find the exact values below. if applicable, click on \undefined.\ cot 31π/6 = sec(-450°) =

Explanation:

Step1: Simplify $\cot\frac{31\pi}{6}$

First, find an equivalent angle. $\frac{31\pi}{6}=5\pi+\frac{\pi}{6}$. Since $\cot(x + n\pi)=\cot x$ for integer $n$, $\cot\frac{31\pi}{6}=\cot(5\pi+\frac{\pi}{6})=\cot\frac{\pi}{6}$. And $\cot\theta=\frac{\cos\theta}{\sin\theta}$, so $\cot\frac{\pi}{6}=\frac{\cos\frac{\pi}{6}}{\sin\frac{\pi}{6}}=\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\sqrt{3}$.

Step2: Simplify $\sec(- 450^{\circ})$

Use the property $\sec(-x)=\sec x$, so $\sec(-450^{\circ})=\sec450^{\circ}$. Also, $450^{\circ}=360^{\circ}+90^{\circ}$, and $\sec(x + 360^{\circ})=\sec x$, then $\sec450^{\circ}=\sec(360^{\circ}+90^{\circ})=\sec90^{\circ}$. Since $\sec\theta=\frac{1}{\cos\theta}$ and $\cos90^{\circ}=0$, $\sec90^{\circ}$ is undefined.

Answer:

$\cot\frac{31\pi}{6}=\sqrt{3}$ $\sec(-450^{\circ})=\text{Undefined}$