# Find all x and y (0 ≤ x < 2π, 0 ≤ y < 2π) so that the following equation is true? (Enter your answers as a comma-separated list.) cosx+isiny=sinx+i

Aug 7, 2018

The four solutions: $z = \left(x , y\right) = x + i y ,$ in the complex plane:
$\left(\frac{\pi}{4} , \frac{1}{2} \pi\right) , \left(\frac{\pi}{4} , \frac{3}{2} \pi\right) , \left(\frac{5}{4} \pi , \frac{1}{2} \pi\right) , \left(\frac{5}{4} \pi , \frac{3}{2} \pi\right)$.
See plots, in the complex plane.

#### Explanation:

Equating the real parts,

$\cos x = \sin x ,$ giving

$\tan x = 1 = \tan \frac{\pi}{4}$,

${\tan}^{- 1} 1 = \frac{\pi}{4}$, and so,

$x = {\left(\tan\right)}^{- 1} \tan \left(\frac{\pi}{4}\right) = n \pi + {\tan}^{- 1} 1 = n \pi + \frac{\pi}{4} ,$

$n = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$. For the answer,

$x = \frac{\pi}{4} , \frac{5}{4} \pi \in \left[0. 2 \pi\right]$;

Equating the imaginary parts,

sin y = 1, and proceeding as for the real parts,

$y = m \pi + {\left(- 1\right)}^{m} \left(\frac{\pi}{2}\right) , m = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots . ,$

#y =pi/2, 3/2pi in [ 0, 2 pi ];

The four solutions

$z = \left(x , y\right) = x + i y ,$ in the complex plane:

$\left(\frac{\pi}{4} , \frac{1}{2} \pi\right) , \left(\frac{\pi}{4} , \frac{3}{2} \pi\right) , \left(\frac{5}{4} \pi , \frac{1}{2} \pi\right) , \left(\frac{5}{4} \pi , \frac{3}{2} \pi\right)$.

See solutions plotted in the graph.

graph{((x-pi/4)^2+(y-pi/2)^2-0.01)((x-pi/4)^2+(y-3pi/2)^2-0.01)((x-5pi/4)^2+(y-pi/2)^2-0.01)((x-5pi/4)^2+(y-3pi/2)^2-0.01)=0[0 14 0 7]}