how many periods of the function y = tan x are there between -\\frac{3\\pi}{4} and \\frac{5\\pi}{4}?

Explanation:

Step1: Recall the period of tangent function

The period of the function $y = \tan x$ is $\pi$.

Step2: Calculate the length of the interval

The length of the interval from $-\frac{3\pi}{4}$ to $\frac{5\pi}{4}$ is $\frac{5\pi}{4}-\left(-\frac{3\pi}{4}\right)=\frac{5\pi + 3\pi}{4}=2\pi$.

Step3: Divide interval length by period

To find the number of periods, divide the length of the interval by the period of the function. So, $\frac{2\pi}{\pi}=2$.

Answer:

2