Trigonometry/Trigonometric Analysis

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

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 $\theta =\pi$
${\begin{aligned}\sin(-\theta )&=-\sin \theta \\\cos(-\theta )&=+\cos \theta \\\tan(-\theta )&=-\tan \theta \\\csc(-\theta )&=-\csc \theta \\\sec(-\theta )&=+\sec \theta \\\cot(-\theta )&=-\cot \theta \end{aligned}}$	${\begin{aligned}\sin({\tfrac {\pi }{2}}-\theta )&=+\cos \theta \\\cos({\tfrac {\pi }{2}}-\theta )&=+\sin \theta \\\tan({\tfrac {\pi }{2}}-\theta )&=+\cot \theta \\\csc({\tfrac {\pi }{2}}-\theta )&=+\sec \theta \\\sec({\tfrac {\pi }{2}}-\theta )&=+\csc \theta \\\cot({\tfrac {\pi }{2}}-\theta )&=+\tan \theta \end{aligned}}$	${\begin{aligned}\sin(\pi -\theta )&=+\sin \theta \\\cos(\pi -\theta )&=-\cos \theta \\\tan(\pi -\theta )&=-\tan \theta \\\csc(\pi -\theta )&=+\csc \theta \\\sec(\pi -\theta )&=-\sec \theta \\\cot(\pi -\theta )&=-\cot \theta \\\end{aligned}}$

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
${\begin{aligned}\sin({\tfrac {\pi }{2}}+\theta )&=+\cos \theta \\\cos({\tfrac {\pi }{2}}+\theta )&=-\sin \theta \\\tan({\tfrac {\pi }{2}}+\theta )&=-\cot \theta \\\csc({\tfrac {\pi }{2}}+\theta )&=+\sec \theta \\\sec({\tfrac {\pi }{2}}+\theta )&=-\csc \theta \\\cot({\tfrac {\pi }{2}}+\theta )&=-\tan \theta \end{aligned}}$	${\begin{aligned}\sin(\pi +\theta )&=-\sin \theta \\\cos(\pi +\theta )&=-\cos \theta \\\tan(\pi +\theta )&=+\tan \theta \\\csc(\pi +\theta )&=-\csc \theta \\\sec(\pi +\theta )&=-\sec \theta \\\cot(\pi +\theta )&=+\cot \theta \\\end{aligned}}$	${\begin{aligned}\sin(2\pi +\theta )&=+\sin \theta \\\cos(2\pi +\theta )&=+\cos \theta \\\tan(2\pi +\theta )&=+\tan \theta \\\csc(2\pi +\theta )&=+\csc \theta \\\sec(2\pi +\theta )&=+\sec \theta \\\cot(2\pi +\theta )&=+\cot \theta \end{aligned}}$

Resources

Practice Questions

1  

$sin{\frac {\pi }{6}}=$

2  

$cos{\frac {\pi }{3}}=$

3  

$tan{\frac {\pi }{4}}=$

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