Sure! Let's evaluate [tex]\(\cos \frac{17\pi}{6}\)[/tex] step by step.
1. Understand the Periodicity of Cosine:
The cosine function is periodic with period [tex]\(2\pi\)[/tex]. This means:
[tex]\[
\cos(\theta) = \cos(\theta + 2k\pi)
\][/tex]
for any integer [tex]\(k\)[/tex].
2. Simplify the Given Angle:
To simplify the expression [tex]\(\frac{17\pi}{6}\)[/tex], we can subtract multiples of [tex]\(2\pi\)[/tex]:
[tex]\[
\frac{17\pi}{6} - 2\pi = \frac{17\pi}{6} - \frac{12\pi}{6} = \frac{5\pi}{6}
\][/tex]
So, [tex]\(\cos \frac{17\pi}{6} = \cos \frac{5\pi}{6}\)[/tex].
3. Evaluate the Simplified Angle:
Now, we need to evaluate [tex]\(\cos \frac{5\pi}{6}\)[/tex].
From the unit circle:
- The angle [tex]\(\frac{5\pi}{6}\)[/tex] is in the second quadrant.
- In the second quadrant, the cosine of an angle is negative.
- The reference angle for [tex]\(\frac{5\pi}{6}\)[/tex] is [tex]\(\pi - \frac{5\pi}{6} = \frac{\pi}{6}\)[/tex].
We know that [tex]\(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\)[/tex].
Since [tex]\(\frac{5\pi}{6}\)[/tex] is in the second quadrant where cosine is negative:
[tex]\[
\cos \frac{5\pi}{6} = -\cos \frac{\pi}{6} = -\frac{\sqrt{3}}{2}
\][/tex]
4. Conclusion:
Therefore, the value of [tex]\(\cos \frac{17\pi}{6}\)[/tex] is:
[tex]\[
\cos \frac{17\pi}{6} = -\frac{\sqrt{3}}{2}
\][/tex]
So, the correct answer is [tex]\(-\frac{\sqrt{3}}{2}\)[/tex].
