Complex Phobia 1

Consider the complex third root of unity $\omega^3 = 1$ , $\implies \omega = e^{\frac {2\pi}3i}$ and we have:

$\begin{aligned} 1 + \omega + \omega^2 & = 0 \\ e^{0i} + e^{\frac {2\pi}3i} + e^{\frac {4\pi}3i} & = 0 &\small \color{#3D99F6} \text{By Euler's formula } e^{xi} = \cos x + i \sin x \\ \implies e^{{\color{#3D99F6}\theta}i} \left(e^{0i} + e^{\frac {2\pi}3i} + e^{\frac {4\pi}3i}\right) & = 0 &\small \color{#3D99F6} \text{where }\theta \text{ is a real constant.} \\ \cos \theta + i \sin \theta + \cos \left( \theta + \frac {2\pi}3 \right) + i \sin \left( \theta + \frac {2\pi}3 \right) + \cos \left( \theta + \frac {4\pi}3 \right) + i \sin \left(\theta + \frac {4\pi}3 \right) & = 0 \\ \cos \theta + \cos \left( \theta + \frac {2\pi}3 \right) + \cos \left( \theta + \frac {4\pi}3 \right) + i \left(\sin \theta + \sin \left( \theta + \frac {2\pi}3 \right) + \sin \left(\theta + \frac {4\pi}3 \right)\right) & = 0 \end{aligned}$

Equating the real and imaginary parts on both sides,

$\implies \cos \theta + \cos \left( \theta + \frac {2\pi}3 \right) + \cos \left( \theta + \frac {4\pi}3 \right) = \sin \theta + \sin \left( \theta + \frac {2\pi}3 \right) + \sin \left(\theta + \frac {4\pi}3 \right) = 0$

Therefore, the general solution for $x, y, z = \theta, \theta + \frac {2\pi}3, \theta + \frac {4\pi}3$ and $\sin \theta \sin \left(\theta + \frac {2\pi}3 \right) \sin \left(\theta + \frac {4\pi}3 \right)$ has no unique solution . The answer is thus $\boxed{\text{None of the others.}}$