Question 1089899

Because of the periodic nature of tan(x), this is a tough one.<br>
When finding domains one looks for things that are undefined in the set of Real numbers. Among the things to avoid are:<ul><li>Zeros in denominators</li><li>Negative radicands of even-numbered roots. For example: {{{sqrt(-8)}}}</li><li>Zero or negative arguments to logarithm functions</li></ul>
Your expression has two of these:<ul><li>A denominator. It is hidden in the tan function! Since {{{tan(x) = (sin(x))/(cos(x))}}} we cannot allow x values which make cos(x) be zero!
cos(x) = 0 when x = {{{pi/2}}}, {{{3*pi/2}}}, {{{5*pi/2}}}, etc. So we must exclude all these values from the domain.</li><li>An even-numbered root: square root. So we cannot allow tan(x) to be negative.
tan(x) < 0 when {{{pi/2 < x < pi}}}, {{{3*pi/2 < x < 2*pi}}}, {{{5*pi/2 < x < 3*pi}}}, .etc.</li></ul>
The domain is rest of the Real numbers: {{{0 <= x < pi/2}}}, {{{pi <= x < 3*pi/2}}}, {{{2*pi <= x < 5*pi/2}}}, {{{3*pi <= x < 7*pi/2}}}, etc. Expressing this succinctly and completely is not easy. The domain is:
{ {{{x: 2*n*pi <= x < (2*n+1)*(pi/2)}}} } 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.