Concept Check Match each angle in Column I with its reference angle in Column II. Choices may be used once, more than once, or not at all. See Example 1. I. II. 5. A. 45° 6. B. 60° 7. -135° C. 82° 8. D. 30° 9. E. 38° 10. F. 32°

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. It is always between 0° and 90°.
2. For negative angles, first find the positive coterminal angle by adding 360° (i.e., \( \theta_{positive} = \theta_{negative} + 360^\circ \)).
3. Determine the quadrant of the positive angle to find the reference angle using these rules: - Quadrant I: reference angle = angle itself - Quadrant II: reference angle = 180° - angle - Quadrant III: reference angle = angle - 180° - Quadrant IV: reference angle = 360° - angle
4. Apply the above to the angle \(-135^\circ\): add 360° to get \(225^\circ\), which lies in Quadrant III, so reference angle = \(225^\circ - 180^\circ = 45^\circ\).
5. Match the calculated reference angle with the options in Column II, and repeat the process for other angles if given.

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.

Angle Measurement and Quadrants

Angles can be positive or negative and are measured from the positive x-axis, moving counterclockwise for positive angles and clockwise for negative ones. Understanding which quadrant an angle lies in helps determine the reference angle and the sign of trigonometric functions.

Calculating Reference Angles for Negative Angles

For negative angles, the reference angle is found by adding the angle to 360° (or 2π radians) to find its positive coterminal angle, then determining the acute angle between this coterminal angle and the x-axis. This process ensures correct matching with the reference angle.