Question

Use the triangle shown on the unit circle to find tanπ3 . (1 point) Responses 1 1 3√2 Start Fraction Start Root 3 End Root over 2 End Fraction 3–√ Start Root 3 End Root 12

Answer 1

To find \(\tan\left(\frac{\pi}{3}\right)\) using the unit circle, we first identify the coordinates of the point on the unit circle corresponding to \(\frac{\pi}{3}\).

The angle \(\frac{\pi}{3}\) radians corresponds to \(60^\circ\). The coordinates of this angle on the unit circle are:
\[
\left(\cos\left(\frac{\pi}{3}\right), \sin\left(\frac{\pi}{3}\right)\right)
\]
where:
- \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\)
- \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\)

The tangent function is defined as:
\[
\tan\theta = \frac{\sin\theta}{\cos\theta}
\]

Now, substituting in our values:
\[
\tan\left(\frac{\pi}{3}\right) = \frac{\sin\left(\frac{\pi}{3}\right)}{\cos\left(\frac{\pi}{3}\right)} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}
\]

When you divide these fractions:
\[
\tan\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \cdot \frac{2}{1} = \sqrt{3}
\]

Thus, the value of \(\tan\left(\frac{\pi}{3}\right)\) is:
\[
\sqrt{3}
\]

If you compare that with your provided options, the correct answer is \(\sqrt{3}\).