Values of $a$ such that the absolute value of complex number $w=(z_1)^n+(z_2)^n$ is maximum.

Question

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

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

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

Answer 1

You did ok: now just do a further step $$ \left\{ \matrix{ b = \pi /4 - a \hfill \cr \sin a + \cos a = \sqrt 2 \sin \left( {\pi /4 + a} \right) = \sqrt 2 \sin \left( {\pi /2 - b} \right) = \sqrt 2 \cos b \hfill \cr \sin a - \cos a = \sqrt 2 \cos \left( {3/4\,\pi - a} \right) = \sqrt 2 \cos \left( {\pi /2 + b} \right) = - \sqrt 2 \sin b \hfill \cr} \right. $$

where the relation between $b$ and $a$ is linear, monotonically decreasing, and thus does not affect the search for a maximum.

Answer 2

$$z_1=\sqrt2e^{i(3\pi/4-a)}$$

$$z_2=\sqrt2e^{i (\pi/4-a)}$$

$$P=z_1^n+z_2^n=(\sqrt2)^n\cos n(\pi/4)\left(e^{in(\pi/2-a)}\right)$$

will be real if $$n(\pi/2-a)=k\pi\iff a=\pi/2-k\pi/n$$

$$|P|=(\sqrt 2)^n\left|\cos \dfrac{n\pi}4\right|$$