To determine the cosine of [tex]\( \frac{7\pi}{6} \)[/tex], let's carefully walk through the steps to find the correct answer.
1. Understanding the Angle:
The given angle is [tex]\( \frac{7\pi}{6} \)[/tex]. This is an angle in radians. To get a better conceptual understanding, let's place this angle in the appropriate quadrant.
- Converting [tex]\( \frac{7\pi}{6} \)[/tex] to degrees:
[tex]\[
\frac{7\pi}{6} \times \frac{180^\circ}{\pi} = \frac{7 \times 180^\circ}{6} = 210^\circ
\][/tex]
2. Locating the Angle:
An angle of [tex]\( 210^\circ \)[/tex] lies in the third quadrant of the unit circle, where both sine and cosine are negative.
3. Reference Angle:
The reference angle for [tex]\( 210^\circ \)[/tex] in the third quadrant is:
[tex]\[
210^\circ - 180^\circ = 30^\circ
\][/tex]
So, [tex]\( \frac{7\pi}{6} \)[/tex] has a reference angle of [tex]\( \frac{\pi}{6} \)[/tex].
4. Cosine Values in Third Quadrant:
The cosine of the reference angle ([tex]\( \frac{\pi}{6} \)[/tex]) is [tex]\( \frac{\sqrt{3}}{2} \)[/tex]. However, since the angle [tex]\( \frac{7\pi}{6} \)[/tex] is in the third quadrant where cosine is negative, we take the negative value of the cosine of the reference angle.
[tex]\[
\cos\left( \frac{7\pi}{6} \right) = -\cos\left( \frac{\pi}{6} \right) = -\frac{\sqrt{3}}{2}
\][/tex]
5. Conclusion:
From the options provided:
[tex]\[
\text{A. } -\frac{1}{2} \\
\text{B. } -\frac{\sqrt{3}}{2} \\
\text{C. } \frac{1}{2} \\
\text{D. } \frac{\sqrt{3}}{2}
\][/tex]
The correct choice matches with:
[tex]\[
\cos\left( \frac{7\pi}{6} \right) = -\frac{\sqrt{3}}{2}
\][/tex]
So the correct answer is B. [tex]\( -\frac{\sqrt{3}}{2} \)[/tex].
