Determine the exact values of $sin(frac{5pi}{6})$ and $cos(frac{5pi}{6})$ in the unit circle

To determine the exact values of $\sin(\frac{5\pi}{6})$ and $\cos(\frac{5\pi}{6})$, we use the unit circle.

For the angle $\frac{5\pi}{6}$, it is in the second quadrant where sine is positive and cosine is negative. The reference angle for $\frac{5\pi}{6}$ is:

$ \pi – \frac{5\pi}{6} = \frac{\pi}{6} $

The values for sine and cosine at $\frac{\pi}{6}$ are known:

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

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

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

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

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

For the angle $frac{5pi}{6}$, it lies in the second quadrant where sine is positive and cosine is negative. The reference angle is:

$ frac{pi}{6} $

Therefore:

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

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

Thus:

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

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

For the angle $frac{5pi}{6}$ in the second quadrant,

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

and

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

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