sin−1(sin(7π6))
Start inside the parentheses by finding sin(7π6).
According to the unit circle at 7π6,
the y coordinate or sine of 7π6 is equal to −12.
Next, substitute −12 into the original problem.
sin−1(−12)
Recall that the range of sin−1 is −π2 to π2.
If you are finding sin−1 of a positive value, the answer will be in the first quadrant between 0 and π2.
If you are finding sin−1 of a negative value, the answer will be in the fourth quadrant between −π2 and 0.
Again using the unit circle, the fourth quadrant angle with a sine of −12 is 11π6. But this is NOT the answer! Because of the restriction on the range, you need to find an angle between −π2 and 0. The angle is then −π6.
Also note that sin−1(sinx) does not automatically "cancel out" and yield x.
