Find the two real numbers r and θ so that $Z1 = \overline{Z0}$

Question

Let $\theta \in [0, \pi]$ and $r \neq 0$

And the two complex numbers:

$Z0 = r(-\cos(\theta) + i\sin(\theta))$

$ Z1 = r^2(\sin (\theta) + i\cos(\theta)) $

The question is: Find the two real numbers r and θ so that $Z1 = \overline{Z0}$

Answer 1

From the given $Z_1=\bar Z_0$, we get,

$$ r^2(\sin \theta+ i\cos\theta) = r(-\cos\theta- i\sin\theta)$$

Rearrange and use $r\ne 0$,

$$r\sin\theta + \cos\theta +i(r\cos\theta +\sin\theta)=0$$

which leads to the system of equations, $$r\sin\theta + \cos\theta =0\tag 1$$

$$r\cos\theta + \sin\theta =0\tag 2$$

(1) + (2)

$$(r+1)(\sin\theta +\cos\theta )= 0$$

Two cases:

1) $r+1=0$. Substitute $r=-1$ into (1) to get $\theta = \frac\pi4+n\pi$.

2) $\sin\theta +\cos\theta= 0$ leads to $\theta = -\frac\pi4+n\pi$ and $r=1$.

Thus, the real numbers are,

$$(r, \theta)= (-1,\frac\pi4),\> (1, \frac{3\pi}4)$$