miércoles, 27 de abril de 2022

The compound angle formulas from the half angle formulas

My goal is not necessarily to give the simplest derivation. I've been trying to rescue the half angle formulas from oblivion and give them the status they deserve by showing its many applications.

Various proofs of the formulas of compound angles are given here. The cosine of the sum of two angles is given by the formula

$$\cos{(\alpha+\beta)}=\cos{\alpha}\cos{\beta}-\sin{\alpha}\sin{\beta}\tag{1}$$

Proof. We take advantage of the cyclic nature of the half angle formulas whose proof can be found here. The proof of the half angle formulas given in the link makes use of the Pythagorean trigonometric identity and the double-angle formula for sine whose proofs (I hope you clicked on the links) are independent of the compound angle formulas.

Suppose $\triangle{ABC}$ is a triangle with sides $|BA|=a$, $|AC|=b$ and $|AB|=c$. Let $\angle{BAC}=2\alpha$, $\angle{CBA}=2\beta$ and $\angle{ACB}=2\gamma$. Let's start with the right-hand side of formula $(1)$. Substituting from the half angle formulas we have

\begin{aligned} \cos{\alpha}\cos{\beta}-\sin{\alpha}\sin{\beta} &= \sqrt{\frac{s(s-a)}{bc}\cdot{\frac{s(s-b)}{ac}}}-\sqrt{\frac{(s-b)(s-c)}{bc}\cdot{\frac{(s-a)(s-c)}{ac}}}\\&=\frac{s}{c}\cdot{\sqrt{\frac{(s-a)(s-b)}{ab}}}-\frac{s-c}{c}\cdot{\sqrt{\frac{(s-a)(s-b)}{ab}}}\\&=\sqrt{\frac{(s-a)(s-b)}{ab}}\cdot{\left(\frac{s}{c}-\frac{s-c}{c}\right)}\\&=\sin{\gamma}\\&=\sin{\left(\frac{\pi}{2}-(\alpha+\beta)\right)}\\&=\cos{(\alpha+\beta)}\end{aligned}

$\square$

The other formulas of compound angles can be obtained similarly.