Rotating a complex number

Question

I have the complex number $3 + i$, and I am asked to get the complex number resulted by rotating the first one by $\frac{\pi}{4}$.

I got the polar form of the first one to get its angle ($18.43°$) but when I add $\frac{\pi}{4}$ to it and try to find its cartesian form, the result is different.

Am I doing something wrong with the procedure? If so, what can I do?

Answer 1

Rotating by $\frac\pi4$ is the same thing as multiplying your number by $\cos\left(\frac\pi4\right)+\sin\left(\frac\pi4\right)i=\frac1{\sqrt2}+\frac i{\sqrt2}$. So, the answer is$$\left(\frac1{\sqrt2}+\frac i{\sqrt2}\right)(3+i)=\sqrt2+2\sqrt2i,$$

Answer 2

Write in polar form: $re^{i\theta}$;

$r=\sqrt{10},$ $\tan(\theta) = 1/3$.

Rotate by $π/4$ counter clockwise.

$\tan(\theta') = \tan(\theta +π/4)=$

$\dfrac{\tan(\theta) + 1}{1-\tan(\theta)}= \dfrac{4/3}{2/3}=2;$

Rotated point: $ z' =re^{i\theta'}, $

$r=\sqrt{10}$ , $\tan(\theta') =2$.

Reverting to $z'= a' +ib':$

$z' = √2(1+2i).$