If sin theta = -0.7660, which of the following represents an approximate value of tan theta, for 180 degrees < theta < 270 degrees?

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°.

230°.