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[tex]$
\cos \left(\frac{7 \pi}{6}\right)=
$[/tex]
\qquad
A. [tex]$-\frac{1}{2}$[/tex]

\qquad
B. [tex]$-\frac{\sqrt{3}}{2}$[/tex]

\qquad
C. [tex]$\frac{1}{2}$[/tex]

\qquad
D. [tex]$\frac{\sqrt{3}}{2}$[/tex]


Sagot :

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].
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