11. If and , then is:
    and
    and
    and
    and

To solve for when , we must first recognize that sine is negative in the third and fourth quadrants. Using a calculator, we can find the reference angle whose sine is . This reference angle is approximately .

In the third quadrant, the angle can be calculated as (which we round to ), and in the fourth quadrant, it is (which we round to ). Thus, the angles satisfying the condition are and .

So, the correct answer is:
and

Now for some extra insights!

Sine values have been used since ancient times for various calculations, especially in navigation and astronomy. Historically, the concept of angles and trigonometric functions developed significantly with the Greeks and then flourished in the Islamic Golden Age, where they created sine tables for more accurate calculations.

To avoid common pitfalls, remember to properly determine which quadrant your angle is in based on the sign of the sine value. Confirming the reference angle using a calculator is essential for both accuracy and efficiency. Practice finding sine values for a variety of angles to become more comfortable with these calculations!