \cos (β)$

In particular, we label the axes with the given variables.

The Tangent Function

The tangent function is periodic, but its graph is not similar to the graphs of sine and cosine. Recall that the tangent of an angle in standard position is defined by

$\tan (θ) = \frac{y}{x}$

Study the figure at right to see that as $θ$ increases from $0 degrees$ to $90 degrees$ , $y$ increases while $x$ remains constant, so the value of $\tan (θ)$ increases.

Right triangles with a common base x and increasing y

Example 3

Sketch a graph of $f (θ) = \tan (θ)$ for $0 degrees less than or equal to θ less than or equal to 180 degrees$ .

You can use your calculator to verify the following values for $\tan θ$ .

$θ$	$0 degrees$	$30 degrees$	$60 degrees$	$70 degrees$	$80 degrees$	$85 degrees$
$\tan (θ)$	$0$	$0.58$	$1.73$	$2.75$	$5.67$	$11.43$

As $θ$ gets closer to $90 degrees$ , $\tan (θ)$ increases very rapidly. Recall that $tan ⁡ open parenthesis 90 degrees close parenthesis$ is undefined, so there is no point on the graph at $θ equals 90 degrees$ . The graph of $f (θ) = \tan (θ)$ for $0 less than or equal to θ less than 90 degrees$ is shown at right.

For $θ$ in the second quadrant, the tangent is negative. The reference angle for each angle in the second quadrant is its supplement, so

$\tan (θ) = - \tan (180^{\circ} - θ)$

as shown at right. For example, you can verify that

$\tan (130^{\circ}) = - \tan (180^{\circ} - 130^{\circ}) = - \tan (50^{\circ}) = - 1.19$

In particular, for values of $θ$ close to $90 degrees$ , the values of $\tan (θ)$ are large negative numbers. We plot several points and sketch the graph in the second quadrant.

$θ$	$100 degrees$	$110 degrees$	$120 degrees$	$150 degrees$	$180 degrees$
$\tan (θ)$	$minus 5.67$	$minus 2.75$	$minus 1.73$	$minus 0.58$	$0$

In the figure at right, note that the graph has a break at $θ equals 90 degrees$ , because $tan ⁡ open parenthesis 90 degrees close parenthesis$ is undefined.

Now let's consider the graph of $f (θ) = \tan (θ)$ in the third and fourth quadrants. The tangent is positive in the third quadrant, and negative in the fourth quadrant. In fact, from the figure below you can see that the angles $θ$ and $180 degrees plus θ$ are vertical angles.

Because $θ$ and $180 degrees plus θ$ have the same reference angle, they have the same tangent. For example,

$\begin{aligned} \tan (200^{\circ}) & = \tan (20^{\circ}) \\ \tan (230^{\circ}) & = \tan (50^{\circ}) \\ \tan (250^{\circ}) & = \tan (70^{\circ}) \end{aligned}$

Thus, the graph of $\tan (θ)$ in the third quadrant is the same as its graph in the first quadrant. Similarly, the graph of the tangent function in the fourth quadrant is the same as its graph in the second quadrant. The completed graph is shown below.

What is the period of the tangent function?
Does the graph of tangent have an amplitude?
For what values of $θ$ is $\tan (θ)$ undefined?
Give the equations of any horizontal or vertical asymptotes for $0 degrees less than or equal to θ less than or equal to 360 degrees$ .

$180 degrees$
No
$90 degrees, 270 degrees$ , and their coterminal angles
The graph has vertical asymptotes at $θ equals 90 degrees$ and $θ equals 270 degrees$

Period, Midline and Amplitude

All sine and cosine graphs have the characteristic "wave" shape we've seen in previous examples. But we can alter the size and frequency of the waves by changing the formula for the function. In the next example we consider three variations of the sine function.

Example 4

Make a table of values and sketch a graph for each of the functions. How does each differ from the graph of $y = \sin (θ)$ ?

