Find the exact values of $ sin( heta) $ and $ cos( heta) $ for $ heta = frac{pi}{4} $

To find the exact values of $ \sin(\theta) $ and $ \cos(\theta) $ for $ \theta = \frac{\pi}{4} $, we use the unit circle definition:

On the unit circle, the coordinates of the point corresponding to $ \theta = \frac{\pi}{4} $ are:

$ ( \cos(\frac{\pi}{4}), \sin(\frac{\pi}{4}) ) $

For $ \theta = \frac{\pi}{4} $, both $ \sin(\frac{\pi}{4}) $ and $ \cos(\frac{\pi}{4}) $ are:

$ \frac{\sqrt{2}}{2} $

Thus, we have:

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

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

Using the unit circle, the coordinates for $ heta = frac{pi}{4} $ are:

$ ( cos(frac{pi}{4}), sin(frac{pi}{4}) ) $

Both values are:

$ frac{sqrt{2}}{2} $

So,

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

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

From the unit circle,

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

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

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