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 \(\sqrt{3}\).
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