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Why are sine, tangent, cosecant, and cotangent called odd functions?

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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 mathematics, an odd function is defined by the property $f (- x) = - f (x)$ , meaning the sign of the function is reversed when the sign of the independent variable is reversed.1

In trigonometry, these functions are classified based on their symmetry. When an angle $A$ is placed in a standard position on a coordinate system with a point $Q$ having coordinates $(x, y)$ , the corresponding point for the negative angle $- A$ has coordinates $(x, - y)$ .2

Because the sine, tangent, cosecant, and cotangent functions depend on the $y$ -coordinate, negating the angle results in a negation of the function's value. Specifically, these functions satisfy the identity $f (- A) = - f (A)$ , which makes them symmetrical about the origin and classifies them as odd functions. In contrast, cosine and secant are considered even functions because they are symmetrical about the $y$ -axis, meaning their values remain unchanged when the sign of the angle is reversed.2