Find the sine, cosine, and tangent values at $45^{circ}$ using the unit circle.

To find the sine, cosine, and tangent values at $45^{\circ}$ (or $\frac{\pi}{4}$ radians) using the unit circle, we look at the coordinates of the corresponding point on the circle.

On the unit circle, at $45^{\circ}$, the coordinates are $\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$. The x-coordinate represents the cosine value, and the y-coordinate represents the sine value.

Thus,

$ \cos(45^{\circ}) = \frac{\sqrt{2}}{2} $

$ \sin(45^{\circ}) = \frac{\sqrt{2}}{2} $

The tangent value is given by the ratio of the sine value to the cosine value:

$ \tan(45^{\circ}) = \frac{\sin(45^{\circ})}{\cos(45^{\circ})} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 $

So, the values are:

$ \cos(45^{\circ}) = \frac{\sqrt{2}}{2} $

$ \sin(45^{\circ}) = \frac{\sqrt{2}}{2} $

$ \tan(45^{\circ}) = 1 $

Using the unit circle, we can determine the sine, cosine, and tangent of $45^{circ}$.

At $45^{circ}$ (or $frac{pi}{4}$ radians), the coordinates are $left( frac{sqrt{2}}{2}, frac{sqrt{2}}{2}
ight)$. The x-coordinate is the cosine value, and the y-coordinate is the sine value.

Therefore:

$ cos(45^{circ}) = frac{sqrt{2}}{2} $

$ sin(45^{circ}) = frac{sqrt{2}}{2} $

For the tangent:

$ an(45^{circ}) = frac{sin(45^{circ})}{cos(45^{circ})} = frac{frac{sqrt{2}}{2}}{frac{sqrt{2}}{2}} = 1 $

At $45^{circ}$ on the unit circle, the coordinates are $left( frac{sqrt{2}}{2}, frac{sqrt{2}}{2}
ight)$. Therefore:

$ cos(45^{circ}) = frac{sqrt{2}}{2} $

$ sin(45^{circ}) = frac{sqrt{2}}{2} $

$ an(45^{circ}) = 1 $

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