$ ext{Find the tangent of } 45^circ ext{ using the unit circle.}$

To find the tangent of 45 degrees using the unit circle, we first locate the point corresponding to 45 degrees on the circle. The coordinates of this point are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.

Recall that the tangent function is defined as the ratio of the y-coordinate to the x-coordinate:

$\tan(45^\circ) = \frac{\text{y-coordinate}}{\text{x-coordinate}} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$

Therefore, $\tan(45^\circ) = 1$.

Using the unit circle to find the tangent of 45 degrees, we identify the coordinates at 45 degrees, which are $(frac{sqrt{2}}{2}, frac{sqrt{2}}{2})$.

The formula for tangent is:

$ an( heta) = frac{sin( heta)}{cos( heta)}$

For $45^circ$, both $sin(45^circ)$ and $cos(45^circ)$ are $frac{sqrt{2}}{2}$. Thus,

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

Hence, $ an(45^circ) = 1$.

From the unit circle, at $45^circ$, the coordinates are $(frac{sqrt{2}}{2}, frac{sqrt{2}}{2})$.

Thus,

$ an(45^circ) = frac{ ext{y}}{ ext{x}} = frac{frac{sqrt{2}}{2}}{frac{sqrt{2}}{2}} = 1$

Therefore, $ an(45^circ) = 1$.

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