SOLUTION: Find each of the following exactly in radians and degrees. Do not use a calculator. sin-1(-√3/2)

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Question 825934: Find each of the following exactly in radians and degrees. Do not use a calculator. sin-1(-√3/2)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
When a problem says "find the exact value" or "do not use a calculator" it usually means that the problem involves special angles.

Here's a solution:
  • With sin%5E%28-1%29%28-sqrt%283%29%2F2%29 we should recognize (even without being told not to use a calculator) that sqrt%283%29%2F2 is a special angle value for sin.
  • We should also know that a reference angle of pi%2F3 has a sin of sqrt%283%29%2F2.
  • Since the angle in question, sin%5E%28-1%29%28-sqrt%283%29%2F2%29, has a negative sin value and since sin is negative in the 3rd and 4th quadrants, we now know that sin%5E%28-1%29%28-sqrt%283%29%2F2%29 terminates in one of these quadrants.
  • We should know that the inverse sin function has a range limited to angles between -pi%2F2 and pi%2F2. So sin%5E%28-1%29%28-sqrt%283%29%2F2%29 will have a value in that range. (Note: This range includes angles which terminate in the 1st quadrant (between 0 and pi%2F2) and the 4th quadrant (between 0 and -pi%2F2).
  • Combining the fact that the angle must terminate in the 3rd or 4th quadrants (because the sin is negative) and the range of the inverse sin function, we should now know that sin%5E%28-1%29%28-sqrt%283%29%2F2%29 is between 0 and -pi%2F2
  • Combining that with the fact that the reference angle is pi%2F3, it should not take long to figure out that:
    -pi%2F3
    is the only angle with the right reference angle for sqrt%283%29%2F2, with a negative sin and in the range of the inverse sin function.
To turn this into degrees, just multiply it by 180%2Fpi.