$y equals 3 sin ⁡ open parenthesis θ close parenthesis$
$y = 3 + \sin (θ)$
$y equals sin ⁡ open parenthesis 3 θ close parenthesis$

We make a table with multiples of

45 degrees

$θ$	$0 degrees$	$45 degrees$	$90 degrees$	$135 degrees$	$180 degrees$	$225 degrees$	$270 degrees$	$315 degrees$	$360 degrees$
$y equals 3 sin ⁡ open parenthesis θ close parenthesis$	$0$	$2.1$	$3$	$2.1$	$0$	$minus 2.1$	$minus 3$	$minus 2.1$	$0$

We plot the points, and connect them with a sine-shaped wave. Compare the graph, shown at right, to the graph of $y = \sin (θ)$ . The graph is like a sine graph, except that it oscillates between a maximum value of $3$ and a minimum value of $minus 3$ . The amplitude of this function is $3$ .

Again, we make a table of values with multiples of

45 degrees

and plot the points.

$θ$	$0 degrees$	$45 degrees$	$90 degrees$	$135 degrees$	$180 degrees$	$225 degrees$	$270 degrees$	$315 degrees$	$360 degrees$
$y = 3 + \sin (θ)$	$3$	$3.7$	$4$	$3.7$	$3$	$2.3$	$2$	$2.1$	$3$

This graph has the same amplitude as $y = \sin (θ)$ , but the entire graph is shifted up by $3$ units, as shown at right. The midline of this function is the line $y equals 3$ .

This time we'll make a table with multiples of

15 degrees

$θ$	$15 degrees$	$30 degrees$	$45 degrees$	$60 degrees$	$75 degrees$	$90 degrees$	$105 degrees$	$120 degrees$	$135 degrees$	$150 degrees$	$165 degrees$	$180 degrees$
$y equals sin ⁡ open parenthesis 3 θ close parenthesis$	$0.7$	$1$	$0.7$	$0$	$minus 0.7$	$minus 1$	$minus 0.7$	$0$	$0.7$	$1$	$0.7$	$0$

You can continue the table for $θ$ between $180 degrees$ and $360 degrees$ , and plot the points to find the graph shown at right. The graph has the same amplitude and midline as $y = \sin (θ)$ , but it completes three cycles from $0 degrees$ to $360 degrees$ instead of one cycle. The period of this graph is one-third of $360 degrees$ , or $120 degrees$ .

The graphs in the previous example illustrate a general rule about sine and cosine graphs.

Note

Amplitude, Period, and Midline

The graph of
$y = A \cos (θ) or y = A \sin (θ)$
has amplitude $| A |$ .
The graph of
$y = \cos (B θ) or y = \sin (B θ)$
has period $\frac{360^{\circ}}{| B |}$ .
The graph of
$y = k + \cos (θ) or y = k + \sin (θ)$
has midline $y = k$ .

Sketch a graph for each of the following functions. Describe how each is different from the graph of $y = \cos θ$ .

$h open parenthesis θ close parenthesis equals 2 cos ⁡ open parenthesis θ close parenthesis$
$g open parenthesis θ close parenthesis equals cos ⁡ open parenthesis 2 θ close parenthesis$
$f (θ) = 2 + \cos (θ)$

The amplitude is 2.
The period is $180 degrees$ .
The midline is $y equals 2$ .

The quantities $A, B,$ and $k$ in the equations above are called parameters, and their values for a particular function give us information about its graph.

State the period, midline, and amplitude of the graph of

$y = 1 - 3 \sin (2 θ)$

and graph the function.

Amplitude $3$ , period $180 degrees$ , midline $y equals 1$

Algebra Refresher

Graph the function.
Give the coordinates of any intercepts, and any maximum or minimum values.

$f (x) = - 6 + \frac{2}{3} x$
$g (x) = 4 - \frac{3}{2} x$
$p (t) = t^{2} - 4$
$q (t) = 9 - t^{2}$
$F (z) = 2 - \sqrt{z}$
$G (z) = \sqrt{4 - z}$

$\underset{―}{}$

Algebra Refresher Answers

2. $open parenthesis 0, minus 6 close parenthesis, open parenthesis 9, 0 close parenthesis$
2. $open parenthesis 0, 4 close parenthesis$ , $(\frac{8}{3}, 0)$
2. $(0, - 4), (- 2, 0), (2, 0), Min: - 4$
2. $open parenthesis 0, 9 close parenthesis, open parenthesis minus 3, 0 close parenthesis, open parenthesis 3, 0 close parenthesis, Max: 9$
2. $open parenthesis 0, 2 close parenthesis, open parenthesis 4, 0 close parenthesis, Max: 2$
2. $open parenthesis 0, 2 close parenthesis, open parenthesis 4, 0 close parenthesis, Min: 0$

