Reference Angle Calculator

How to Calculate Reference Angles

A reference angle is the acute angle (an angle between 0° and 90°) formed by the terminal side of an angle in standard position and the x-axis. It's a useful concept in trigonometry that simplifies many calculations.

Formula

The reference angle θ' for an angle θ can be calculated using the following formula:

\[ \theta' = \min(|\theta \bmod 360°|, |360° - (\theta \bmod 360°)|) \]

Where:

• θ' is the reference angle
• θ is the given angle
• mod is the modulo operation

Calculation Steps

1. Convert the given angle to radians if it's in degrees.
2. Take the absolute value of the angle and find its modulo with 2π (or 360° if working in degrees).
3. If the result is greater than π (or 180°), subtract it from 2π (or 360°).
4. If the result is greater than π/2 (or 90°), subtract it from π (or 180°).
5. The resulting angle is the reference angle.
6. Convert back to degrees if needed.

Example

Let's calculate the reference angle for 225°:

1. Given angle: θ = 225°
2. 225° mod 360° = 225°
3. 225° > 180°, so 360° - 225° = 135°
4. 135° > 90°, so 180° - 135° = 45°
5. Therefore, the reference angle is 45°

Visual Representation

This diagram illustrates the reference angle for 225°. The blue line represents the original angle of 225°, and the red line shows the reference angle of 45°.