Coterminal Angle Calculator

Two angles are considered coterminal when they are drawn in standard position and share the same terminal side. Because a full rotation around a circle equals 360 degrees or 2π radians, you can find coterminal angles by repeatedly adding or subtracting full rotations. Mathematically, for any angle θ in degrees, its coterminal angles are expressed as θ + 360° × k, where k is any integer. Similarly, in radians, the formula is θ + 2π × k. To find a standard coterminal angle between 0° and 360° (or 0 and 2π), we apply the modulo operation to reduce the angle within a single rotation.

Let's calculate the standard coterminal angle of 1000° that falls between 0° and 360°. First, determine how many full 360° rotations fit into 1000° by dividing 1000 by 360, which gives approximately 2.77 full circles. Take the integer part, 2, and multiply it by 360° to find the total degrees in those rotations: 2 × 360° = 720°. Next, subtract this value from the original angle: 1000° - 720° = 280°. Thus, 280° is the unique positive coterminal angle of 1000° within the standard circle.

Always verify whether your problem requires answers in degrees or radians before starting calculations to avoid conversion mistakes. When dealing with large negative angles, add 360° (or 2π radians) repeatedly until the resulting value falls within the standard 0° to 360° range. Keep fractional multiples of pi in exact form rather than converting to decimals too early to maintain higher mathematical precision.