Section 4.2 Summary

Vocabulary

Input variable
Output variable
Periodic function
Period
Midline
Amplitude
Asymptote

Concepts

We use the notation $y = f (x)$ to indicate that $y$ is a function of $x$ , that is, $x$ is the input variable and $y$ is the output variable.
Periodic functions are used to model phenomena that exhibit cyclical behavior.
The trigonometric ratios $\sin (θ)$ and $\cos (θ)$ are functions of the angle $θ$ .
The period of the sine function is $360 degrees$ . Its midline is the horizontal line $y equals 0$ , and the amplitude of the sine function is 1.
The graph of the cosine function has the same period, midline, and amplitude as the graph of the sine function. However, the locations of the intercepts and of the maximum and minimum values are different.
The tangent function has period $180 degrees$ . It is undefined at odd multiples of $90 degrees$ , and is increasing on each interval of its domain.
Note
Amplitude, Period, and Midline
1. The graph of
  $y = A \cos (θ) or y = A \sin (θ)$
  has amplitude $| A |$ .
2. The graph of
  $y = \cos (B θ) or y = \sin 9$
  has period $\frac{360^{\circ}}{| B |}$ .
3. The graph of
  $y = k + \cos (θ) or y = k + \sin (θ)$
  has midline $y = k$ .

Study Questions

Use the figure to help you fill in the blanks.
1. As $θ$ increases from $0 degrees$ to $90 degrees$ , $f (θ) = \sin (θ)$ ______ from ____ to ____.
2. As $θ$ increases from $90 degrees$ to $180 degrees$ , $f (θ) = \sin (θ)$ ______ from ____ to ____.
3. As $θ$ increases from $180 degrees$ to $270 degrees$ , $f (θ) = \sin (θ)$ ______ from ____ to ____.
4. As $θ$ increases from $270 degrees$ to $360 degrees$ , $f (θ) = \sin (θ)$ ______ from ____ to ____.
Use the figure to help you fill in the blanks.
1. As $θ$ increases from $0 degrees$ to $90 degrees$ , $f (θ) = \cos (θ)$ ______ from ____ to ____.
2. As $θ$ increases from $90 degrees$ to $180 degrees$ , $f (θ) = \cos (θ)$ ______ from ____ to ____.
3. As $θ$ increases from $180 degrees$ to $270 degrees$ , $f (θ) = \cos (θ)$ ______ from ____ to ____.
4. As $θ$ increases from $270 degrees$ to $360 degrees$ , $f (θ) = \cos (θ)$ ______ from ____ to ____.
List several ways in which the graph of $y = \tan θ$ is different from the graphs of $y = \sin (θ)$ and $y = \cos (θ)$ .
State the period, midline, and amplitude of the graph of $H = 5 + 0.2 \cos (3 α)$ .

Skills

Sketch graphs of the sine and cosine functions #1-4, 9-10, 19-22
Find the coordinates of points on a sine or cosine graph #5-8, 37-42
Use function notation #11-18
Graph the tangent function #23-24
Write an equation for a sine or cosine graph #25-30, 49-66
Graph a sine or cosine function and state the period, midline, and amplitude #31-36, 43-48

Homework 4.2

Prepare a graph with the horizontal axis scaled from $0 degrees$ to $360 degrees$ in multiples of $45 degrees$ .
Sketch a graph of $f (θ) = \sin (θ)$ by plotting points for multiples of $45 degrees$ .

Prepare a graph with the horizontal axis scaled from $0 degrees$ to $360 degrees$ in multiples of $45 degrees$ .
Sketch a graph of $f (θ) = \cos (θ)$ by plotting points for multiples of $45 degrees$ .

Prepare a graph with the horizontal axis scaled from $0 degrees$ to $360 degrees$ in multiples of $30 degrees$ .
Sketch a graph of $f (θ) = \cos (θ)$ by plotting points for multiples of $30 degrees$ .

Prepare a graph with the horizontal axis scaled from $0 degrees$ to $360 degrees$ in multiples of $30 degrees$ .
Sketch a graph of $f (θ) = \sin (θ)$ by plotting points for multiples of $30 degrees$ .

