Question

find the measure of the reference angle for a 580° angle.

Explanation:

Step1: Reduce the angle to 0 - 360°

To find the coterminal angle within 0° to 360°, we subtract 360° from 580° until the angle is in this range.
$580^{\circ}-360^{\circ} = 220^{\circ}$

Step2: Determine the quadrant of 220°

220° is in the third quadrant (180° < 220° < 270°). For angles in the third quadrant, the reference angle is calculated as the angle minus 180°.

Step3: Calculate the reference angle

Reference angle = $220^{\circ}-180^{\circ}=40^{\circ}$

Answer:

$40$