Given a point P on the unit circle at an angle of $ heta = frac{3pi}{4} $, find the coordinates of point P. Then, determine the value of $ cos(2 heta) $ and $ sin(2 heta) $.

When $ \theta = \frac{3\pi}{4} $, the coordinates of point P on the unit circle are given by $ (\cos(\theta), \sin(\theta)) $.

First, we need to calculate these values:

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

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

So, the coordinates of point P are $ \left( -\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) $.

Next, we determine $ \cos(2\theta) $ and $ \sin(2\theta) $ using the double-angle formulas:

$ \cos(2\theta) = \cos(2 \cdot \frac{3\pi}{4}) = \cos \left( \frac{6\pi}{4} \right) = \cos \left( \frac{3\pi}{2} \right) = 0 $

$ \sin(2\theta) = \sin(2 \cdot \frac{3\pi}{4}) = \sin \left( \frac{6\pi}{4} \right) = \sin \left( \frac{3\pi}{2} \right) = -1 $

We start by locating point P on the unit circle at an angle of $ heta = frac{3pi}{4} $.

The coordinates are:

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

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

Thus, point P is $ left( -frac{sqrt{2}}{2}, frac{sqrt{2}}{2}
ight) $.

We then use the double-angle identities:

$ cos(2 heta) = cos(2 cdot frac{3pi}{4}) = cos left( frac{3pi}{2}
ight) = 0 $

$ sin(2 heta) = sin(2 cdot frac{3pi}{4}) = sin left( frac{3pi}{2}
ight) = -1 $

Coordinates of P at $ heta = frac{3pi}{4} $ are:

$ left( -frac{sqrt{2}}{2}, frac{sqrt{2}}{2}
ight) $

For $ cos(2 heta) $ and $ sin(2 heta) $:

$ cos(2cdotfrac{3pi}{4}) = 0 $

$ sin(2cdotfrac{3pi}{4}) = -1 $

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