Math / Trigonometry

QuestionQuestion 14 (1 point) Determine the exact value of $\tan \frac{7 \pi}{6}$ . a) $-\frac{1}{\sqrt{3}}$ b) $\frac{1}{\sqrt{3}}$ C) $-\frac{\sqrt{3}}{2}$ d) $\frac{\sqrt{3}}{2}$

Studdy Solution

STEP 1

1. We are asked to find the exact value of $\tan \frac{7\pi}{6}$ .
2. The angle $\frac{7\pi}{6}$ is in radians.
3. The options provided are possible values for $\tan \frac{7\pi}{6}$ .

STEP 2

1. Identify the reference angle for $\frac{7\pi}{6}$ .
2. Determine the quadrant in which $\frac{7\pi}{6}$ lies.
3. Use the reference angle and quadrant to find the exact value of $\tan \frac{7\pi}{6}$ .

STEP 3

Identify the reference angle for $\frac{7\pi}{6}$ :
The angle $\frac{7\pi}{6}$ is equivalent to $210^\circ$ in degrees. The reference angle is the angle it makes with the x-axis, which is:
$210^\circ - 180^\circ = 30^\circ$
Thus, the reference angle is $\frac{\pi}{6}$ .

STEP 4

Determine the quadrant in which $\frac{7\pi}{6}$ lies:
Since $\frac{7\pi}{6}$ is greater than $\pi$ but less than $\frac{3\pi}{2}$ , it lies in the third quadrant.

STEP 5

Use the reference angle and quadrant to find the exact value of $\tan \frac{7\pi}{6}$ :
In the third quadrant, the tangent function is positive. The tangent of the reference angle $\frac{\pi}{6}$ is:
$\tan \frac{\pi}{6} = \frac{1}{\sqrt{3}}$
Since $\tan$ is positive in the third quadrant:
$\tan \frac{7\pi}{6} = \frac{1}{\sqrt{3}}$
The exact value of $\tan \frac{7\pi}{6}$ is:
$\boxed{\frac{1}{\sqrt{3}}}$

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