Determine the exact values of $ an( heta) $ for $ heta = frac{5pi}{6} $, $ heta = frac{3pi}{4} $, and $ heta = frac{7pi}{4} $ from the unit circle.

To determine the exact values of $ \tan(\theta) $ for the given angles using the unit circle, we need to recall the tangent function and its relation to sine and cosine:

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$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $

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1. For $ \theta = \frac{5\pi}{6} $:

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$ \sin(\frac{5\pi}{6}) = \frac{1}{2}, \quad \cos(\frac{5\pi}{6}) = -\frac{\sqrt{3}}{2} $

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Therefore:

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$ \tan(\frac{5\pi}{6}) = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} $

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2. For $ \theta = \frac{3\pi}{4} $:

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$ \sin(\frac{3\pi}{4}) = \frac{\sqrt{2}}{2}, \quad \cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2} $

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Therefore:

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$ \tan(\frac{3\pi}{4}) = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = -1 $

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3. For $ \theta = \frac{7\pi}{4} $:

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$ \sin(\frac{7\pi}{4}) = -\frac{\sqrt{2}}{2}, \quad \cos(\frac{7\pi}{4}) = \frac{\sqrt{2}}{2} $

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Therefore:

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$ \tan(\frac{7\pi}{4}) = \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = -1 $

To determine the exact values of $ an( heta) $, we use the definition:

$ an( heta) = frac{sin( heta)}{cos( heta)} $

For $ heta = frac{5pi}{6} $:

$ sin(frac{5pi}{6}) = frac{1}{2}, quad cos(frac{5pi}{6}) = -frac{sqrt{3}}{2} $

$ an(frac{5pi}{6}) = frac{1}{2} cdot frac{2}{-sqrt{3}} = -frac{sqrt{3}}{3} $

For $ heta = frac{3pi}{4} $:

$ an(frac{3pi}{4}) = frac{sqrt{2}}{2} cdot frac{2}{-sqrt{2}} = -1 $

For $ heta = frac{7pi}{4} $:

$ an(frac{7pi}{4}) = frac{-sqrt{2}}{2} cdot frac{2}{sqrt{2}} = -1 $

Using the unit circle, we find: