To find the exact value of [tex]\( \cos \left(\frac{7 \pi}{6}\right) \)[/tex] using the unit circle, we'll go through a few steps. Here’s a detailed step-by-step solution:
1. Understanding the angle: The angle given is [tex]\( \frac{7 \pi}{6} \)[/tex]. Let's first convert this angle to degrees for better understanding:
[tex]\[
\frac{7 \pi}{6} \text{ radians} \times \frac{180^\circ}{\pi} = \frac{7 \times 180^\circ}{6} = 210^\circ
\][/tex]
2. Locate the angle on the unit circle: The angle [tex]\( 210^\circ \)[/tex] lies in the third quadrant. Angles in the third quadrant are between [tex]\( 180^\circ \)[/tex] and [tex]\( 270^\circ \)[/tex].
3. Reference angle: To find the cosine of [tex]\( 210^\circ \)[/tex], we determine its reference angle. The reference angle is the smallest angle between the terminal side of the given angle and the x-axis.
[tex]\[
210^\circ - 180^\circ = 30^\circ
\][/tex]
Thus, the reference angle is [tex]\( 30^\circ \)[/tex].
4. Cosine of reference angle: From trigonometric values, we know that
[tex]\[
\cos 30^\circ = \frac{\sqrt{3}}{2}
\][/tex]
5. Sign of cosine in the third quadrant: In the third quadrant, the cosine value is negative because the x-coordinates of points in this quadrant are negative.
6. Conclusion: Given that the cosine of the reference angle [tex]\( 30^\circ \)[/tex] is [tex]\( \frac{\sqrt{3}}{2} \)[/tex], and noting that cosine is negative in the third quadrant,
[tex]\[
\cos \left( 210^\circ \right) = \cos \left( \frac{7 \pi}{6} \right) = -\frac{\sqrt{3}}{2}
\][/tex]
Thus, the exact value of [tex]\( \cos \left( \frac{7 \pi}{6} \right) \)[/tex] is [tex]\( -\frac{\sqrt{3}}{2} \)[/tex].
