Question

use the image to answer the question.
diagram: unit circle with a right triangle, point ((\frac{sqrt{3}}{2}, \frac{1}{2})), legs (\frac{sqrt{3}}{2}) (x-axis) and (\frac{1}{2}) (vertical), hypotenuse 1, angle (\theta) at origin
use the triangle shown on the unit circle to find (\tan\theta)
(1 point)
multiple-choice options (partially visible: (\frac{sqrt{3}}{3}), (1), (sqrt{3}), etc.)

Explanation:

Step1: Recall the definition of tangent

In a right triangle, the tangent of an angle \( \theta \) is defined as the ratio of the opposite side to the adjacent side, i.e., \( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \).

Step2: Identify the opposite and adjacent sides

From the unit circle triangle, the length of the opposite side to \( \theta \) is \( \frac{1}{2} \) and the length of the adjacent side is \( \frac{\sqrt{3}}{2} \).

Step3: Calculate \( \tan\theta \)

Substitute the values into the formula: \( \tan\theta = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} \). When dividing fractions, we multiply by the reciprocal, so \( \tan\theta=\frac{1}{2}\times\frac{2}{\sqrt{3}}=\frac{1}{\sqrt{3}} \). Rationalizing the denominator, we get \( \tan\theta = \frac{\sqrt{3}}{3} \) (since \( \frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3} \) after multiplying numerator and denominator by \( \sqrt{3} \)). Wait, but let's check again. Wait, maybe I misread the options. Wait, the point is \( (\frac{\sqrt{3}}{2},\frac{1}{2}) \), so in the right triangle, the opposite side (y - coordinate) is \( \frac{1}{2} \), adjacent (x - coordinate) is \( \frac{\sqrt{3}}{2} \). So \( \tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3} \). But looking at the options, maybe there was a typo? Wait, no, maybe I made a mistake. Wait, no, the options: let's see, the first option is \( \frac{\sqrt{3}}{3} \)? Wait, the user's options: let's re - check. The options are: \( \frac{\sqrt{3}}{3} \), 1, \( \sqrt{3} \), \( \frac{\sqrt{3}}{1} \)? Wait, no, the last option is \( \frac{\sqrt{3}}{3} \)? Wait, no, the original problem's options: maybe I misread. Wait, the point is \( (\frac{\sqrt{3}}{2},\frac{1}{2}) \), so \( x = \frac{\sqrt{3}}{2} \), \( y=\frac{1}{2} \). In the unit circle, for an angle \( \theta \), \( \tan\theta=\frac{y}{x} \). So \( \tan\theta=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3} \). So the correct option is the one with \( \frac{\sqrt{3}}{3} \).

Answer:

\( \boldsymbol{\frac{\sqrt{3}}{3}} \) (assuming the first option is \( \frac{\sqrt{3}}{3} \))