For Problems 5–8, give the coordinates of each point on the graph of $f (θ) = \sin (θ)$ or $f (θ) = \cos (θ)$ .

$(- 225^{\circ}, \frac{1}{\sqrt{2}})$
$(- 135^{\circ}, \frac{- 1}{\sqrt{2}})$
$(- 90^{\circ}, - 1)$
$(45^{\circ}, \frac{1}{\sqrt{2}})$
$open parenthesis 180 degrees, 0 close parenthesis$
$(315^{\circ}, \frac{- 1}{\sqrt{2}})$

$(- 240^{\circ}, \frac{- 1}{2})$
$(- 210^{\circ}, \frac{- \sqrt{3}}{2})$
$(- 60^{\circ}, \frac{- 1}{2})$
$(30^{\circ}, \frac{\sqrt{3}}{2})$
$(120^{\circ}, \frac{- 1}{2})$
$open parenthesis 270 degrees, 0 close parenthesis$

Make a short table of values like the one shown, and sketch the function by hand. Be sure to label the $x$ -axis and $y$ -axis appropriately.

$θ$	$0 degrees$	$90 degrees$	$180 degrees$	$270 degrees$	$360 degrees$
$f (θ)$

$f (θ) = \sin (θ)$
$f (θ) = \cos (θ)$

$θ$	$0 degrees$	$90 degrees$	$180 degrees$	$270 degrees$	$360 degrees$
$f (θ)$	$0$	$1$	$0$	$minus 1$	$0$

$θ$	$0 degrees$	$90 degrees$	$180 degrees$	$270 degrees$	$360 degrees$
$f (θ)$	$1$	$0$	$minus 1$	$0$	$1$

One of these graphs is $y = A \sin (k θ)$ , and the other is $y = A \cos (k θ)$ . Explain how you know which is which.

For Problems 11–18, evaluate the expression for $f (θ) = \sin (θ)$ and $g (θ) = \cos (θ)$ .

$3 plus f open parenthesis 30 degrees close parenthesis$

$\frac{7}{2}$

$3 f open parenthesis 30 degrees close parenthesis$

$4 g open parenthesis 225 degrees close parenthesis minus 1$

$- 2 \sqrt{2} - 1$

$- 4 + 2 g (225^{\circ})$

$minus 2 f open parenthesis 3 θ close parenthesis$ , for $θ equals 90 degrees$

$2$

$6 f (\frac{θ}{2})$ , for $θ equals 90 degrees$

$8 - 5 g (\frac{θ}{3})$ , for $θ equals 360 degrees$

$\frac{21}{2}$

$1 minus 4 g open parenthesis 4 θ close parenthesis$ , for $θ equals 135 degrees$

The graph shows your height as a function of angle as you ride the Ferris wheel. For each location $A$ – $E$ on the Ferris wheel, mark the corresponding point on the graph.

The graph shows your height as a function of angle as you ride the Ferris wheel. For each location $F$ – $J$ on the graph, mark the corresponding point on the Ferris wheel.

The graph shows the horizontal displacement of your foot from the center of the chain gear as you pedal a bicycle. For each location $K$ – $O$ on the chain gear, mark the corresponding point on the graph.

The graph shows the horizontal displacement of your foot from the center of the chain gear as you pedal a bicycle. For each location $P$ – $T$ on the graph, mark the corresponding point on the chain gear.

Fill in the table for values of

\tan (θ)

. Round your answers to three decimal places.

$θ$	$81 degrees$	$82 degrees$	$83 degrees$	$84 degrees$	$85 degrees$	$86 degrees$	$87 degrees$	$88 degrees$	$89 degrees$
$\tan (θ)$

What happens to $\tan (θ)$ as $θ$ increases toward $90 degrees$ ?

Fill in the table for values of

\tan (θ)

. Round your answers to three decimal places.

$θ$	$99 degrees$	$98 degrees$	$97 degrees$	$96 degrees$	$95 degrees$	$94 degrees$	$93 degrees$	$92 degrees$	$91 degrees$
$\tan (θ)$

What happens to $\tan (θ)$ as $θ$ decreases toward $90 degrees$ ?
What value does your calculator give for $tan ⁡ open parenthesis 90 degrees close parenthesis$ ? Why?

