Solve rcos(θ)+irsin(θ)=6-3i

$$r \cos ( \theta ) + i r \sin ( \theta ) = 6 - 3 i$$

Solve for r

$$r=\left(6-3i\right)e^{-i\theta }$$

Steps for Solving Linear Equation

Combine all terms containing r.

$$\left(\cos(\theta )+i\sin(\theta )\right)r=6-3i$$

Divide both sides by \cos(\theta )+i\sin(\theta ).

$$\frac{\left(\cos(\theta )+i\sin(\theta )\right)r}{\cos(\theta )+i\sin(\theta )}=\frac{6-3i}{\cos(\theta )+i\sin(\theta )}$$

Dividing by \cos(\theta )+i\sin(\theta ) undoes the multiplication by \cos(\theta )+i\sin(\theta ).

$$r=\frac{6-3i}{\cos(\theta )+i\sin(\theta )}$$

Solve for θ

$$\theta =-i\ln(\frac{6-3i}{r})+2\pi n_{1}$$

$$n_{1}\in \mathrm{Z}$$

$$r\neq 0$$