Trig Identities

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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. Cofunction and reduction identities
  4. 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.