An angle's reference angle is the measure of the smallest, positive, acute angle t formed by the terminal side of the angle t and the horizontal axis.
Given the value of x to be
[tex]x=\frac{-31\pi}{6}[/tex]
To get the reference angle of x, we will need to add an even multiple of pi to x
So we will have
[tex]6\pi+(-\frac{31\pi}{6})=\frac{5\pi}{6}[/tex]
We have the reference angle to be:
[tex]\frac{5\pi}{6}[/tex]
Part B
To find sin x
we will have
[tex]\begin{gathered} \\ \sin x=\sin \frac{5\pi}{6}=\frac{1}{2} \\ \text{Therefore } \\ \sin x=\frac{1}{2} \end{gathered}[/tex]
Similarly, we will have tan x to be
[tex]\begin{gathered} \tan x=\tan \frac{5\pi}{6}=-\frac{\sqrt[]{3}}{3} \\ \end{gathered}[/tex]
For sec x
[tex]\begin{gathered} \sec x=\frac{1}{\cos x} \\ so\text{ let us check cos x} \\ \cos x=\cos \frac{5\pi}{6}=-\frac{\sqrt[]{3}}{2} \\ \\ \text{Next,}we\text{ can compute} \\ \sec x=\frac{1}{\cos x}=\frac{1}{-\frac{\sqrt[]{3}}{2}}=-\frac{2}{\sqrt[]{3}} \end{gathered}[/tex]
Simplifying further
[tex]\sec x=\frac{-2\sqrt[]{3}}{3}[/tex]
