Determine the values of $sin$ and $cos$ at a given angle on the unit circle

Given an angle of $\theta = \frac{5\pi}{6}$, determine the values of $\sin(\theta)$ and $\cos(\theta)$ using the unit circle.

First, recognize that $\theta = \frac{5\pi}{6}$ is located in the second quadrant.

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

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

Knowing the values from the unit circle:

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

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

Since $\theta = \frac{5\pi}{6}$ lies in the second quadrant, we have:

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

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

Consider the angle $ heta = frac{7pi}{4}$. Calculate $sin( heta)$ and $cos( heta)$ using the unit circle.

First, note that $ heta = frac{7pi}{4}$ is in the fourth quadrant.

In the fourth quadrant, the sine value is negative and the cosine value is positive.

The reference angle for $frac{7pi}{4}$ is: $2pi – frac{7pi}{4} = frac{pi}{4}$.

From the unit circle values:

$sinleft(frac{pi}{4}
ight) = frac{sqrt{2}}{2}$

$cosleft(frac{pi}{4}
ight) = frac{sqrt{2}}{2}$

Since $ heta = frac{7pi}{4}$ is in the fourth quadrant, we have:

$sinleft(frac{7pi}{4}
ight) = -frac{sqrt{2}}{2}$

$cosleft(frac{7pi}{4}
ight) = frac{sqrt{2}}{2}$

For the angle $ heta = frac{2pi}{3}$, find $sin( heta)$ and $cos( heta)$ on the unit circle.

Since $ heta = frac{2pi}{3}$ is in the second quadrant:

$sinleft(frac{2pi}{3}
ight) = sinleft(pi – frac{pi}{3}
ight) = sinleft(frac{pi}{3}
ight) = frac{sqrt{3}}{2}$

$cosleft(frac{2pi}{3}
ight) = cosleft(pi – frac{pi}{3}
ight) = -cosleft(frac{pi}{3}
ight) = -frac{1}{2}$

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