# Formulas

## Trig Identities # Cofunction and Reduction Identities

• Angles (arguments of functions): $$\alpha$$, $$\beta$$
Trigonometric functions: $$\sin \alpha,$$ $$\cos \alpha,$$ $$\tan \alpha,$$ $$\cot \alpha$$
1. Cofunction and reduction identities allow to transform trigonometric functions of the angles of the form $$90^\circ \pm \alpha,$$ $$180^\circ \pm \alpha,$$ $$270^\circ \pm \alpha$$ or $$360^\circ \pm \alpha$$ to trigonometric functions of the elementary angle $$\alpha$$. For example, the following formulas are known cofunction identities:
$$\cos {(90^\circ – \alpha)} = \sin \alpha,$$ $$\sin {(90^\circ – \alpha)} = \cos \alpha$$.
2. Table of cofunction and reduction identities
The angle $$\beta$$ denotes an initial compound angle that includes the elementary angle $$\alpha$$. Using these identities, it is possible to pass from the angle $$\beta$$ to $$\alpha$$.
3. The cofunction and reduction identities can be easily memorized using the following simple rules:
−  If the initial compound angle contains the angles $$180^\circ$$ or $$360^\circ$$, the function name does not change (reduction formulas). If the initial angle includes the angles $$90^\circ$$ or $$270^\circ$$, then the function changes to its cofunction, i.e. the sine changes to cosine, tangent to cotangent and vice-versa (cofunction formulas).
−  The sign of the right side must correspond to the sign of the function in the left side assuming that the angle $$\alpha$$ is acute.