Coterminal Angle Calculator

How to Calculate Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position on a coordinate plane. They differ by multiples of 360° (or 2π radians). Understanding coterminal angles is crucial in trigonometry and circular functions.

Formula

The formula for finding coterminal angles is:

\[ \theta_{coterminal} = \theta \pm 360°n \]

Where:

• θ = The original angle
• n = Any integer

Calculation Steps

1. Ensure the angle is in degrees. If it's in radians, convert it to degrees using the formula: degrees = radians × (180° / π).
2. To find the positive coterminal angle between 0° and 360°:
   • Divide the angle by 360° and take the remainder.
   • If the remainder is negative, add 360° to get the positive equivalent.
3. To find the negative coterminal angle:
   • Subtract 360° from the positive coterminal angle found in step 2.

Example

Let's find the coterminal angles for 405°:

1. The angle is already in degrees, so no conversion is needed.
2. Find the positive coterminal angle: \[ 405° \div 360° = 1 \text{ remainder } 45° \] So, the positive coterminal angle is 45°.
3. Find the negative coterminal angle: \[ 45° - 360° = -315° \]

Therefore, the coterminal angles for 405° are 45° and -315°.

Visual Representation

This diagram illustrates the coterminal angles for 405°. The blue arc represents the original angle, while the green and red labels show the positive and negative coterminal angles, respectively.