Calculate the sine and cosine values for the angle $frac{pi}{4}$ on the unit circle.

To find the sine and cosine values for the angle $\frac{\pi}{4}$ on the unit circle, we use the fact that the unit circle has a radius of 1 and the coordinates of the point on the unit circle corresponding to this angle are $(\cos\theta, \sin\theta)$.

For $\theta = \frac{\pi}{4}$, the coordinates are:

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

We know from trigonometric identities:

$ \cos\frac{\pi}{4} = \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2} $

Thus, the cosine and sine values for the angle $\frac{\pi}{4}$ are both $\frac{\sqrt{2}}{2}$.

To determine the sine and cosine of $frac{pi}{4}$ on the unit circle:

1. Convert the angle to radians: $frac{pi}{4}$.

2. Identify the coordinates on the unit circle: $(cos heta, sin heta)$.

3. For $ heta = frac{pi}{4}$, we use:

$ cosleft(frac{pi}{4}
ight) = sinleft(frac{pi}{4}
ight) = frac{sqrt{2}}{2} $

Thus, the sine and cosine values for the angle $frac{pi}{4}$ are $frac{sqrt{2}}{2}$.

For the angle $frac{pi}{4}$ on the unit circle, the coordinates $(cos heta, sin heta)$ are:

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

So, $cosfrac{pi}{4} = frac{sqrt{2}}{2}$ and $sinfrac{pi}{4} = frac{sqrt{2}}{2}$.

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