Find the exact trigonometric values of $cosleft(frac{5pi}{6}
ight)$ and $sinleft(frac{5pi}{6}
ight)$ from the unit circle

To find the exact values of $\cos\left(\frac{5\pi}{6}\right)$ and $\sin\left(\frac{5\pi}{6}\right)$, we refer to the unit circle.

For the angle $\frac{5\pi}{6}$:

The reference angle is $\pi – \frac{5\pi}{6} = \frac{\pi}{6}$

On the unit circle, the coordinates for $\frac{\pi}{6}$ are $(\cos(\frac{\pi}{6}), \sin(\frac{\pi}{6}))$ = (\frac{\sqrt{3}}{2}, \frac{1}{2})$

Since $\frac{5\pi}{6}$ is in the second quadrant, $\cos(\frac{5\pi}{6})$ is negative and $\sin(\frac{5\pi}{6})$ is positive:

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

Using the unit circle, we find the reference angle for $frac{5pi}{6}$ is $frac{pi}{6}$. The coordinates for $frac{pi}{6}$ are:

$(cos(frac{pi}{6}), sin(frac{pi}{6})) = (frac{sqrt{3}}{2}, frac{1}{2})$

Because $frac{5pi}{6}$ is in the 2nd quadrant, where cosine is negative and sine is positive:

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

For $frac{5pi}{6}$, the reference angle is $frac{pi}{6}$. The coordinates are:

$(cos(frac{pi}{6}), sin(frac{pi}{6})) = (frac{sqrt{3}}{2}, frac{1}{2})$

Since $frac{5pi}{6}$ is in the second quadrant, where cosine is negative:

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

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