If sin(theta) = - 0.7660 => angle theta = inverse sine of - 0.766.
When you plug that into the calculator you get theta ≈ - 50°.
A negative angle means that it is below from the x-axis and that means that the answer of the calculator is 360° - 50° = 310°.
As you know that the sine function is negative in third and fourth quadrants you have that the angle theta may be 180° + 50° = 230° and 360° - 50 = 310°.
The problem states that the angle is in the third quadrant (180 < theta < 270), so there is only one answer. Theta ≈ 230°.
Which you can verifiy when you calculate sine of 230°.
Answer: 230°.
