Question

A curved line with triangle upper X upper Y upper Z inscribed beneath it are plotted on the first quadrant of a coordinate plane.
Use the triangle shown on the unit circle to find tanπ3 .

(1 point)
Responses

3√2
Start Fraction Start Root 3 End Root over 2 End Fraction

1
1

3–√
Start Root 3 End Root

12

Answer 1

To find \(\tan\left(\frac{\pi}{3}\right)\), we can use the unit circle.

On the unit circle, the angle \(\frac{\pi}{3}\) corresponds to \(60^\circ\). The coordinates for this angle on the unit circle are \((\frac{1}{2}, \frac{\sqrt{3}}{2})\).

The tangent of an angle is defined as the ratio of the y-coordinate to the x-coordinate:

\[
\tan\left(\frac{\pi}{3}\right) = \frac{\text{y-coordinate}}{\text{x-coordinate}} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \frac{\sqrt{3}}{2} \cdot \frac{2}{1} = \sqrt{3}
\]

Therefore, the correct value for \(\tan\left(\frac{\pi}{3}\right)\) is:

\[
\sqrt{3}
\]

Among the options provided, the closest representation is:

\[
3-\sqrt{3}
\]

However, it seems there may be a confusion in the options presented. The correct value should simply be \(\sqrt{3}\).