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Reference Angles and Calculations

Sep 9, 2025

Overview

This lecture explains how to find reference angles for both positive and negative angles in degrees and radians, including key formulas and example calculations.

Reference Angles in Degrees

  • The reference angle is the acute angle (0°–90°) between the terminal side of a given angle and the x-axis.
  • In Quadrant I, the reference angle is the angle itself.
  • In Quadrant II, reference angle = 180° – angle.
  • In Quadrant III, reference angle = angle – 180°.
  • In Quadrant IV, reference angle = 360° – angle.
  • For negative angles, add 360° to get the coterminal angle and then apply the relevant formula.

Example Calculations (Degrees)

  • Reference angle for 120° (Quadrant II): 180° – 120° = 60°.
  • Reference angle for 210° (Quadrant III): 210° – 180° = 30°.
  • Reference angle for 150° (Quadrant II): 180° – 150° = 30°.
  • Reference angle for 315° (Quadrant IV): 360° – 315° = 45°.
  • Reference angle for –150°: Coterminal angle = –150° + 360° = 210°; 210° – 180° = 30°.
  • Reference angle for –240°: Coterminal angle = –240° + 360° = 120°; 180° – 120° = 60°.

Reference Angles in Radians

  • For common unit circle angles, reference angle equals the denominator's base angle (e.g., 2Ï€/3, 4Ï€/3, 5Ï€/3 all have reference angle Ï€/3).
  • For uncommon angles, convert radians to degrees to identify quadrant and apply the formula.
  • After finding the reference angle in degrees, convert it back to radians.

Example Calculations (Radians)

  • Reference angle for 3Ï€/5: Convert to degrees → 108° (Quadrant II); 180° – 108° = 72°; Convert to radians: 2Ï€/5.
  • Reference angle for 9Ï€/8: 9Ï€/8 → 202.5° (Quadrant III); 202.5° – 180° = 22.5°; Convert to radians: Ï€/8.
  • Reference angle for –8Ï€/9: –8Ï€/9 → –160°; add 360° → 200° (Quadrant III); 200° – 180° = 20°; Convert to radians: Ï€/9.

Key Terms & Definitions

  • Reference Angle — The smallest angle between the terminal side of an angle and the x-axis, always positive and less than 90° (or Ï€/2 radians).
  • Coterminal Angle — An angle that shares the same terminal side as the given angle, found by adding or subtracting multiples of 360° (or 2Ï€ radians).

Action Items / Next Steps

  • Practice finding reference angles for positive and negative angles in both degrees and radians.
  • Work through additional textbook exercises on reference angles.
  • Memorize quadrant rules and conversion steps.

Full transcript