To find the value of [tex]\(\sec \frac{13 \pi}{6}\)[/tex], let's proceed step-by-step.
1. Convert the given angle to a more familiar range:
[tex]\[
\frac{13 \pi}{6}
\][/tex]
Since [tex]\(\frac{12 \pi}{6} = 2\pi\)[/tex], we have:
[tex]\[
\frac{13 \pi}{6} = 2\pi + \frac{\pi}{6}
\][/tex]
Thus, [tex]\(\frac{13 \pi}{6}\)[/tex] is equivalent to [tex]\(\frac{\pi}{6}\)[/tex] in terms of periodicity because [tex]\(2\pi\)[/tex] is a full rotation in radians.
2. Evaluate the secant function:
The secant function is defined as:
[tex]\[
\sec x = \frac{1}{\cos x}
\][/tex]
Therefore:
[tex]\[
\sec \frac{13 \pi}{6} = \sec \left(\frac{\pi}{6}\right)
\][/tex]
3. Determine [tex]\(\cos \frac{\pi}{6}\)[/tex]:
From trigonometric values, we know:
[tex]\[
\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}
\][/tex]
4. Calculate the secant value:
[tex]\[
\sec \frac{\pi}{6} = \frac{1}{\cos \frac{\pi}{6}} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}
\][/tex]
5. Numerical approximation:
The numerical value of [tex]\(\sec \frac{\pi}{6}\)[/tex] turns out to be approximately:
[tex]\[
1.1547005383792517
\][/tex]
So, the value of [tex]\(\sec \frac{13 \pi}{6}\)[/tex] is indeed:
[tex]\[
1.1547005383792517
\][/tex]
