Welcome to the Lesson of Analytical Trigonometry
Part of the School of Olympiads

This topic deals with the analytical aspects of Trigonometry. Widely this topic covers Trigonometric Identities and Equations. And important part of this topic is trigonometry through Complex Numbers by the use of De Moivre's Law and its application.

Function Inverse function Reciprocal Inverse reciprocal
sine sin arcsine arcsin cosecant csc arccosecant arccsc
cosine cos arccosine arccos secant sec arcsecant arcsec
tangent tan arctangent arctan cotangent cot arccotangent arccot

# Theorems

## Identities

### Basic Relationships

Pythagorean trigonometric identity $\sin^2 \theta + \cos^2 \theta = 1\,$ $\tan \theta = \frac{\sin \theta}{\cos \theta}$

Each trigonometric function in terms of the other five.
Function $(\sin \theta)$ $(\cos \theta)$ $(\tan \theta)$ $(\csc \theta)$ $(\sec \theta)$ $(\cot \theta)$
$\sin \theta =$ $\sin \theta\$ $\pm\sqrt{1 - \cos^2 \theta}\$ $\pm\frac{\tan \theta}{\sqrt{1 + \tan^2 \theta}}\$ $\frac{1}{\csc \theta}\$ $\pm\frac{\sqrt{\sec^2 \theta - 1}}{\sec \theta}\$ $\pm\frac{1}{\sqrt{1 + \cot^2 \theta}}\$
$\cos \theta =$ $\pm\sqrt{1 - \sin^2\theta}\$ $\cos \theta\$ $\pm\frac{1}{\sqrt{1 + \tan^2 \theta}}\$ $\pm\frac{\sqrt{\csc^2 \theta - 1}}{\csc \theta}\$ $\frac{1}{\sec \theta}\$ $\pm\frac{\cot \theta}{\sqrt{1 + \cot^2 \theta}}\$
$\tan \theta =$ $\pm\frac{\sin \theta}{\sqrt{1 - \sin^2 \theta}}\$ $\pm\frac{\sqrt{1 - \cos^2 \theta}}{\cos \theta}\$ $\tan \theta\$ $\pm\frac{1}{\sqrt{\csc^2 \theta - 1}}\$ $\pm\sqrt{\sec^2 \theta - 1}\$ $\frac{1}{\cot \theta}\$
$\csc \theta =$ $\frac{1}{\sin \theta}\$ $\pm\frac{1}{\sqrt{1 - \cos^2 \theta}}\$ $\pm\frac{\sqrt{1 + \tan^2 \theta}}{\tan \theta}\$ $\csc \theta\$ $\pm\frac{\sec \theta}{\sqrt{\sec^2 \theta - 1}}\$ $\pm\sqrt{1 + \cot^2 \theta}\$
$\sec \theta =$ $\pm\frac{1}{\sqrt{1 - \sin^2 \theta}}\$ $\frac{1}{\cos \theta}\$ $\pm\sqrt{1 + \tan^2 \theta}\$ $\pm\frac{\csc \theta}{\sqrt{\csc^2 \theta - 1}}\$ $\sec \theta\$ $\pm\frac{\sqrt{1 + \cot^2 \theta}}{\cot \theta}\$
$\cot \theta =$ $\pm\frac{\sqrt{1 - \sin^2 \theta}}{\sin \theta}\$ $\pm\frac{\cos \theta}{\sqrt{1 - \cos^2 \theta}}\$ $\frac{1}{\tan \theta}\$ $\pm\sqrt{\csc^2 \theta - 1}\$ $\pm\frac{1}{\sqrt{\sec^2 \theta - 1}}\$ $\cot \theta\$

Historic Shorthands
Name(s) Abbreviation(s) Value
versed sine, versine $\textrm{versin}\,\theta\$
$\textrm{vers}\,\theta\$
$1 - \cos \theta\$
versed cosine, vercosine,
coversed sine, coversine
$\textrm{vercos}\,\theta\$
$\textrm{coversin}\,\theta\$
$\textrm{cvs}\,\theta\$
$1 - \sin \theta\$
haversed sine, haversine $\textrm{haversin}\,\theta\$
$\textrm{hav}\,\theta\$
$\tfrac{1}{2} \textrm{versin}\ \theta\$
haversed cosine, havercosine,
hacoversed sine, hacoversine,
cohaversed sine, cohaversine
$\textrm{havercos}\,\theta\$
$\textrm{hacoversin}\,\theta\$
$\textrm{cohav}\,\theta\$
$\tfrac{1}{2} \textrm{vercos}\,\theta\$
exterior secant, exsecant $\textrm{exsec}\,\theta\$ $\sec \theta - 1\$
exterior cosecant, excosecant $\textrm{excsc}\,\theta\$ $\csc \theta - 1\$

Symmetries
Reflected in $\theta=0$ Reflected in $\theta= \pi/2$
(co-function identities)
Reflected in $\theta= \pi$
\begin{align} \sin(-\theta) &= -\sin \theta \\ \cos(-\theta) &= +\cos \theta \\ \tan(-\theta) &= -\tan \theta \\ \csc(-\theta) &= -\csc \theta \\ \sec(-\theta) &= +\sec \theta \\ \cot(-\theta) &= -\cot \theta \end{align} \begin{align} \sin(\tfrac{\pi}{2} - \theta) &= +\cos \theta \\ \cos(\tfrac{\pi}{2} - \theta) &= +\sin \theta \\ \tan(\tfrac{\pi}{2} - \theta) &= +\cot \theta \\ \csc(\tfrac{\pi}{2} - \theta) &= +\sec \theta \\ \sec(\tfrac{\pi}{2} - \theta) &= +\csc \theta \\ \cot(\tfrac{\pi}{2} - \theta) &= +\tan \theta \end{align} \begin{align} \sin(\pi - \theta) &= +\sin \theta \\ \cos(\pi - \theta) &= -\cos \theta \\ \tan(\pi - \theta) &= -\tan \theta \\ \csc(\pi - \theta) &= +\csc \theta \\ \sec(\pi - \theta) &= -\sec \theta \\ \cot(\pi - \theta) &= -\cot \theta \\ \end{align}

Periodicity and Shifts
Shift by π/2 Shift by π
Period for tan and cot
Shift by 2π
Period for sin, cos, csc and sec
\begin{align} \sin(\theta + \tfrac{\pi}{2}) &= +\cos \theta \\ \cos(\theta + \tfrac{\pi}{2}) &= -\sin \theta \\ \tan(\theta + \tfrac{\pi}{2}) &= -\cot \theta \\ \csc(\theta + \tfrac{\pi}{2}) &= +\sec \theta \\ \sec(\theta + \tfrac{\pi}{2}) &= -\csc \theta \\ \cot(\theta + \tfrac{\pi}{2}) &= -\tan \theta \end{align} \begin{align} \sin(\theta + \pi) &= -\sin \theta \\ \cos(\theta + \pi) &= -\cos \theta \\ \tan(\theta + \pi) &= +\tan \theta \\ \csc(\theta + \pi) &= -\csc \theta \\ \sec(\theta + \pi) &= -\sec \theta \\ \cot(\theta + \pi) &= +\cot \theta \\ \end{align} \begin{align} \sin(\theta + 2\pi) &= +\sin \theta \\ \cos(\theta + 2\pi) &= +\cos \theta \\ \tan(\theta + 2\pi) &= +\tan \theta \\ \csc(\theta + 2\pi) &= +\csc \theta \\ \sec(\theta + 2\pi) &= +\sec \theta \\ \cot(\theta + 2\pi) &= +\cot \theta \end{align}

# Resources

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