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How do negative angles affect sine, cosine, and tangent?

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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 trigonometry, negative angles are generated by clockwise rotation, whereas positive angles are generated by counterclockwise rotation. When an angle $A$ is represented by a point $(x, y)$ on the terminal side in a standard coordinate system, the corresponding negative angle $- A$ is represented by the point $(x, - y)$ .12

This geometric relationship determines how trigonometric functions behave with negative inputs:

Cosine and Secant: These are even functions, meaning they are symmetrical about the $y$ -axis. Consequently, the sign of the angle does not change the function value: $\cos (- A) = \cos A$ and $\sec (- A) = \sec A$ .2
Sine, Tangent, Cosecant, and Cotangent: These are odd functions, meaning they are symmetrical about the origin. For these functions, the negative sign is preserved: $\sin (- A) = - \sin A$ and $\tan (- A) = - \tan A$ .23

In summary, the cosine and secant functions remain unchanged by a negative angle, while the sine, tangent, cosecant, and cotangent functions negate their output.2