The function is given to be:
[tex]f(x)=\cot (x+\frac{\pi}{6})[/tex]
A full period of a tan graph is π. Hence, we will make sure our table of values will cover more than π units for the x-axis.
To get the first value we can use in the table of values, we will equate the function to 0 and solve for x:
[tex]\begin{gathered} \cot (x+\frac{\pi}{6})=0 \\ x+\frac{\pi}{6}=\cot ^{-1}0 \\ x+\frac{\pi}{6}=\frac{\pi}{2} \\ \therefore \\ x=\frac{\pi}{2}-\frac{\pi}{6} \\ x=\frac{\pi}{3} \end{gathered}[/tex]
We will take intervals of π/6, such that we will use values of x to be:
[tex]\begin{gathered} x_1=\frac{\pi}{3}+\frac{\pi}{6}=\frac{\pi}{2} \\ x_2=\frac{\pi}{2}+\frac{\pi}{6}=\frac{2\pi}{3} \\ x_3=\frac{2\pi}{3}+\frac{\pi}{6}=\frac{5\pi}{6} \\ x_4=\frac{5\pi}{6}+\frac{\pi}{6}=\pi \\ x_5=\pi+\frac{\pi}{6}=\frac{7\pi}{6} \\ x_6=\frac{7\pi}{6}+\frac{\pi}{6}=\frac{4\pi}{3} \end{gathered}[/tex]
If we substitute these values into the function, we can prepare a table as shown below:
Note that there is a vertical asymptote at x = 5π/6.
Using this table, the graph is drawn as shown below:
