In Exercises 44–48, find the reference angle for each angle.
265°

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Identify the quadrant in which the angle 265° lies. Since 265° is between 180° and 270°, it is in the third quadrant.

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 \( \theta_r \) is found by subtracting 180° from the given angle: \( \theta_r = \theta - 180^\circ \).

Substitute the given angle into the formula: \( \theta_r = 265^\circ - 180^\circ \).

Simplify the expression to find the reference angle.

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주요 개념

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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 an angle in the first quadrant.

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 angle is measured counterclockwise for positive angles and clockwise for negative angles, which helps determine the quadrant where the terminal side lies.

Quadrants and Angle Measurement

The coordinate plane is divided into four quadrants, each spanning 90°. Knowing the quadrant of an angle helps find the reference angle by subtracting the angle from the nearest x-axis boundary (0°, 90°, 180°, 270°, or 360°), depending on the quadrant.

In Exercises 44–48, find the reference angle for each angle.265°