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Chapter 4: Problem 109

Explain how reference angles are used to evaluate trigonometric functions. Give an example with your description.

Short Answer

Expert verified
Reference angles are used in trigonometry to simplify the process of evaluating trigonometric functions. They relate any angle to an acute angle in the first quadrant, making calculations easier. The functions of an angle and its reference angle have the same magnitude but may differ in sign. For instance, for a 130 degrees angle, its reference angle would be 40 degrees.

Step by step solution

01

Understanding Reference Angles

A reference angle is the acute version (less than 90 degrees) of any given angle located in the first quadrant of a coordinate plane. It is always related to the given angle by being its horizontal or vertical projection on the x or y-axis.
02

Applying Reference Angles to Trigonometric Functions

Reference angles are particularly useful when calculating trigonometric functions because the sine, cosine, and tangent of angles that differ by multiples of 180 degrees are the same. In other words, the sine, cosine, and tangent of an angle and its reference angle are equal in magnitude, but may differ in sign.
03

Example with Reference Angles

Consider an angle of 130 degrees. Its reference angle is \(90^\circ\) less, which is \(40^\circ\). The sine, cosine, and tangent of 130 degrees can be determined by finding the sine, cosine, and tangent of 40 degrees and adjusting the sign accordingly.

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