Question

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`

Answer 1

The four solutions: `z = ( x, y ) = x + i y,` in the complex plane:
`( pi/4, 1/2 pi ), ( pi/4, 3/2 pi ), ( 5/4 pi, 1/2 pi ), ( 5/4 pi, 3/2 pi )`.
See plots, in the complex plane.

Explanation:

Equating the real parts,

`cos x = sin x,` giving

`tan x = 1 = tan pi/4`,

`tan^(-1)1 = pi/4`, and so,

`x = (tan)^(-1)tan (pi/4) = npi+tan^(-1)1 = npi +pi/4,`

`n = 0, +-1, +-2, +-3, ...`. For the answer,

`x = pi/4, 5/4pi in [ 0. 2 pi ]`;

Equating the imaginary parts,

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

`y = mpi +(-1)^m(pi/2), m= 0, +-1, +-2, +-3, ....,`

For the answer,

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

The four solutions

`z = ( x, y ) = x + i y,` in the complex plane:

`( pi/4, 1/2 pi ), ( pi/4, 3/2 pi ), ( 5/4 pi, 1/2 pi ), ( 5/4 pi, 3/2 pi )`.

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]}