In Exercises 35–60, find the reference angle for each angle. 160°

Verified step by step guidance

1. Identify the quadrant in which the angle 160° lies. Since 160° is between 90° and 180°, it is in the second quadrant.
2. Recall that the reference angle is the acute angle formed between the terminal side of the given angle and the x-axis.
3. For angles in the second quadrant, the reference angle \( \theta_r \) is calculated by subtracting the angle from 180°: \( \theta_r = 180^\circ - \theta \).
4. Substitute the given angle into the formula: \( \theta_r = 180^\circ - 160^\circ \).
5. Simplify the expression to find the reference angle.

Key Concepts

Reference Angle

A reference angle is the acute angle formed between the terminal side of a given angle and the x-axis. It is always positive and less than or equal to 90°, used to simplify trigonometric calculations by relating any angle to a corresponding acute angle.

Standard Position of an Angle

An angle in standard position has its vertex at the origin and its initial side along the positive x-axis. The terminal side rotates counterclockwise for positive angles and clockwise for negative angles, determining the angle's quadrant and aiding in finding the reference angle.

Quadrants and Angle Measurement

The coordinate plane is divided into four quadrants, each spanning 90°. Knowing which quadrant an angle lies in helps determine how to calculate its reference angle, as the reference angle depends on the difference between the given angle and the nearest x-axis boundary.