document.write( "Question 1089899: Find the domain:\r
\n" ); document.write( "\n" ); document.write( "Y= sqrt. tan(x) \n" ); document.write( "
Algebra.Com's Answer #704413 by MathLover1(20850)\"\" \"About 
You can put this solution on YOUR website!

\n" ); document.write( "Because of the periodic nature of tan(x), this is a tough one.

\n" ); document.write( "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

\n" ); document.write( "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!
    \n" ); document.write( "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.
    \n" ); document.write( "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.

\n" ); document.write( "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:
\n" ); document.write( "{ \"x%3A+2%2An%2Api+%3C=+x+%3C+%282%2An%2B1%29%2A%28pi%2F2%29\" } where n is any integer.
\n" ); document.write( "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.
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