Topics
3A. Trigonometric and Polar Functions
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United States of America
AP Precalculus

3.4 Sine and cosine function graphs

Lesson
Practice
Lesson

Introduction

Learning objective

  • 3.4.A Construct representations of the sine and cosine functions using the unit circle.

Graphing the sine function

Recall the unit circle, which we can use to evaluate exact trigonometric ratios for \left( \cos \theta, \sin \theta \right):

The unit circle with the special angles labeled. Starting from the positive x axis then moving counterclockwise, the special angles are: 0, pi over 6, pi over 4, pi over 3, pi over 2, 2 pi over 3, 3 pi over 4, 5 pi over 6, pi, 7 pi over 6, 5 pi over 4, 4 pi over 3, 3 pi over 2, 5 pi over 3, 7 pi over 4, and 11 pi over 6. Speak to your teacher for more details.

The graph of trigonometric functions has the angles in the unit circle to represent the x-axis. For example:

Degree90\degree180\degree270\degree360\degree
Radian\dfrac{\pi}{2}\pi\dfrac{3\pi}{2}2\pi

The coordinates in the unit circle that ranges from -1 to 1 is used to represent the y-axis. For example:

Decimals-1-0.500.51
Fractions-1-\dfrac120\dfrac121

Exploration

Explore the applet by moving the slider.

Loading interactive...
  1. Move the slider to 45 \degree. How does the sine of this point on the unit circle relate to the point on the graphed curve?
  2. What do you notice about the relationship between the location of a point on the unit circle and the location of a point on the curve?
  3. When is the curve negative? How does this relate to the point on the unit circle?

Recall that sine is represented by the vertical leg of the right triangle positioned in the unit circle, or the y-coordinate.

Plotting the corresponding values of \sin \theta will result to the following graph.

A unit circle drawn on a four quadrant coordinate plane. A radius of the unit circle is also drawn in the first quadrant. The radius makes an angle theta from the positive x axis. A right triangle with the radius as the hypotenuse and a leg on the x axis is shown. The vertical leg of the triangle is labeled sine theta. The first and fourth quadrant of another coordinate system is shown with the origin located at point (1,0) of the first coordinate system. A sine function is graphed in the second coordinate system. Speak to your teacher for more details.

Notice that as we move through values of \theta, the graph of f \left( \theta \right)= \sin \theta will oscillate accordingly between -1 and 1.

Examples

Example 1

Consider the function f \left( \theta \right) = \sin \theta.

a

Complete the table with values in exact form:

\theta0\dfrac{\pi}{6}\dfrac{\pi}{2}\dfrac{5 \pi}{6}\pi\dfrac{7 \pi}{6}\dfrac{3 \pi}{2}\dfrac{11 \pi}{6}2 \pi
\sin \theta
Worked Solution
Create a strategy

Since the function f \left( \theta \right)= \sin \theta is represented by the y-coordinate on the unit circle, we can use the y-coordinates at each angle on the unit circle to complete the table of values.

Apply the idea
\theta0\dfrac{\pi}{6}\dfrac{\pi}{2}\dfrac{5 \pi}{6}\pi\dfrac{7 \pi}{6}\dfrac{3 \pi}{2}\dfrac{11 \pi}{6}2 \pi
\sin \theta0\dfrac{1}{2}1\dfrac{1}{2}0-\dfrac{1}{2}-1-\dfrac{1}{2}0
b

Sketch a graph for f \left( \theta \right) = \sin \theta on the domain [-2\pi, 2\pi].

Worked Solution
Create a strategy

Plot the points for [0, 2 \pi] using the table.

Then, we can work backwards along the unit circle to each negative value of \theta which is coterminal to a positive angle between 0 and 2 \pi and match the values of sine from the unit circle.

Apply the idea
-1π
1π
\theta
-1
1
f \left( \theta \right)

For the function from -2\pi to 0, \theta= \dfrac{-\pi}{2} is coterminal to \alpha= \dfrac{3 \pi}{2}, and since \sin \left( \dfrac{3\pi}{2} \right)=-1, \sin \left( \dfrac{-\pi}{2} \right) = -1 also. The following coterminal angles give the exact values:

\sin \left( -\pi \right) = \sin \left( \pi \right) = 0

\sin \left( \dfrac{-3 \pi}{2} \right) = \sin \left( \dfrac{\pi}{2} \right) = 1

\sin \left( -2 \pi \right) = \sin \left( 2 \pi \right) = 0

-1π
1π
\theta
-1
1
f \left( \theta \right)
c

State the sign of \sin \left( \dfrac{- \pi}{12} \right).

Worked Solution
Create a strategy

Determine which quadrant \dfrac{- \pi}{12} is located in on the coordinate plane, then determine the sign of the y-coordinate at that point.

