In Exercises 44–48, find the reference angle for each angle.
-410°

Verified step by step guidance

1. Understand that the reference angle is the acute angle formed between the terminal side of the given angle and the x-axis.
2. Since the given angle is negative (-410°), first find its positive coterminal angle by adding 360° repeatedly until the angle is between 0° and 360°: \(-410° + 360° = -50°\), then add 360° again: \(-50° + 360° = 310°\).
3. Now, with the positive coterminal angle 310°, determine which quadrant it lies in. Since 310° is between 270° and 360°, it lies in the fourth quadrant.
4. For angles in the fourth quadrant, the reference angle \(\theta_r\) is calculated as \(\theta_r = 360° - \theta\), where \(\theta\) is the positive coterminal angle. So, \(\theta_r = 360° - 310°\).
5. Calculate the difference to find the reference angle, which will be an acute angle between 0° and 90°.

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.

Coterminal Angles

Coterminal angles differ by full rotations of 360°. To find a coterminal angle within the standard 0° to 360° range, add or subtract multiples of 360°. This helps in simplifying angles like -410° by bringing them into a familiar range.

Quadrants and Angle Positioning

The position of an angle in the coordinate plane (quadrants I-IV) determines how to calculate its reference angle. Knowing which quadrant the angle's terminal side lies in guides the subtraction or addition needed to find the reference angle.