Find the exact values of trigonometric functions at angles on the unit circle

Consider the angles $ \theta = \frac{5\pi}{6} $, $ \theta = \frac{7\pi}{4} $, and $ \theta = \frac{2\pi}{3} $ on the unit circle. We need to find the exact values of $ \sin(\theta) $, $ \cos(\theta) $, and $ \tan(\theta) $ for each angle.

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

$ \sin\left( \frac{5\pi}{6} \right) = \frac{1}{2} $

$ \cos\left( \frac{5\pi}{6} \right) = -\frac{\sqrt{3}}{2} $

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

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

$ \sin\left( \frac{7\pi}{4} \right) = -\frac{\sqrt{2}}{2} $

$ \cos\left( \frac{7\pi}{4} \right) = \frac{\sqrt{2}}{2} $

$ \tan\left( \frac{7\pi}{4} \right) = -1 $

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

$ \sin\left( \frac{2\pi}{3} \right) = \frac{\sqrt{3}}{2} $

$ \cos\left( \frac{2\pi}{3} \right) = -\frac{1}{2} $

$ \tan\left( \frac{2\pi}{3} \right) = -\sqrt{3} $

Consider the angles $ heta = frac{5pi}{6} $, $ heta = frac{7pi}{4} $, and $ heta = frac{2pi}{3} $. Find the exact values of $ sin( heta) $, $ cos( heta) $, and $ an( heta) $ for each.

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

$ sinleft( frac{5pi}{6}
ight) = frac{1}{2} $

$ cosleft( frac{5pi}{6}
ight) = -frac{sqrt{3}}{2} $

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

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

$ sinleft( frac{7pi}{4}
ight) = -frac{sqrt{2}}{2} $

$ cosleft( frac{7pi}{4}
ight) = frac{sqrt{2}}{2} $

$ anleft( frac{7pi}{4}
ight) = -1 $

For $ heta = frac{2pi}{3} $:

$ sinleft( frac{2pi}{3}
ight) = frac{sqrt{3}}{2} $

$ cosleft( frac{2pi}{3}
ight) = -frac{1}{2} $

$ anleft( frac{2pi}{3}
ight) = -sqrt{3} $

Find the exact values of $ sin( heta) $, $

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