Trigonometry & Inverse Trigonometric Functions

Overview

Trigonometry models circular motion, angular harmonics, and triangular geometry. In competitive mathematics, mastery spans compound angle telescopic products, solution of trigonometric equations without extraneous roots, and properties of triangles (circumradius $R$, inradius $r$, exradii $r_1, r_2, r_3$).

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1. Compound & Multiple Angle Master Identities

$$\begin{array}{rl}\sin (A\pm B)& =\sin A\cos B\pm \cos A\sin B\\ \cos (A\pm B)& =\cos A\cos B\mp \sin A\sin B\\ \tan (A\pm B)& =\frac{\tan A\pm \tan B}{1\mp \tan A\tan B}\\ \sin 2A& =2\sin A\cos A=\frac{2\tan A}{1+{\tan }^{2}A}\\ \cos 2A& ={\cos }^{2}A-{\sin }^{2}A=2{\cos }^{2}A-1=1-2{\sin }^{2}A=\frac{1-{\tan }^{2}A}{1+{\tan }^{2}A}\\ \tan 3A& =\frac{3\tan A-{\tan }^{3}A}{1-3{\tan }^{2}A}\end{array}$$

Product of Cosines Cascade

$$\prod _{k=0}^{n-1}\cos (2^k\theta )=\cos \theta \cos (2\theta )\cos (4\theta )\dots \cos (2^{n-1}\theta )=\frac{\sin (2^n\theta )}{2^n\sin \theta }$$

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2. Properties of Triangles (Solutions of Triangles - SOT)

In $\mathrm{△}ABC$ with side lengths $a,b,c$, semi-perimeter $s=\frac{a+b+c}{2}$, and area $\mathrm{\Delta }$:

• Sine Rule: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$
• Cosine Rule: $\cos A=\frac{b^2+c^2-a^2}{2bc}$
• Projection Formula: $a=b\cos C+c\cos B$
• Area: $\mathrm{\Delta }=\frac{1}{2}ab\sin C=\sqrt{s(s-a)(s-b)(s-c)}=\frac{abc}{4R}=rs$
• Inradius: $r=\frac{\mathrm{\Delta }}{s}=(s-a)\tan \frac{A}{2}=4R\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$
• Exradii: $r_1=\frac{\mathrm{\Delta }}{s-a}=s\tan \frac{A}{2}=4R\sin \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}$

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3. Inverse Trigonometric Functions (ITF)

Function	Domain	Principal Value Branch (Range)
$\arcsin x$ (${\sin }^{-1}x$)	$[-1,1]$	$[-\frac{\pi }{2},\frac{\pi }{2}]$
$\arccos x$ (${\cos }^{-1}x$)	$[-1,1]$	$[0,\pi ]$
$\arctan x$ (${\tan }^{-1}x$)	$\mathbb{R}$	$(-\frac{\pi }{2},\frac{\pi }{2})$
$\text{arccot }x$ (${\cot }^{-1}x$)	$\mathbb{R}$	$(0,\pi )$

Master Addition Formula for $\arctan$

$$\arctan x+\arctan y=\{\begin{array}{ll}\arctan \left(\frac{x+y}{1-xy}\right)& \text{if }xy<1\\ \pi +\arctan \left(\frac{x+y}{1-xy}\right)& \text{if }x>0,y>0,xy>1\\ -\pi +\arctan \left(\frac{x+y}{1-xy}\right)& \text{if }x<0,y<0,xy>1\end{array}$$

Telescoping Series of Inverse Tangents

$$\arctan \left(\frac{x_{k+1}-x_k}{1+x_{k+1}{x}_k}\right)=\arctan (x_{k+1})-\arctan (x_k)$$