quality:corruption, stylistic:style_transfer
Polar coordinates

Sine and cosine are used to connect the real and imaginary parts of a complex number with its polar coordinates (r, θ) {\displaystyle (r,\theta)}: z = r (cos (θ) + i sin (θ)), {\displaystyle z=r(\cos(\theta)+i\sin(\theta)),} and the real and imaginary parts are: Re (z) = r cos (θ), Im (z) = r sin (θ), {\displaystyle {\begin{aligned}\operatorname {Re} (z)&=r\cos(\theta),\\\operatorname {Im} (z)&=r\sin(\theta),\end{aligned}}} where r {\displaystyle r} and φ {\displaystyle \varphi } represents the magnitude and angle of the complex number z {\displaystyle z}.

For any real number θ {\displaystyle \theta }, Euler's formula in terms of polar coordinates is stated as z = r e i θ {\textstyle z=re^{i\theta }}.