Find the cosine and sine of the angle $ frac{5pi}{6} $ using the unit circle.

To find the cosine and sine of the angle $ \frac{5\pi}{6} $, we can use the unit circle. The angle $ \frac{5\pi}{6} $ is in the second quadrant, where the cosine is negative and the sine is positive.

First, find the reference angle:

$ \text{Reference angle} = \pi – \frac{5\pi}{6} = \frac{\pi}{6} $

For the angle $ \frac{\pi}{6} $, cosine and sine values are:

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

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

Since $ \frac{5\pi}{6} $ is in the second quadrant:

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

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

The angle $ frac{5pi}{6} $ lies in the second quadrant. In this quadrant, cosine is negative and sine is positive.

To determine the reference angle:

$ ext{Reference angle} = pi – frac{5pi}{6} = frac{pi}{6} $

Values for $ frac{pi}{6} $ are:

$ cos left( frac{pi}{6}
ight) = frac{sqrt{3}}{2} $

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

Therefore, in the second quadrant:

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

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

For $ frac{5pi}{6} $, which is in the second quadrant:

$ ext{Reference angle} = pi – frac{5pi}{6} = frac{pi}{6} $

Since $ cos left( frac{pi}{6}
ight) = frac{sqrt{3}}{2} $ and $ sin left( frac{pi}{6}
ight) = frac{1}{2} $:

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

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

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