The Compound Angle Formulae

We know how to find the sine, cosine, and tangent of certain individual angles, such as $30^\circ$ , $45^\circ$ , and $60^\circ$ . But what if we need to find the sine of $75^\circ$ ?

The compound angle formulae are a set of equations that we can use to calculate this using the values we already know. This formula works because $75^\circ$ can be written as the sum of $30^\circ$ and $45^\circ$ , angles for which we know these trigonometric values.

We can use the same approach to then calculate the sine, cosine and tangent for any angle, if we can write it as the sum of two angles that we already have these values for. The compound angle formulae are written as:

\begin{aligned} \sin{\left(x+y\right)} &= \sin{\left(x\right)}\cos{\left(y\right)} + \cos{\left(x\right)}\sin{\left(y\right)} \\ \cos{\left(x+y\right)} &= \cos{\left(x\right)}\cos{\left(y\right)} - \sin{\left(x\right)}\sin{\left(y\right)} \end{aligned}

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