ANSWER
[tex]-\frac{2\sqrt{3}}{3}[/tex]
EXPLANATION
We want to find the value of:
[tex]\sec(\frac{31\pi}{6})[/tex]
First, we can rewrite the angle by reducing it by 4π i.e. two full rotations of 2π to make the angle between 0 and 2π. The angle then becomes:
[tex]\sec(\frac{7\pi}{6})[/tex]
This is the same as:
[tex]\frac{1}{\cos(\frac{7\pi}{6})}[/tex]
Now, we can apply the reference angle by finding the equivalent value of the angle in the first quadrant:
[tex]\frac{1}{\cos(\frac{7\pi}{6})}\Rightarrow\frac{1}{-\cos(\frac{\pi}{6})}[/tex]
The value is negative because the reference angle is in the third quadrant and cosine is negative in the third quadrant.
Now, solving this, we have:
[tex]\begin{gathered} \frac{1}{-\frac{\sqrt{3}}{2}}\Rightarrow-\frac{2}{\sqrt{3}} \\ \\ -\frac{2}{\sqrt{3}}*\frac{\sqrt{3}}{\sqrt{3}} \\ \\ -\frac{2\sqrt{3}}{3} \end{gathered}[/tex]
That is the answer.