$θ$	$81 degrees$	$82 degrees$	$83 degrees$	$84 degrees$	$85 degrees$	$86 degrees$	$87 degrees$	$88 degrees$	$89 degrees$
$\tan (θ)$	$6.314$	$7.115$	$8.144$	$9.514$	$11.43$	$14.301$	$19.081$	$28.636$	$57.29$

$tan ⁡ open parenthesis θ close parenthesis approaches ∞$

$θ$	$99 degrees$	$98 degrees$	$97 degrees$	$96 degrees$	$95 degrees$	$94 degrees$	$93 degrees$	$92 degrees$	$91 degrees$
$\tan (θ)$	$minus 6.314$	$minus 7.115$	$minus 8.144$	$minus 9.514$	$minus 11.43$	$minus 14.301$	$minus 19.081$	$minus 28.636$	$minus 57.29$

$tan ⁡ open parenthesis θ close parenthesis approaches minus ∞$
The calculator gives an error message because $tan ⁡ open parenthesis 90 degrees close parenthesis$ is undefined.

Fill in the table with exact values of

\tan (θ)

. Then give decimal approximations to two places.

$θ$	$0 degrees$	$30 degrees$	$45 degrees$	$60 degrees$	$90 degrees$	$120 degrees$	$135 degrees$	$150 degrees$	$180 degrees$
$\tan (θ)$ (exact)
$\tan (θ)$ (approx.)

Fill in the table with exact values of

\tan (θ)

. Then give decimal approximations to two places.

$θ$	$180 degrees$	$210 degrees$	$225 degrees$	$240 degrees$	$270 degrees$	$300 degrees$	$315 degrees$	$330 degrees$	$360 degrees$
$\tan (θ)$ (exact)
$\tan (θ)$ (approx.)

Plot the points from the tables and sketch a graph of $f (θ) = \tan (θ)$ .

Write an equation for a sine function with amplitude 6.

$y equals 6 sin ⁡ open parenthesis θ close parenthesis$

Write an equation for a cosine function with amplitude $\frac{1}{2}$ .

Write an equation for a cosine function with midline $minus 5$ .

$y = \cos (θ) - 5$

Write an equation for a sine function with midline 2.

Write an equation for a sine function with period $90 degrees$ .

$y equals sin ⁡ open parenthesis 4 θ close parenthesis$

Write an equation for a cosine function with period $720 degrees$ .

For Problems 31–36,

Graph the function.
State the amplitude, period and midline of the function.

$y equals 3 cos ⁡ open parenthesis θ close parenthesis$

$y equals 4 sin ⁡ open parenthesis θ close parenthesis$

$y = 3 + \sin (θ)$

$y = - 2 + \cos (θ)$

$y equals cos ⁡ open parenthesis 3 θ close parenthesis$

$y equals sin ⁡ open parenthesis 2 θ close parenthesis$

For Problems 37–42, give the coordinates of the points on the graph.

$f (θ) = - 3 \cos (θ)$

$A open parenthesis 0 degrees, minus 3 close parenthesis$ , $B (135^{\circ}, \frac{3}{\sqrt{2}})$ , $C (300^{\circ}, \frac{- 3}{2})$

$f (θ) = - 4 \sin (θ)$

$f open parenthesis θ close parenthesis equals sin ⁡ open parenthesis 4 θ close parenthesis$

$P open parenthesis 112.5 degrees, 1 close parenthesis$ , $Q open parenthesis 180 degrees, 0 close parenthesis$ , $R open parenthesis 337.5 degrees, minus 1 close parenthesis$

$f open parenthesis θ close parenthesis equals cos ⁡ open parenthesis 3 θ close parenthesis$

$f (θ) = - 3 + \cos (θ)$

$X (45^{\circ}, - 3 + \frac{1}{\sqrt{2}})$ , $Y open parenthesis 90 degrees, minus 3 close parenthesis$ , $Z open parenthesis 300 degrees, minus 2 close parenthesis$

$f (θ) = 1 + \sin (θ)$

For Problems 43–48, graph the function using technology. State the amplitude, period, and midline.

