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ZingPath: Angles and their Measures

The Reference Angle

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Angles and their Measures

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The Reference Angle


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Learners identify reference angles and use the properties to find trigonometric ratios.

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Now You Know

After completing this tutorial, you will be able to complete the following:

  • Define and identify reference angles.
  • Evaluate trigonometric ratios of an angle by using its reference angle.
  • Evaluate trigonometric ratios of an angle by using its reference angle, given a different trigonometric ratio of the same angle.

Everything You'll Have Covered

The reference angle is used to find trigonometric ratios of both obtuse and negative angles. If ? is an angle in standard position, the reference angle of ? is the positive acute angle between the terminal side of ? and the x-axis. The sine, cosine, and tangent have the same absolute value on both an angle and its reference angle.

To find the reference angle of a given angle ?, measure the angle formed by the terminal side of ? and the x-axis. For example, if , the reference angle is

The sign of the tangent, sine, or cosine of an angle measured with its reference angle is dependent on the quadrant of that angle.

, the reference angle is

The Cartesian coordinate plane is divided into four quadrants, numbered counterclockwise from I to IV. For example, the angle lies in the second quadrant. The following table records the sign of each trigonometric ratio as a function of quadrant.

Note, if one remembers the signs of the x and y coordinates of any point in a given quadrant, the signs of the tangent, sine and cosine should be easy to determine. Recall that the x-coordinate represents the cosine, the y-coordinate represents the sine and the ratio of the x-coordinate to the y-coordinate represents the tangent of the angle in a the unit circle.

To find the trigonometric ratios of an obtuse angle, find the reference angle, calculate the trigonometric ratios of the reference angle, and then determine the correct sign to the for the ratio based on the quadrant.

For example, let . This angle is located in the second quadrant and its reference angle is . Since , and the x-coordinates are negative in the second quadrant, we . Also, since the y-coordinates are positive, . The table above can be used for reference.

This Activity Object introduces a four-step process to calculate sine, cosine, and tangent by means of the reference angle.

Tutorial Details

Approximate Time 30 Minutes
Pre-requisite Concepts the concept of angle, find the measure of an angle in both degrees and radians, write angles in standard position, and use the unit circle.
Course Algebra-2
Type of Tutorial Skills Application
Key Vocabulary reference angles, properties, trigonometric rations