Find the exact values of $sin(frac{3pi}{4})$, $cos(frac{3pi}{4})$, and $ an(frac{3pi}{4})$ using the unit circle.

We are asked to find the exact values of $\sin(\frac{3\pi}{4})$, $\cos(\frac{3\pi}{4})$, and $\tan(\frac{3\pi}{4})$ using the unit circle.

First, we locate the angle $\frac{3\pi}{4}$ on the unit circle: it is in the second quadrant.

The reference angle for $\frac{3\pi}{4}$ is $\frac{\pi}{4}$ (45 degrees). In the second quadrant, the sine value is positive, and the cosine value is negative.

Thus, we have:

$\sin(\frac{3\pi}{4}) = \sin(\pi – \frac{\pi}{4}) = \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$

$\cos(\frac{3\pi}{4}) = \cos(\pi – \frac{\pi}{4}) = -\cos(\frac{\pi}{4}) = -\frac{\sqrt{2}}{2}$

$\tan(\frac{3\pi}{4}) = \frac{\sin(\frac{3\pi}{4})}{\cos(\frac{3\pi}{4})} = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = -1$

By using the unit circle, we need to find the values of $sin(frac{3pi}{4})$, $cos(frac{3pi}{4})$, and $ an(frac{3pi}{4})$.

The angle $frac{3pi}{4}$ is situated in the second quadrant, where sine is positive and cosine is negative.

We can use the reference angle $frac{pi}{4}$:

$sin(frac{3pi}{4}) = sin(pi – frac{pi}{4}) = sin(frac{pi}{4}) = frac{sqrt{2}}{2}$

$cos(frac{3pi}{4}) = cos(pi – frac{pi}{4}) = -cos(frac{pi}{4}) = -frac{sqrt{2}}{2}$

$ an(frac{3pi}{4}) = frac{sin(frac{3pi}{4})}{cos(frac{3pi}{4})} = frac{frac{sqrt{2}}{2}}{-frac{sqrt{2}}{2}} = -1$

To find $sin(frac{3pi}{4})$, $cos(frac{3pi}{4})$, and $ an(frac{3pi}{4})$ using the unit circle:

The angle $frac{3pi}{4}$ is in the second quadrant.

$sin(frac{3pi}{4}) = frac{sqrt{2}}{2}$

$cos(frac{3pi}{4}) = -frac{sqrt{2}}{2}$

$ an(frac{3pi}{4}) = -1$

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