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Reference Angles and Angles in the Unit Circle

Formed between the terminal side of an angle and the closest part of the x-axis.

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Reference Angles and Angles in the Unit Circle

When you walk into math class one day, your teacher has a surprise for the class. You're going to play series of games related to the things you've been learning about in class. For the first game, your teacher hands each group of students a spinner with an "x" and "y" axis marked. The game is to see how many angles you identify correctly. However, in this game, you are supposed to give what is called the "reference angle". You spin your spinner three times. Each picture below shows one of the spins:

Can you correctly identify the reference angles for these pictures?

Reference Angles

Consider the angle . If we graph this angle in standard position, we see that the terminal side of this angle is a reflection of the terminal side of , across the axis.

Notice that makes a angle with the negative axis. Therefore we say that is the reference angle for . Formally, the reference angle of an angle in standard position is the angle formed with the closest portion of the axis. Notice that is the reference angle for many angles. For example, it is the reference angle for and for .

In general, identifying the reference angle for an angle will help you determine the values of the trig functions of the angle.

 

Identifying Reference Angles  

Graph each of the following angles and identify their reference angles.

a.

 makes a  angles with the negative axis. Therefore the reference angle is .

b.

 makes a  angle with the negative axis. Therefore the reference angle is 

c.

 is a full rotation of , plus an additional . So this angle is co-terminal with , and  is its reference angle. 

Determining the Value of Trigonometric Functions 

1. Find the ordered pair for and use it to find the value of .

As we found in part b under the question above, the reference angle for is . The figure below shows and the three other angles in the unit circle that have as a reference angle.

The terminal side of the angle represents a reflection of the terminal side of over both axes. So the coordinates of the point are . The coordinate is the sine value, so .

Just as the figure above shows and three related angles, we can make similar graphs for and .

Knowing these ordered pairs will help you find the value of any of the trig functions for these angles.

2. Find the value of

 

Using the graph above, you will find that the ordered pair is . Therefore the cotangent value is

We can also use the concept of a reference angle and the ordered pairs we have identified to determine the values of the trig functions for other angles.

Examples

Example 1

Earlier, you were asked if you can correctly identify the reference angles in the pictures. 

Since you know how to measure reference angles now, upon examination of the spinners, you know that the first angle is , the second angle is , and the third angle is .

Example 2

Graph and identify its reference angle.

The graph of looks like this:

and since the angle makes a angle with the negative "x" axis, the reference angle is .

Example 3

Graph and identify its reference angle.

The graph of looks like this:

and since the angle makes a angle with the positive "x" axis, the reference angle is .

Example 4

Find the ordered pair for and use it to find the value of cos .

 Since the reference angle is , we know that the coordinates for the point on the unit circle are . This is the same as the value for , except the "x" coordinate is negative instead of positive. Knowing this,

Review

  1. Graph and identify its reference angle.
  2. Graph and identify its reference angle.
  3. Graph and identify its reference angle.

Calculate each value using the unit circle and special right triangles.

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.18. 

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Vocabulary

Reference Angle

A reference angle is the angle formed between the terminal side of the angle and the closest of either the positive or negative x-axis.

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