Apply the idea

By splitting each semicircle of the unit circle into 12 equal parts, we can see that starting along the x-axis and moving to \dfrac{-\pi}{12}, the angle will be in the fourth quadrant.

The unit circle with each semi circle split into 12 equal parts. The 1/12 part below in the fourth quadrant and below the positive x axis is highlighted and labeled negative pi over 12.

Since the sign of the y-coordinate is negative in the fourth quadrant, \sin \left( \dfrac{- \pi}{12} \right) is negative.

Idea summary

The graph of f \left( \theta \right) = \sin \theta relates closely to the y-coordinate of points on the unit circle.

Graphing the cosine function

Exploration

Explore the applet by moving the slider.

Loading interactive...
  1. Move the slider to 45 \degree. How does the cosine of this point on the unit circle relate to the point on the curved graph?
  2. What do you notice about the relationship between the location of a point on the unit circle and the location of a point on the curve?
  3. When is the curve negative? How does this relate to the point on the unit circle?

Recall that cosine is represented by the horizontal leg of the right triangle positioned in the unit circle, or the x-coordinate.

Plotting the corresponding values of \cos \theta will result to the following graph.

A unit circle drawn on a four quadrant coordinate plane. A radius of the unit circle is also drawn in the first quadrant. The radius makes an angle theta from the positive x axis. A right triangle with the radius as the hypotenuse and a leg on the x axis is shown. The vertical leg of the triangle is labeled cosine theta. The first and fourth quadrant of another coordinate system is shown with the origin located at point (1,0) of the first coordinate system. A cosine function is graphed in the second coordinate system. Speak to your teacher for more details.

As we move through values of \theta, the graph of f \left( \theta \right) = \cos \theta will oscillate accordingly between -1 and 1.

Examples

Example 2

Consider the function f \left(\theta \right) = \cos \theta.

a

Complete the table with values in exact form:

\theta0\dfrac{\pi}{3}\dfrac{\pi}{2}\dfrac{2 \pi}{3}\pi\dfrac{4 \pi}{3}\dfrac{3 \pi}{2}\dfrac{5 \pi}{3}2 \pi
\cos \theta
Worked Solution
Create a strategy

Since the function f \left( \theta \right) = \cos \theta is represented by the x-coordinate on the unit circle, we can use the x-coordinates at each angle on the unit circle to complete the table of values.

Apply the idea
\theta0\dfrac{\pi}{3}\dfrac{\pi}{2}\dfrac{2 \pi}{3}\pi\dfrac{4 \pi}{3}\dfrac{3 \pi}{2}\dfrac{5 \pi}{3}2 \pi
\cos \theta1\dfrac{1}{2}0-\dfrac{1}{2}-1-\dfrac{1}{2}0\dfrac{1}{2}1
b

Sketch a graph for f \left( \theta \right) = \cos \theta on the domain [-2 \pi, 2 \pi].

Worked Solution
Create a strategy

Plot the points for [0, 2 \pi] using the table.

Then, we can work backwards along the unit circle to each negative value of \theta which is coterminal to a positive angle between 0 and 2 \pi and match the values of cosine from the unit circle.

Apply the idea
-1π
1π
\theta
-1
1
f \left( \theta \right)

For the function from -2\pi to 0, \theta= \dfrac{-\pi}{2} is coterminal to \alpha= \dfrac{3 \pi}{2}, and since \cos \left( \dfrac{3\pi}{2} \right)=0, \cos \left( \dfrac{-\pi}{2} \right) = 0 also. The following coterminal angles give the exact values:

\cos \left( -\pi \right) = \cos \left( \pi \right) = -1

\cos \left( \dfrac{-3 \pi}{2} \right) = \cos \left( \dfrac{\pi}{2} \right) = 0

\cos \left( -2 \pi \right) = \cos \left( 2 \pi \right) = 1

-1π
1π
\theta
-1
1
f \left( \theta \right)
c

State the sign of \cos \left( \dfrac{-\pi}{12} \right).

Worked Solution
Create a strategy

Using a revolution of the coordinate plane split into 24 equal parts, we can see that \dfrac{-\pi}{12} is in the fourth quadrant. We can use this to determine the sign of cosine at this angle.

The unit circle with each semi circle split into 12 equal parts. The 1/12 part below in the fourth quadrant and below the positive x axis is highlighted and labeled negative pi over 12.
Apply the idea

Since the sign of the x-coordinate is positive in the fourth quadrant, \cos \left( \dfrac{- \pi}{12} \right) is positive.

Idea summary

The graph of f \left( \theta \right) = \cos \theta relates closely to the x-coordinate of points on the unit circle.

Outcomes

3.4.A

Construct representations of the sine and cosine functions using the unit circle.

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