# complex numbers, polar form

#### Vali

##### Junior Member
I have these complex numbers: $$\displaystyle z_1=\sin(a)-\cos(a)+i(\sin(a)+\cos(a))$$ and $$\displaystyle z_2=\sin(a)+\cos(a)+i(\sin(a)-\cos(a))$$

I need to find $$\displaystyle a$$ values such that the absolute value of complex number $$\displaystyle w=(z_1)^n+(z_2)^n$$ is maximum.The right answer is $$\displaystyle \frac{k\pi}{n}+\frac{\pi}{2}$$

My try: I wrote $$\displaystyle (\sin(a)+\cos(a)=\sqrt{2}\sin(\frac{\pi}{4}+a)$$ and $$\displaystyle \sin(a)-\cos(a)=\sqrt{2}\cos(\frac{3\pi}{4}-a)$$ so $$\displaystyle z_1=\sqrt{2}[\cos(\frac{3\pi}{4}-a)+i\sin(\frac{pi}{4}+a)]$$ but the polar form it's not good.How to convert it to the polar form ?How to approach this exercise?