Answer:
Step-by-step explanation:
From the above question, we are given a table with various values and we are asked to find the probability where:
-5≤ x ≤ 5
This means we are to find
x ≤ 5 and x ≥ -5
One way to do this is to notice
![\dfrac{17\pi}{12}=\dfrac\pi6+\dfrac{5\pi}4](https://tex.z-dn.net/?f=%5Cdfrac%7B17%5Cpi%7D%7B12%7D%3D%5Cdfrac%5Cpi6%2B%5Cdfrac%7B5%5Cpi%7D4)
Then
![\tan\dfrac{17\pi}{12}=\tan\left(\dfrac\pi6+\dfrac{5\pi}4\right)=\dfrac{\tan\frac\pi6+\tan\frac{5\pi}4}{1-\tan\frac\pi6\tan\frac{5\pi}4}](https://tex.z-dn.net/?f=%5Ctan%5Cdfrac%7B17%5Cpi%7D%7B12%7D%3D%5Ctan%5Cleft%28%5Cdfrac%5Cpi6%2B%5Cdfrac%7B5%5Cpi%7D4%5Cright%29%3D%5Cdfrac%7B%5Ctan%5Cfrac%5Cpi6%2B%5Ctan%5Cfrac%7B5%5Cpi%7D4%7D%7B1-%5Ctan%5Cfrac%5Cpi6%5Ctan%5Cfrac%7B5%5Cpi%7D4%7D)
We have
![\tan\dfrac\pi6=\dfrac{\sin\frac\pi6}{\cos\frac\pi6}=\dfrac{\frac12}{\frac{\sqrt3}2}=\dfrac1{\sqrt3}](https://tex.z-dn.net/?f=%5Ctan%5Cdfrac%5Cpi6%3D%5Cdfrac%7B%5Csin%5Cfrac%5Cpi6%7D%7B%5Ccos%5Cfrac%5Cpi6%7D%3D%5Cdfrac%7B%5Cfrac12%7D%7B%5Cfrac%7B%5Csqrt3%7D2%7D%3D%5Cdfrac1%7B%5Csqrt3%7D)
and since
has a period of
,
![\tan\dfrac{5\pi}4=\tan\left(\pi+\dfrac\pi4\right)=\tan\dfrac\pi4=1](https://tex.z-dn.net/?f=%5Ctan%5Cdfrac%7B5%5Cpi%7D4%3D%5Ctan%5Cleft%28%5Cpi%2B%5Cdfrac%5Cpi4%5Cright%29%3D%5Ctan%5Cdfrac%5Cpi4%3D1)
and so
![\tan\dfrac{17\pi}{12}=\dfrac{\frac1{\sqrt3}+1}{1-\frac1{\sqrt3}}=\dfrac{1+\sqrt3}{\sqrt3-1}=2+\sqrt3](https://tex.z-dn.net/?f=%5Ctan%5Cdfrac%7B17%5Cpi%7D%7B12%7D%3D%5Cdfrac%7B%5Cfrac1%7B%5Csqrt3%7D%2B1%7D%7B1-%5Cfrac1%7B%5Csqrt3%7D%7D%3D%5Cdfrac%7B1%2B%5Csqrt3%7D%7B%5Csqrt3-1%7D%3D2%2B%5Csqrt3)