Question

use the figure of the first quadrant of the unit circle to find the exact circular function value. $\tan\frac{\pi}{3}$ $\tan\frac{\pi}{3}=\square$ (simplify your answer, including any ra

Explanation:

Step1: Recall the formula for tangent

The formula for the tangent of an angle in the unit circle is $\tan\theta=\frac{y}{x}$, where $(x,y)$ is the point on the unit - circle corresponding to the angle $\theta$.

Step2: Identify the point for $\theta = \frac{\pi}{3}$

For the angle $\theta=\frac{\pi}{3}$, the point on the unit circle is $(\frac{1}{2},\frac{\sqrt{3}}{2})$. So, $x = \frac{1}{2}$ and $y=\frac{\sqrt{3}}{2}$.

Step3: Calculate $\tan\frac{\pi}{3}$

Substitute $x$ and $y$ into the tangent formula: $\tan\frac{\pi}{3}=\frac{y}{x}=\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}$.
When dividing by a fraction, we multiply by its reciprocal. So, $\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\frac{\sqrt{3}}{2}\times\frac{2}{1}=\sqrt{3}$.

Answer:

$\sqrt{3}$