We begin by recalling a very important identity from trigonometry,

\[ \cos^2{\theta} + \sin^2{\theta} = 1\,.\]Additionally, beginning from \((1)\) and dividing through by \(\cos^2{\theta}\),

\[\begin{aligned} \frac{\cos^2{\theta}}{\cos^2{\theta}} + \frac{\sin^2{\theta}}{\cos^2{\theta}} &= \frac{1}{\cos^2{\theta}} \\ \\ 1 + \tan^2{\theta} &= \sec^2{\theta}\,. \\ \end{aligned}\]Another set of identities worth knowing are,

\[\begin{aligned} \sin(\alpha \pm \beta) &= \sin \alpha \cos\beta \pm \sin\beta\cos\alpha \\ \cos(\alpha \mp \beta) &= \cos \alpha \cos\beta \pm \sin\beta\sin\alpha\,. \\ \end{aligned}\]Recall that a circle of radius \(r\) is described in cartesian coordinates as,

\[ x^2 + y^2 = r^2 \,.\]Extending from \((4)\), we can describe polar coordinates in terms of cartesian coordinates as follows.

\[\begin{aligned} x &= r \cos{\theta} \\ y &= r \sin{\theta} \\ r &= \sqrt{x^2 + y^2} \\ \tan\theta &= \frac{y}{x} \end{aligned}\]
© Daniel Marvin. Last modified: July 18, 2021.