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

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Question 825935: Find each of the following exactly in radians and degrees. Do not use a calculator. cot-1(√3/3)
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 the hint of "do not use a calculator", we should look into the special angle values for cot when we see cot%5E%28-1%29%28sqrt%283%29%2F3%29.
  • It should not take long to find that a reference angle of pi%2F3 has a cot value of sqrt%283%29%2F3.
  • Since the angle in question, cot%5E%28-1%29%28sqrt%283%29%2F3%29, has a positive cot value and since cot is positive in the 1st and 3rd, we now know that cot%5E%28-1%29%28sqrt%283%29%2F3%29 terminates in one of these quadrants.
  • We should know that the inverse cot function has a range limited to angles between 0 and pi. So cot%5E%28-1%29%28sqrt%283%29%2F3%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 2nd quadrant (between pi%2F2 and pi).
  • Combining the fact that the angle must terminate in the 1st or 3rd quadrants (because the cot is positive) and the range of the inverse cot function, we should now know that cot%5E%28-1%29%28sqrt%283%29%2F3%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-pi%2F3+=+%282pi%29%2F3
    is the only angle with the right reference angle for a cot of sqrt%283%29%2F3, with a positive sin and in the range of the inverse cot function.
To turn this into degrees, just multiply it by 180%2Fpi.

P.S. There is what appears to be a multiplication symbol between the exponent and the argument of cot. Ignore it. Do not use it yourself. It should not be there. It is just a flaw in algebra.com's formula drawing software.