Solve: $(\cos x+i\sin x)(\cos 2x+i\sin 2x)(\cos 5x+i\sin 5x)={i+1\over \sqrt 2 }$

Question

Can you please give me a hint for the following exercise: $$(\cos x+i\sin x)(\cos 2x+i\sin 2x)(\cos 5x+i\sin 5x)={i+1\over \sqrt 2 }$$

Thank you!

Answer 1

Note that the top is given by $e^{ix}\cdot e^{2ix}\cdot e^{5ix} = e^{8ix} = \frac{1+i}{\sqrt{2}}$ and so $\cos(8x) = \sin(8x) = 1/\sqrt{2}$ implies $8x = \pi/4$ implies $x = \pi/32 + \pi\cdot n/4$ where $n \in \mathbb{Z}$. This was edited to consider less trivial solutions - if your goal is a complete solution set.

Answer 2

Hint: convert to exponential form

Answer 3

Maybe Euler's Formula? $\cos(x)+i\sin(x)=e^{ix}$