Use reference angles to find the exact value of the foliowing expresision $$ an \frac{7 \pi}{3} $$ Select the correct chaice below and if necessary, fill in the answer box to camptere your chotce A. $$ \frac{7 \pi}{3}= $$

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Use reference angles to find the exact value of the foliowing expresision $$ 	an \frac{7 \pi}{3} $$ Select the correct chaice below and if necessary, fill in the answer box to camptere your chotce A. $$ \frac{7 \pi}{3}= $$

UpStudy AI · Accepted Answer

To find the exact value of $$ \tan \frac{7 \pi}{3} $$, we first need to simplify the angle $$ \frac{7 \pi}{3} $$ to find its reference angle.

1. **Convert $$ \frac{7 \pi}{3} $$ to a more manageable angle**: Since $$ \frac{7 \pi}{3} $$ is greater than $$ 2\pi $$, we can subtract $$ 2\pi $$ (which is equivalent to $$ \frac{6\pi}{3} $$) to find a coterminal angle.

$$
   \frac{7 \pi}{3} - 2\pi = \frac{7 \pi}{3} - \frac{6 \pi}{3} = \frac{1 \pi}{3} = \frac{\pi}{3}
   $$

2. **Determine the reference angle**: The angle $$ \frac{\pi}{3} $$ is already in the first quadrant, so it is its own reference angle.

3. **Calculate $$ \tan \frac{\pi}{3} $$**: The tangent of $$ \frac{\pi}{3} $$ is a known value.

$$
   \tan \frac{\pi}{3} = \sqrt{3}
   $$

Since $$ \frac{7 \pi}{3} $$ is in the same direction as $$ \frac{\pi}{3} $$ (both angles are positive and in the first quadrant), the tangent value remains the same.

Thus, the exact value of $$ \tan \frac{7 \pi}{3} $$ is:

$$
\boxed{\sqrt{3}}
$$
The exact value of $$ \tan \frac{7 \pi}{3} $$ is $$ \sqrt{3} $$.

Use reference angles to find the exact value of the foliowing expresision \( \tan \frac{7 \pi}{3} \) Select the correct chaice below and if necessary, fill in the answer box to camptere your chotce A. \( \frac{7 \pi}{3}= \)

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