Determine the values of trigonometric functions for specific angles on the unit circle

Given the angle $ \theta = \frac{5\pi}{4} $ radians, find the values of $ \sin(\theta) $, $ \cos(\theta) $, and $ \tan(\theta) $.

The coordinates of $ \frac{5\pi}{4} $ on the unit circle are $ \left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right) $.

So, $ \sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} $, $ \cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} $, and

$ \tan\left(\frac{5\pi}{4}\right) = \frac{\sin\left(\frac{5\pi}{4}\right)}{\cos\left(\frac{5\pi}{4}\right)} = 1. $

Given the angle $ heta = frac{7pi}{6} $ radians, find the values of $ sin( heta) $, $ cos( heta) $, and $ an( heta) $.

The coordinates of $ frac{7pi}{6} $ on the unit circle are $ left(-frac{sqrt{3}}{2}, -frac{1}{2}
ight) $.

So, $ sinleft(frac{7pi}{6}
ight) = -frac{1}{2} $, $ cosleft(frac{7pi}{6}
ight) = -frac{sqrt{3}}{2} $, and

$ anleft(frac{7pi}{6}
ight) = frac{sinleft(frac{7pi}{6}
ight)}{cosleft(frac{7pi}{6}
ight)} = frac{1}{sqrt{3}}. $

Given the angle $ heta = frac{3pi}{2} $ radians, find the values of $ sin( heta) $, $ cos( heta) $, and $ an( heta) $.

The coordinates of $ frac{3pi}{2} $ on the unit circle are $ left(0, -1
ight) $.

So, $ sinleft(frac{3pi}{2}
ight) = -1 $, $ cosleft(frac{3pi}{2}
ight) = 0 $, and

$ anleft(frac{3pi}{2}
ight) $ is undefined.

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