Britannica Sources used in this answer

What are coterminal angles in trigonometry?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

Diagram of a right triangle labeled with sides a, b, and hypotenuse c, alongside formulas for the six trigonometric functions-sine, cosine, tangent, cosecant, secant, and cotangent-each defined by ratios of the triangle's sides. — trigonometry Based on the definitions, various simple relationships exist among the functions. For example, csc A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A, and tan A = sin A/cos A.

In trigonometry, coterminal angles are angles that share the same initial and terminal sides when positioned on a coordinate system, despite having different angular measures.12

Because these angles represent the same rotation of a line segment about a vertex, they differ only by an integral number of complete revolutions. Specifically, all angles coterminal with an angle $A$ can be expressed as:12

$A \pm 360^{\circ} n or A \pm 2 π n$

where $n$ is an integer representing the number of full rotations. For example, an angle of $30^{\circ}$ and an angle of $390^{\circ}$ are coterminal because they differ by exactly one $360^{\circ}$ revolution.12

A key property of these angles is that any trigonometric function will yield the same value for all coterminal angles. For instance, $\sin (A \pm 360^{\circ} n) = \sin A$ , which reflects the periodic nature of trigonometric functions.3