At which angles are cotangent and cosecant undefined?

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, the cotangent and cosecant functions are undefined for angles whose terminal side falls on the x-axis of a rectangular coordinate system. Because these functions involve division by zero at these positions, they do not have defined values at these specific angles.1

In terms of rotation, these functions are undefined at angles of $0^{\circ}$ , $180^{\circ}$ , and any integer multiple of $180^{\circ}$ (or $0$ , $π$ , and any integer multiple of $π$ radians). At these points, the terminal side of the angle lies along the x-axis, resulting in a coordinate where the y-value is zero, which makes the ratios defining cotangent and cosecant mathematically undefined.1