Find the sine, cosine, and tangent values of the angle ( frac{pi}{4} ) on the unit circle.

To find the sine, cosine, and tangent values of the angle \( \frac{\pi}{4} \) on the unit circle, we need to recall the coordinates of the corresponding point on the unit circle. The coordinates of the point corresponding to the angle \( \frac{\pi}{4} \) are \( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \).

The sine value is the y-coordinate: $ \sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} $

The cosine value is the x-coordinate: $ \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} $

The tangent value is the ratio of the sine and cosine: $ \tan \left( \frac{\pi}{4} \right) = \frac{ \sin \left( \frac{\pi}{4} \right) }{ \cos \left( \frac{\pi}{4} \right) } = \frac{ \frac{\sqrt{2}}{2} }{ \frac{\sqrt{2}}{2} } = 1 $

First, identify the coordinates for the angle ( frac{pi}{4} ) on the unit circle, which are ( left( frac{sqrt{2}}{2}, frac{sqrt{2}}{2}
ight) ).

Using these coordinates, we get:

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

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

For the tangent value:

$ an left( frac{pi}{4}
ight) = frac{ sin left( frac{pi}{4}
ight) }{ cos left( frac{pi}{4}
ight) } = 1 $

The angle ( frac{pi}{4} ) corresponds to the coordinates ( left( frac{sqrt{2}}{2}, frac{sqrt{2}}{2}
ight) ) on the unit circle.

Thus,

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

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

$ an left( frac{pi}{4}
ight) = 1 $

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