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Evaluate [tex]\cos \frac{17 \pi}{6}[/tex]

A. [tex]\frac{\sqrt{3}}{2}[/tex]
B. [tex]-\frac{\sqrt{3}}{2}[/tex]
C. [tex]\frac{1}{2}[/tex]
D. [tex]-\frac{1}{2}[/tex]


Sagot :

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].
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