Trigonometric Functions of Multiple Angles

Multiple and submultiple angles

Multiple angles. If $\theta$ is an angle, then $2\theta,3\theta,4\theta,\ldots$ (integer multiples of

$\theta$) are called multiple angles.

Submultiple angles. Conversely, $\theta$ itself can be viewed as a multiple of a smaller angle, e.g. $\theta =2\times\dfrac{\theta}{2}$; here $\dfrac{\theta}{2},\dfrac{\theta}{3},\ldots$ are called submultiple angles of

$\theta$.

Every formula for a multiple angle is derived by writing it as a SUM (e.g. $2\theta=\theta+\theta$,

$3\theta=2\theta+\theta$) and applying the compound-angle formulas of section 3.1. Reading the same formulas …