Coterminal Angles

The coterminal angles are the angles that have the same initial side and the same terminal sides. We determine the coterminal angle of a given angle by adding or subtracting 360° or 2π to it. In trigonometry, the coterminal angles have the same values for the functions of sin, cos, and tan.

Once you have understood the concept, you will differentiate between coterminal angles and reference angles, as well as be able to solve problems with the coterminal angles formula.

What Are Coterminal Angles?

Coterminal angles are the angles that have the same initial side and share the terminal sides. These angles occupy the standard position, though their values are different. They are on the same sides, in the same quadrant and their vertices are identical. When the angles are moved clockwise or anticlockwise the terminal sides coincide at the same angle. An angle is a measure of the rotation of a ray about its initial point. The original ray is called the initial side and the final position of the ray after its rotation is called the terminal side of that angle.

Consider 45°. Its standard position is in the first quadrant because its terminal side is also present in the first quadrant. Look at the image.

• On full rotation anticlockwise, 45° reaches its terminal side again at 405°. 405° coincides with 45° in the first quadrant.
• On full rotation clockwise, 45° reaches its terminal side again at -315°. -315° coincides with 45° in the first quadrant.

Thus 405° and -315° are coterminal angles of 45°.