Question

use the triangle shown on the unit circle to find (\tan \frac{pi}{3}). (1 point)

Explanation:

Step1: Recall the definition of tangent

In a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. For an angle \(\theta\) in a right triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).

Step2: Identify the opposite and adjacent sides for \(\frac{\pi}{3}\)

From the unit circle triangle, for the angle \(\frac{\pi}{3}\) (or \(60^\circ\)):

• The length of the opposite side (relative to \(\frac{\pi}{3}\)) is \(\frac{\sqrt{3}}{2}\) (the \(y\)-coordinate of point \(Y\)).
• The length of the adjacent side (relative to \(\frac{\pi}{3}\)) is \(\frac{1}{2}\) (the \(x\)-coordinate of point \(Z\), which is the adjacent side to the angle at \(X\)).

Step3: Calculate \(\tan\frac{\pi}{3}\)

Using the tangent ratio: \(\tan\frac{\pi}{3}=\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}\)
Simplify the fraction: \(\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\frac{\sqrt{3}}{2}\times\frac{2}{1}=\sqrt{3}\)

Answer:

\(\sqrt{3}\) (Assuming one of the options is \(\sqrt{3}\), for example, if the options are as follows (from the partial view): if one of the options is "B. \(\sqrt{3}\)" then the answer is B. \(\sqrt{3}\))