# Trigonometry/Trigonometric Analysis

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

### Angle Sum Identities
## Complex Numbers, De Moivre's Law and Argand Plane
# Resources
## Textbooks
## Active participants

The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this division, you can list your name here (this can help small divisions grow and the participants communicate better; for large divisions a list of active participants is not needed).