The period of a tangent function y = atan(bx) is the distance between any two consecutive vertical asymptotes. And it is given by:
[tex]period=\frac{\pi}{|b|}[/tex]
So, given the function y = tan x, we have that b = 1, therefore the period is:
[tex]\text{period}=\frac{\pi}{|1|}=\pi[/tex]
Next, between the given points:
[tex]\begin{gathered} -\frac{5\pi}{2}=-2.5\pi \\ and \\ \frac{7\pi}{2}=3.5\pi \end{gathered}[/tex]
There are:
[tex]3.5\pi-(-2.5\pi)=6\pi[/tex]
Since the period is π, so:
[tex]\frac{6\pi}{\pi}=6[/tex]
Answer: 6
