Find the cosine and sine of an angle on the unit circle at $frac{5pi}{6}$ radians.

To find the cosine and sine values for $\frac{5\pi}{6}$ radians, first recognize that $\frac{5\pi}{6}$ is in the second quadrant of the unit circle.

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

Next, find the reference angle for $\frac{5\pi}{6}$, which is $\pi – \frac{5\pi}{6} = \frac{\pi}{6}$.

We know the sine and cosine values for the angle $\frac{\pi}{6}$:

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

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

Since $\frac{5\pi}{6}$ is in the second quadrant, the cosine value will be negative.

Therefore:

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

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

To find the sine and cosine values for $frac{5pi}{6}$ radians:

1. Recognize that $frac{5pi}{6}$ is in the second quadrant where sine is positive and cosine is negative.

2. Determine the reference angle by subtracting from $pi$: $pi – frac{5pi}{6} = frac{pi}{6}.$

3. Use known values for $frac{pi}{6}$:

$sinleft( frac{pi}{6}
ight) = frac{1}{2},$ $cosleft( frac{pi}{6}
ight) = frac{sqrt{3}}{2}.$

4. Adjust for the second quadrant:

$sinleft( frac{5pi}{6}
ight) = frac{1}{2},$ $cosleft( frac{5pi}{6}
ight) = -frac{sqrt{3}}{2}.$

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

$ ext{Reference angle: } frac{pi}{6}.$

$sinleft( frac{5pi}{6}
ight) = frac{1}{2},$ $cosleft( frac{5pi}{6}
ight) = -frac{sqrt{3}}{2}.$

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