Find the exact value of the trigonometric functions for the angle $ heta = frac{5pi}{6}$ using the unit circle.

First, locate the angle $\theta = \frac{5\pi}{6}$ on the unit circle. This angle is in the second quadrant.

In the second quadrant, sine is positive and cosine is negative.

The reference angle for $\theta = \frac{5\pi}{6}$ is $\frac{\pi}{6}$.

From the unit circle, $\sin(\frac{\pi}{6}) = \frac{1}{2}$ and $\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}$.

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

Hence, the exact values are:

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

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

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

Locate $ heta = frac{5pi}{6}$ on the unit circle, which corresponds to a second quadrant angle.

The reference angle is $frac{pi}{6}$ where $sin(frac{pi}{6}) = frac{1}{2}$ and $cos(frac{pi}{6}) = frac{sqrt{3}}{2}$.

Thus, in the second quadrant, $sin(frac{5pi}{6}) = frac{1}{2}$ and $cos(frac{5pi}{6}) = -frac{sqrt{3}}{2}$.

So, the values are:

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

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

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

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

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

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

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

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