Summary

This chapter's results fall into a small number of formula families; gathering them here is a working reference, not a re-teach.

Compound-angle (sum and difference) identities.

Double, triple and half-angle identities. With $A$ the full angle:

The half-angle forms are the same relations with $A$ replaced by $\theta/2$ throughout — e.g. $\sin\theta=2\sin\dfrac\theta2\cos\dfrac\theta2$, $\cos\theta=\cos^2\dfrac\theta2-\sin^2\dfrac\theta2=2\cos^2\dfrac\theta2-1=1-2\sin^2\dfrac\theta2$, $\tan\theta=\dfrac{2\tan(\theta/2)}{1-\tan^2(\theta/2)}$.

Product-to-sum / sum-to-product. Products of sines and cosines convert to sums (and back) via the standard identities built from the compound-angle formulas above, e.g. $2\sin A\cos B=\sin(A+B)+\sin(A-B)$ and $\sin C+\sin D=2\sin\dfrac{C+D}2\cos\dfrac{C-D}2$, together with the analogous cosine pairings — these are what let a sum of trigonometric terms collapse to a single product (or the reverse) in a simplification.

General solutions of the basic trigonometric equations ($n\in\mathbb Z$):

• $\sin\theta=\sin\alpha,\ \alpha\in\left[-\dfrac\pi2,\dfrac\pi2\right] \implies \theta=n\pi+(-1)^n\alpha$
• $\cos\theta=\cos\alpha,\ \alpha\in[0,\pi] \implies \theta=2n\pi\pm\alpha$
• $\tan\theta=\tan\alpha,\ \alpha\in\left(-\dfrac\pi2,\dfrac\pi2\right) \implies \theta=n\pi+\alpha$

Solving a triangle — the three laws. For a triangle $ABC$ with sides $a,b,c$ opposite $A,B,C$ and circumradius $R$:

• Law of sines: $\dfrac a{\sin A}=\dfrac b{\sin B}=\dfrac c{\sin C}=2R$.
• Law of cosines: $\cos A=\dfrac{b^2+c^2-a^2}{2bc}$, and cyclically for $B$ and $C$.
• Law of tangents: $\tan\dfrac{A-B}2=\dfrac{a-b}{a+b}\cot\dfrac C2$, and cyclically.

Other standard triangle identities.

• Projection formula: $a=b\cos C+c\cos B$ (and cyclically for $b,c$).
• Area: $\triangle=\dfrac12ab\sin C=\dfrac12bc\sin A=\dfrac12ac\sin B$.
• Heron's formula: $\triangle=\sqrt{s(s-a)(s-b)(s-c)}$, with semi-perimeter $s=\dfrac{a+b+c}2$. …