Finding Specific $ an$ Values on the Unit Circle

To find the exact $\tan$ values at specific angles on the unit circle, consider the following:

1. $\theta = \frac{\pi}{4}$
At this angle, $\tan(\theta) = \tan\left(\frac{\pi}{4}\right) = 1$

2. $\theta = \frac{2\pi}{3}$
At this angle, $\tan(\theta) = \tan\left(\frac{2\pi}{3}\right) = -\sqrt{3}$

3. $\theta = \frac{7\pi}{6}$
At this angle, $\tan(\theta) = \tan\left(\frac{7\pi}{6}\right) = \frac{1}{\sqrt{3}}$

To determine the $ an$ value of angles on the unit circle, follow these steps:

1. For $ heta = frac{pi}{6}$, $ anleft(frac{pi}{6}
ight) = frac{1}{sqrt{3}}$

2. For $ heta = frac{3pi}{4}$, $ anleft(frac{3pi}{4}
ight) = -1$

3. For $ heta = frac{5pi}{3}$, $ anleft(frac{5pi}{3}
ight) = -sqrt{3}$

Calculate the $ an$ values for the given angles:

1. $ heta = frac{pi}{3}$: $ anleft(frac{pi}{3}
ight) = sqrt{3}$

2. $ heta = frac{5pi}{6}$: $ anleft(frac{5pi}{6}
ight) = -frac{1}{sqrt{3}}$

3. $ heta = frac{11pi}{6}$: $ anleft(frac{11pi}{6}
ight) = -frac{1}{sqrt{3}}$

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