SOLUTION: What is the domain of the square root of tanx?

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Question 207460: What is the domain of the square root of tanx?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Because of the periodic nature of tan(x), this is a tough one.

When finding domains one looks for things that are undefined in the set of Real numbers. Among the things to avoid are:
  • Zeros in denominators
  • Negative radicands of even-numbered roots. For example: sqrt%28-8%29
  • Zero or negative arguments to logarithm functions

Your expression has two of these:
  • A denominator. It is hidden in the tan function! Since tan%28x%29+=+%28sin%28x%29%29%2F%28cos%28x%29%29 we cannot allow x values which make cos(x) be zero!
    cos(x) = 0 when x = pi%2F2, 3%2Api%2F2, 5%2Api%2F2, etc. So we must exclude all these values from the domain.
  • An even-numbered root: square root. So we cannot allow tan(x) to be negative.
    tan(x) < 0 when pi%2F2+%3C+x+%3C+pi, 3%2Api%2F2+%3C+x+%3C+2%2Api, 5%2Api%2F2+%3C+x+%3C+3%2Api, .etc.

The domain is rest of the Real numbers: 0+%3C=+x+%3C+pi%2F2, pi+%3C=+x+%3C+3%2Api%2F2, 2%2Api+%3C=+x+%3C+5%2Api%2F2, 3%2Api+%3C=+x+%3C+7%2Api%2F2, etc. Expressing this succinctly and completely is not easy. The domain is:
{ x%3A+2%2An%2Api+%3C=+x+%3C+%282%2An%2B1%29%2A%28pi%2F2%29 } where n is any integer.
Think about various integers. Subsitute them in for n above and see if you can recognize that you get one of the intervals listed or suggested by the "etc." list above.