Find theta given known cos and sin....

[SammyS]

X (theta) must be on 4 Quadrant, so 2x must on 8 quadrants, 540<2x<=720
X 1=5/4 π
X2 = 7/4 π
Both added 2π to get x at 8 quadrants
X1 = 13/4 π
X2 = 15/4 π

Both satisfied the domain
How to decide which one is the answer?

Those actually should written be something like

2x₁ = (5/4) π
2x₂ = (7/4) π​

Then adding 2π gives (two new possibilities for x):

2x₃ = (13/4) π
2x₄ = (15/4) π​

Solve for each x_k .