\part{Advanced}


\chapter{Advanced trigonometry}
\section{Recirpocal trigonometric functions}
- cosecant
- secant
- cotangent
\[ 1 + \cot^2 (\theta) = \sec^2 (\theta)\]


\begin{theorem}[Angle sum identities]
\[\sin(\theta + \phi) = \sin(\theta) \cos(\phi) + \sin(\phi) \cos(\theta) \]
\[\cos(\theta + \phi) = \cos(\theta) \cos(\phi) - \sin(\phi) \sin(\theta) \]
\end{theorem}[Angle sum identities]

These directly leads to the following by taking $\theta=\phi$
\begin{corollary}[Double angle identities]
\[\sin(2\theta ) = 2\sin(\theta) \cos(\theta) \]
\[\cos(2\theta ) = \cos^2 (\theta) - \sin^2 (\theta) \]
\end{corollary}

\section{Chebyshev polynomials}


\begin{theorem}[Auxiliary angle theorem]
	\[a\cos(\omega \theta)+b\sin(\omega\theta)\]
	\[\sqrt{a^2+b^2} \cos (\omega \theta + \tan^{-1}(-\frac{b}{a}))\]
	\[\sqrt{a^2+b^2} \sin (\omega \theta + \tan^{-1}(\frac{a}{b}))\]
\end{theorem}

\chapter{Other curvilinear coordinate systems}

\section{Polar coordinates}
\[x=r\cos(\theta)\]
\[y=r\sin(\theta)\]
\[r=\sqrt{x^2+y^2}\]
\[\theta=\tan^{-1}(\frac{y}{x})\]

\subsection{Polar equations}
Polar coordinates allow us to create algebraic equations based on radius and angle rather than horizontal and vertical distances from the origin!
- ellipse
- focus
- locus
\section{Spherical coordinates}
\[x=\rho\sin(\phi)\cos(\theta)\]
\[x=\rho\sin(\phi)\sin(\theta)\]
\[x=\rho\cos(\phi)\]
\[\rho=\sqrt{x^2+y^2+z^2}\]
\[\theta=\]


\chapter{Vectors}

Vecotr, however we can can discuss their usage and properties without going deep into the formalism of linear algebra.

\section{Vector laws}
\section{Norm}
\section{Dot product}
Useful 'tool' for angle related analysis between 2 vectors
\subsection{Vector projection}
\section{Cross product}
Anothee useful 'tool' for finding perpendicular vector to a pair of vectors.


\chapter{Circle geometry}

Some of the oldest deductions in mathematics.
- Thale's theorem
- inscribed angle theorem
- alternate segment theorem
- intersecting chords theorem
- ptolemy's theorem
