Questions: Use the figure of the first quadrant of the unit circle to find the exact circular function value. tan(pi/6) tan(pi/6)= (Simplify your answer, including any radicals. Use integ fractions for any numbers in the expression.)

$Use the figure of the first quadrant of the unit circle to find the exact circular function value. tan(pi/6) tan(pi/6)= (Simplify your answer, including any radicals. Use integ fractions for any numbers in the expression.)$

Transcript text: Use the figure of the first quadrant of the unit circle to find the exact circular function value. \[ \tan \frac{\pi}{6} \] \[ \tan \frac{\pi}{6}= \] (Simplify your answer, including any radicals. Use integ fractions for any numbers in the expression.)

Solution

Solution Steps

Step 1: Find the coordinates associated with $\frac{\pi}{6}$ on the unit circle.

On the unit circle, the angle $\frac{\pi}{6}$ corresponds to the point $(\frac{\sqrt{3}}{2}, \frac{1}{2})$.

Step 2: Calculate the tangent.

The tangent of an angle is defined as the ratio of the y-coordinate to the x-coordinate of the point on the unit circle corresponding to that angle. Therefore, $$ \tan\left(\frac{\pi}{6}\right) = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{2} \cdot \frac{2}{\sqrt{3}} = \frac{1}{\sqrt{3}}$$

Step 3: Rationalize the denominator

Multiplying the numerator and denominator by $\sqrt{3}$ yields $$\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}.$$