The bisection formulae

From the duplication formulae we derive the bisection formulae, which express $\sin(\alpha/2)$ and $\cos(\alpha/2)$ in terms of $\cos\alpha$.

Property — Bisection formulae

\cos\!\left(\frac{\alpha}{2}\right) = \pm\sqrt{\frac{1+\cos\alpha}{2}}.$$ The sign "$\pm$" must be chosen case by case according to the quadrant in which $\alpha/2$ lies: if it is in the first or second quadrant, $\sin(\alpha/2)\ge 0$; if it is in the first or fourth, $\cos(\alpha/2)\ge 0$. For the tangent: $$\tan\!\left(\frac{\alpha}{2}\right) = \frac{1-\cos\alpha}{\sin\alpha} = \frac{\sin\alpha}{1+\cos\alpha},$$ equivalent forms obtained by multiplying numerator and denominator by the respective conjugate.

Visualisation: the angle $\alpha/2$ is half of the angle $\alpha$, read on the same unit circle; the bisection formulae express the sine and cosine of $P_{\alpha/2}$ starting from the $x$-coordinate of $P_\alpha$ (that is, $\cos\alpha$).

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Topics: Trigonometric formulae
Concepts: Unit circle · Bisection formulae · Duplication formulae
Functions: Cosine · Sine · Tangent
Methods: Bisection formulae
Skills: Using formulae