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1.18: Reference Angles and Angles in the Unit Circle

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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?

At the end of this Concept, you'll know what reference angles are and be able to identify them in the pictures above.

Watch This

James Sousa: Determining Trig Function Values Using Reference Angles and Reference Triangles

Guidance

Consider the angle 150^\circ . 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 30^\circ , across the y- axis.

Notice that 150^\circ makes a 30^\circ angle with the negative x- axis. Therefore we say that 30^\circ is the reference angle for 150^\circ . Formally, the reference angle of an angle in standard position is the angle formed with the closest portion of the x- axis. Notice that 30^\circ is the reference angle for many angles. For example, it is the reference angle for 210^\circ and for -30^\circ .

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

Example A

Graph each angle and identify its reference angle.

a. 140^\circ

b. 240^\circ

c. 380^\circ

Solution:

a. 140^\circ makes a 40^\circ angle with the negative x- axis. Therefore the reference angle is 40^\circ .

b. 240^\circ makes a 60^\circ with the negative x- axis. Therefore the reference angle is 60^\circ .

c. 380^\circ is a full rotation of 360^\circ , plus an additional 20^\circ . So this angle is co-terminal with 20^\circ , and 20^\circ is its reference angle.

Example B

Find the ordered pair for 240^\circ and use it to find the value of \sin 240^\circ .

Solution: \sin 240^\circ = \frac{-\sqrt{3}}{2}

As we found in Example A, the reference angle for 240^\circ is 60^\circ . The figure below shows 60^\circ and the three other angles in the unit circle that have 60^\circ as a reference angle.

The terminal side of the angle 240^\circ represents a reflection of the terminal side of 60^\circ over both axes. So the coordinates of the point are \left( -\frac{1}{2}, -\frac{\sqrt{3}}{2} \right ) . The y- coordinate is the sine value, so \sin 240^\circ = -\frac{\sqrt{3}}{2} .

Just as the figure above shows 60^\circ and three related angles, we can make similar graphs for 30^\circ and 45^\circ .

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

Example C

Find the value of \cot 300^\circ

Solution: \cot 300^\circ =  -\frac{1}{\sqrt{3}}

Using the graph above, you will find that the ordered pair is \left ( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right ) . Therefore the cotangent value is \cot 300^\circ = \frac{x}{y} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{2} \times -\frac{2}{\sqrt{3}} = -\frac{1}{\sqrt{3}}

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.

Vocabulary

Reference Angle: A reference angle is the angle formed between the terminal side of an angle and the closest of either the positive or negative 'x' axis.

Guided Practice

1. Graph 210^\circ and identify its reference angle.

2. Graph 315^\circ and identify its reference angle.

3. Find the ordered pair for 150^\circ and use it to find the value of cos 150^\circ .

Solutions:

1. The graph of 210^\circ looks like this:

and since the angle makes a 30^\circ angle with the negative "x" axis, the reference angle is 30^\circ .

2. The graph of 315^\circ looks like this:

and since the angle makes a 45^\circ angle with the positive "x" axis, the reference angle is 45^\circ .

3. Since the reference angle is 30^\circ , we know that the coordinates for the point on the unit circle are \left( -\frac{\sqrt{3}}{2},\frac{1}{2} \right) . This is the same as the value for 30^\circ , except the "x" coordinate is negative instead of positive. Knowing this,

\cos 150^\circ = \frac{adjacent}{hypotenuse} = \frac{-\frac{\sqrt{3}}{2}}{1} = -\frac{\sqrt{3}}{2}

Concept Problem Solution

Since you know how to measure reference angles now, upon examination of the spinners, you know that the first angle is 30^\circ , the second angle is 45^\circ , and the third angle is 60^\circ .

Practice

  1. Graph 100^\circ and identify its reference angle.
  2. Graph 200^\circ and identify its reference angle.
  3. Graph 290^\circ and identify its reference angle.

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

  1. \sin 120^\circ
  2. \cos 120^\circ
  3. \csc 120^\circ
  4. \cos 135^\circ
  5. \sin 135^\circ
  6. \tan 135^\circ
  7. \sin 210^\circ
  8. \cos 210^\circ
  9. \cot 210^\circ
  10. \sin 225^\circ
  11. \cos 225^\circ
  12. \sec 225^\circ

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Date Created:

Sep 26, 2012

Last Modified:

May 27, 2014
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