Question

$f\left(x\right)=\cot x$

(a) Solve $f\left(x\right)=-\sqrt{3}$.

(b) For what values of x is $f\left(x\right)>-\sqrt{3}$ on the interval $[0,\mathrm{\pi }]$?

Short answer

(a) The solution is $\left\{x|x=-\frac{\mathrm{\pi }}{6}+n\mathrm{\pi },\mathrm{n}\mathrm{is}\mathrm{an}\mathrm{integer}\right\}$.

(b) The solution set is $\left\{-\frac{\mathrm{\pi }}{6},\frac{2\mathrm{\pi }}{3}\right\}$.

Step-by-step solution

Step 1. Given Information 

$f\left(x\right)=\cot x$

(a) Solve $f\left(x\right)=-\sqrt{3}$

(b) For what values of x is $f\left(x\right)>-\sqrt{3}$on the interval $[0,\mathrm{\pi }]$?

Part (a) Step 1. The given equation is f(x)=cotx

Putting $f\left(x\right)=-\sqrt{3}$

$$\begin{gathered} -\sqrt{3}=\cot x \\ \cot x=-\sqrt{3} \end{gathered}$$

Part (a) Step 2. The solution for f(x)=-3.

$\theta =-\frac{\mathrm{\pi }}{6}$

The solution is $\left\{x|x=-\frac{\mathrm{\pi }}{6}+n\mathrm{\pi },\mathrm{n}\mathrm{is}\mathrm{an}\mathrm{integer}\right\}$.

Step (b) Part 1. The values of x is f(x)>-3 on the interval [0,π]

$\theta =-\frac{\mathrm{\pi }}{6},\frac{2\mathrm{\pi }}{3}$

The solution set is$\left\{-\frac{\mathrm{\pi }}{6},\frac{2\mathrm{\pi }}{3}\right\}$.