What is how to find reference angle?

In trigonometry, a reference angle is the acute angle formed between the terminal side of an angle in standard position and the x-axis. Reference angles help us find trigonometric values for angles beyond the range of 0 to 90 degrees (or 0 to π/2 radians). Here's how to find them:

• Quadrant I (0° < θ < 90° or 0 < θ < π/2): The reference angle is simply the angle itself.
  • Reference Angle = θ
• Quadrant II (90° < θ < 180° or π/2 < θ < π): The reference angle is found by subtracting the angle from 180° (or π radians).
  • Reference Angle = 180° - θ (degrees)
  • Reference Angle = π - θ (radians)
• Quadrant III (180° < θ < 270° or π < θ < 3π/2): The reference angle is found by subtracting 180° (or π radians) from the angle.
  • Reference Angle = θ - 180° (degrees)
  • Reference Angle = θ - π (radians)
• Quadrant IV (270° < θ < 360° or 3π/2 < θ < 2π): The reference angle is found by subtracting the angle from 360° (or 2π radians).
  • Reference Angle = 360° - θ (degrees)
  • Reference Angle = 2π - θ (radians)

For angles greater than 360° (or 2π radians) or less than 0°:

1. Find a coterminal angle between 0° and 360° (or 0 and 2π radians) by adding or subtracting multiples of 360° (or 2π radians).
2. Then, determine the quadrant of the coterminal angle and use the appropriate formula above.

Important Considerations:

• The reference angle is always positive and acute (between 0° and 90° or 0 and π/2 radians).
• The trigonometric functions of an angle and its reference angle have the same absolute value. The sign depends on the quadrant of the original angle (using the ASTC rule: All Students Take Calculus, which indicates which trigonometric functions are positive in each quadrant).