In Exercises 35–60, find the reference angle for each angle.
-150°

Verified step by step guidance

1. Identify the quadrant in which the angle -150° lies. Since the angle is negative, measure it clockwise from the positive x-axis. -150° corresponds to rotating 150° clockwise, which places the terminal side in the third quadrant.
2. Recall that the reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. For angles in the third quadrant, the reference angle is found by subtracting 180° from the angle if the angle is positive, or by subtracting the absolute value of the angle from 180° if the angle is negative.
3. Calculate the reference angle using the formula for negative angles in the third quadrant: Reference angle = 180° - |angle|. Substitute the given angle: Reference angle = 180° - 150°.
4. Simplify the expression to find the measure of the reference angle in degrees, which will be an acute angle between 0° and 90°.
5. Express the reference angle as a positive acute angle, which represents the smallest angle between the terminal side of the given angle and the x-axis.

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 its acute counterpart.

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. Understanding this helps determine the quadrant in which the terminal side lies, which is essential for finding the correct reference angle.

Quadrants and Angle Measures

The coordinate plane is divided into four quadrants, each corresponding to a range of angle measures. Knowing the quadrant of an angle (e.g., -150° lies in the third quadrant when converted to positive measure) helps in calculating the reference angle by subtracting from the nearest x-axis angle.