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?
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?