### Instruction: What is the period of $y = \tan \frac{x}{2}$?### Response: To find the period of a trigonometric function, I need to figure out how much I have to shift the input variable x by to get the same output value y.
For the standard tangent function y = tan x, the period is pi, because tan (x + pi) = tan x for any x.
But here, I have a different function, y = tan (x/2).
This means that the input x is scaled down by a factor of 2 before it goes into the tangent function.
So, if I want to shift x by some amount T to get the same output, I need to have tan ((x + T)/2) = tan (x/2) for any x.
This means that (x + T)/2 and x/2 must differ by a multiple of pi, since that's the period of the standard tangent function.
In other words, there must be some integer n such that (x + T)/2 = x/2 + n pi.
Solving for T, I get T = 2 n pi.
But I want the smallest positive value of T that works, so I choose n = 1.
This gives me T = 2 pi.