Search results:
1.
Polar representation of a complex number
$a+bi=r\left(\cos \psi + i \sin \psi \right)$
2.
Product in polar representation
$z_1 \cdot z_2 = r_1\left(\cos\psi_1+i\sin\psi_1\right) \cdot r_2\left(\cos\psi_2+i\sin\psi_2\right) = \\ = r_1r_2\left(\cos\left(\psi_1+\psi_2\right)+i\sin\left(\psi_1+\psi_2\right) \right)$
3.
Conjugate in polar form
$\overline{r\left(\cos\psi+i\sin\psi\right)}=r\left[\cos(-\psi)+i\sin(-\psi\right)]$
4.
Inverse in polar form
$\dfrac{1}{r\left(\cos\psi + i \sin \psi\right)}=\dfrac{1}{r}\left[ \cos(-\psi)+i\sin(-\psi)\right]$
5.
Qotient in polar form
$\dfrac{z_1}{z_2} = \dfrac{r_1\left(\cos\psi_1+i\sin\psi_1\right)}{r_2\left(\cos\psi_2+i\sin\psi_2\right)}=\dfrac{r_1}{r_2}\left(\cos(\psi_1-\psi_2) + i\sin(\psi_1-\psi_2)\right)$
6.
Power in ploar form
$z^n= \left[r \left(\cos\psi+i\sin\psi\right)\right]^n= r^n \left[\cos(n\psi)+i\sin(n\psi)\right]$
7.
De Moivre Formula
$\left( \cos\psi + i \sin\psi\right)^n= \cos(n\psi) + i \sin(n\psi)$
8.
N-th root of complex number
$\sqrt[n]z=\sqrt[n]{r(\cos\psi+i\sin\psi)} = \sqrt[n]r \left(\cos\dfrac{\psi+2\pi k}{n} + \sin\dfrac{\psi+2\pi k}{n} \right) \\[1.2 em] k=0,1,2,\dots,n-1$
10.
Distance in polar coordinates
$d=\sqrt{r_1^2+r_2^2-2r_1r_2\cos\left(\phi_2-\phi_1\right)}$
11.
Rectangular coordinates to polar
$r=\sqrt{x^2+y^2} ,\, \phi= \tan\dfrac{y}{x}$
13.
Line in prametric form
$\begin{aligned}
x &= a_1+tb_1 \\[0.3 em]
y &= a_2+tb_2
\end{aligned}$
14.
Parametirc equation of a circle
$\begin{aligned}
x &= a + R\cos t \\
y &= b + R\sin t \\
\end{aligned} ~~~~~~ 0 \le t \le 2\pi $