Find the exact values of $ sin( heta) $, $ cos( heta) $, and $ an( heta) $ at $ heta = frac{3pi}{4} $

To find the exact values of $ \sin(\theta) $, $ \cos(\theta) $, and $ \tan(\theta) $ at $ \theta = \frac{3\pi}{4} $, we use the unit circle:

For $ \theta = \frac{3\pi}{4} $, the corresponding point on the unit circle is in the second quadrant where both $ \sin(\theta) $ and $ \cos(\theta) $ have specific values:

$ \sin(\frac{3\pi}{4}) $: The sine value is given by:

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

$ \cos(\frac{3\pi}{4}) $: The cosine value is given by:

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

$ \tan(\frac{3\pi}{4}) $: The tangent value is given by:

$ \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 $

To find the exact values of $ sin(frac{3pi}{4}) $, $ cos(frac{3pi}{4}) $, and $ an(frac{3pi}{4}) $, we use the unit circle:

For $ heta = frac{3pi}{4} $:

$ sin(frac{3pi}{4}) $:

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

$ cos(frac{3pi}{4}) $:

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

$ an(frac{3pi}{4}) $:

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

At $ heta = frac{3pi}{4} $:

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

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