$y = 3 + 4 \cos (θ)$

amp $equals 4$ , period $equals 360 degrees$ , midline: $y equals 3$

$y = - 4 + 3 \sin (θ)$

$y equals 5 sin ⁡ open parenthesis 2 θ close parenthesis$

amp $equals 5$ , period $equals 180 degrees$ , midline: $y equals 0$

$y equals 6 cos ⁡ open parenthesis 4 θ close parenthesis$

$f (θ) = - 4 + 3 \sin (3 θ)$

amp $equals 3$ , period $equals 120 degrees$ , midline: $y = - 4$

$f (θ) = 2 + 4 \cos (3 θ)$

For Problems 49–56,

State the amplitude, period, and midline for the graph.
Write an equation for the graph using sine or cosine.

amp $equals 1$ , period $equals 90 degrees$ , midline: $y equals 0$
$y equals sin ⁡ open parenthesis 4 θ close parenthesis$

amp $equals 1$ , period $equals 360 degrees$ , midline: $y equals 3$
$y = 3 + \cos (θ)$

amp $equals 4$ , period $equals 360 degrees$ , midline: $y = - 2$
$y = - 2 + 4 \sin (θ)$

amp $equals 2$ , period $equals 120 degrees$ , midline: $y equals 2$
$y = 2 + 2 \cos (3 θ)$

For Problems 57–62, write the equation of a sine or cosine function with the given properties.

Midline $y = - 4$ , amplitude $6$ , period $120 degrees$

$y = - 4 + 6 \sin (3 θ)$ (Answers vary)

Midline $y equals 5$ , amplitude $\frac{3}{2}$ , period $180 degrees$

Maximum points at $open parenthesis 0 degrees, 5 close parenthesis$ and $open parenthesis 360 degrees, 5 close parenthesis$ , minimum point at $open parenthesis 180 degrees, 1 close parenthesis$

$y = 3 + 2 \cos (θ)$ (Answers vary)

Maximum point at $open parenthesis 90 degrees, 1 close parenthesis$ , minimum point at $open parenthesis 270 degrees, minus 3 close parenthesis$

Horizontal intercepts at $45 degrees$ and $135 degrees$ , vertical intercept at $open parenthesis 0 degrees, 12 close parenthesis$

$y equals 12 cos ⁡ open parenthesis 2 θ close parenthesis$ (Answers vary)

Horizontal intercepts at $30 degrees$ and $90 degrees$ , vertical intercept at $open parenthesis 0 degrees, minus 8 close parenthesis$

For Problems 63–66, the table describes a sine or cosine function. Find an equation for the function.

$θ$	$0 degrees$	$45 degrees$	$90 degrees$	$135 degrees$	$180 degrees$	$225 degrees$	$270 degrees$	$315 degrees$	$360 degrees$
$f (θ)$	$7$	$5.56$	$2$	$minus 1.54$	$minus 3$	$minus 1.54$	$2$	$5.54$	$7$

$y = 2 + 5 \cos (θ)$

$θ$	$0 degrees$	$45 degrees$	$90 degrees$	$135 degrees$	$180 degrees$	$225 degrees$	$270 degrees$	$315 degrees$	$360 degrees$
$f (θ)$	$1$	$3.12$	$4$	$3.12$	$1$	$minus 1.12$	$minus 2$	$minus 1.12$	$1$

$θ$	$0 degrees$	$45 degrees$	$90 degrees$	$135 degrees$	$180 degrees$	$225 degrees$	$270 degrees$	$315 degrees$	$360 degrees$
$f (θ)$	$0$	$minus 2.83$	$minus 4$	$minus 2.83$	$0$	$2.83$	$4$	$2.83$	$0$

$y = - 4 \sin (θ)$

$θ$	$0 degrees$	$45 degrees$	$90 degrees$	$135 degrees$	$180 degrees$	$225 degrees$	$270 degrees$	$315 degrees$	$360 degrees$
$f (θ)$	$minus 9$	$minus 6.36$	$0$	$6.36$	$9$	$6.36$	$0$	$minus 6.36$	$minus 9$

Trigonometry by Katherine Yoshiwara (yoshiwarabooks.org), GNU Free Documentation License 1.2 or later. Adapted for the XYZ HTML edition with the authors' permission (recorded 2026-07-04). License: GFDL-1.2-or-later.