Find the exact values of $sin( heta)$ and $cos( heta)$ using the unit circle.

To find the exact values of $\sin(\theta)$ and $\cos(\theta)$ using the unit circle, let’s consider $\theta = \frac{5\pi}{6}$.

First, we know that $\frac{5\pi}{6}$ is in the second quadrant.

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

Using the reference angle $\frac{\pi}{6}$, we have:

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

and

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

To determine $sin( heta)$ and $cos( heta)$ using the unit circle for $ heta = frac{5pi}{6}$, follow these steps:

Identify the quadrant: $frac{5pi}{6}$ is in the second quadrant where sine is positive and cosine is negative.

Find the reference angle: $pi – frac{5pi}{6} = frac{pi}{6}$.

Therefore,

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

and

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

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

In the second quadrant,

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

and

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

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