Find the sine, cosine, and tangent of the angle $frac{pi}{4}$ on the unit circle

First, let’s find the coordinates of the angle $\frac{\pi}{4}$ on the unit circle. The unit circle has a radius of 1, and the coordinates at an angle $\theta$ are $(\cos(\theta), \sin(\theta))$.

For $\theta = \frac{\pi}{4}$:

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

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

Therefore, the coordinates are $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$.

Now, the tangent is the ratio of sine to cosine:

$\tan\left(\frac{\pi}{4}\right) = \frac{\sin\left(\frac{\pi}{4}\right)}{\cos\left(\frac{\pi}{4}\right)} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$

To determine the sine, cosine, and tangent for the angle $frac{pi}{4}$ on the unit circle, we start with the unit circle definition. The unit circle has a radius of 1, and the coordinates $(x,y)$ at an angle $ heta$ are $(cos( heta), sin( heta))$.

When $ heta = frac{pi}{4}$, the coordinates are:

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

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

The tangent of $frac{pi}{4}$ is given by:

$ anleft(frac{pi}{4}
ight) = frac{sinleft(frac{pi}{4}
ight)}{cosleft(frac{pi}{4}
ight)} = 1$

On the unit circle, for angle $frac{pi}{4}$:

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

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

Therefore,

$ anleft(frac{pi}{4}
ight) = 1$

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