To find the exact value of $\tan\left(\frac{\pi}{3}\right)$, we proceed as follows:

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Step 1: Recall the tangent value for $\frac{\pi}{3}$

The angle $\frac{\pi}{3}$ is one of the standard angles on the unit circle. From trigonometric tables or the properties of the 30-60-90 triangle:

$\tan\left(\frac{\pi}{3}\right) = \sqrt{3}.$

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Step 2: Express the result with a rational denominator

Since the result $\sqrt{3}$ is not in rational form, we express it with a rational denominator:

1. To write $\sqrt{3}$ as a fraction, we multiply by $\frac{\sqrt{3}}{\sqrt{3}}$ (rationalizing the denominator):

$\sqrt{3} = \frac{\sqrt{3} \cdot \sqrt{3}}{\sqrt{3}} = \frac{3}{\sqrt{3}}.$

Thus:

$\tan\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{1} \quad \text{(already simplified).}$

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Final Answer:

The exact value of $\tan\left(\frac{\pi}{3}\right)$ is:

$\boxed{\sqrt{3}}$

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Additional Explanation:

If you want to see this value derived from the sine and cosine of $\frac{\pi}{3}$:

1. The sine and cosine values for $\frac{\pi}{3}$ are:

$\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}, \quad \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}.$

2. The tangent is the ratio of sine to cosine:

$\tan\left(\frac{\pi}{3}\right) = \frac{\sin\left(\frac{\pi}{3}\right)}{\cos\left(\frac{\pi}{3}\right)} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}.$

Simplify:

$\tan\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \cdot \frac{2}{1} = \sqrt{3}.$

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Do you want a deeper explanation of this derivation? Let me know!

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Related Questions:

1. What is the exact value of $\tan\left(\frac{\pi}{4}\right)$?
2. Find $\tan\left(\frac{\pi}{6}\right)$ and express it with a rational denominator.
3. Derive the values of sine and cosine for $\frac{\pi}{3}$ from the unit circle.
4. Prove that $\tan\left(\theta\right) = \frac{\sin\theta}{\cos\theta}$ for any angle $\theta$.
5. Determine $\tan\left(\frac{5\pi}{6}\right)$ using symmetry properties of the unit circle.

Tip:

When working with trigonometric functions, always reference the unit circle and standard triangles (30-60-90 and 45-45-90) for exact values.