\chapter{Trigonmomety}
\section{Trigonometric functions}
\subsection{Congruence between right angle triangles}
\subsection{Trigonometric functions}

\begin{definition}[Sine function (unit circle)]
Consider a right angled triangle formed by  $(0,0)$ to $(x,\sqrt{1-x^2})$ to $(x,0)$ back to $(0,0)$. let $\theta$ be the angle formed at $(0,0)$, the \emph{sine function} is defined as such.
\[\sin (\theta) = \sqrt{1-x^2}\]
\end{definition}

\begin{definition}[Cosine function (unit circle)]
Consider a right angled triangle formed by  $(0,0)$ to $(x,\sqrt{1-x^2})$ to $(x,0)$ back to $(0,0)$. let $\theta$ be the angle formed at $(0,0)$, the \emph{cosine function} is defined as such.
\[\cos (\theta) = x\]
\end{definition}

\begin{definition}[Tangent function (unit circle)]
Consider a right angled triangle formed by  $(0,0)$ to $(x,\sqrt{1-x^2})$ to $(x,0)$ back to $(0,0)$. let $\theta$ be the angle formed at $(0,0)$, the \emph{tangent function} is defined as such.
	\[\tan (\theta) = \frac{\sqrt{1-x^2}}{x}\]
\end{definition}

The 3 main trigonometic functions repesent ratios of sides of right angled triangles.


\begin{proposition}
\[\sin (\theta) = \cos (\pi - \theta)\]
\[\sin^2 (\theta) + \cos^2 (\theta) = 1\]
\end{proposition}

\subsection{Recirpocal trigonometric functions}
- cosecant
- secant
- cotangent

\begin{proposition}
\[ 1 + \cot^2 (\theta) = \sec^2 (\theta)\]
\end{proposition}

\section{Inverse trigonometric functions}
- arcsine
- arccosine
- arctangent

\begin{proposition}
\[ \sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2}\]
\end{proposition}

\begin{proposition}
	\[ \cos (\sin^{-1}(x)) =  \sin (\cos^{-1}(x)) = \sqrt{1-x^2}\]
	\[ \tan (\sin^{-1}(x)) =  \frac{x}{\sqrt{1-x^2}}\]
	\[ \tan (\cos^{-1}(x)) =  \frac{\sqrt{1-x^2}}{x}\]
	\[ \sin(\tan^{-1}(x)) =  \frac{x}{\sqrt{1+x^2}}\]
	\[ \cos(\tan^{-1}(x)) =  \frac{1}{\sqrt{1+x^2}}\]
\end{proposition}


\section{Theorems of trigonometry}

- sine rule
- cosine rule

-radians
-
 
\subsection{Angle sum identities}

\[\sin(\theta + \phi) = \sin(\theta) \cos(\phi) + \sin(\phi) \cos(\theta) \]
\[\cos(\theta + \phi) = \cos(\theta) \cos(\phi) - \sin(\phi) \sin(\theta) \]


\[\sin(2\theta ) = 2\sin(\theta) \cos(\theta) \]
\[\cos(2\theta ) = \cos^2 (\theta) - \sin^2 (\theta) \]

Allude to Chebyshev polynomials