Is the tangent function (like in trig) and tangent lines the same? So, a 45 degree angle in the unit circle has a tan value of 1. Does that mean the slope of a tangent line from that point is also 1? Or is something different entirely?

The $\tan$ function can be described four different ways that I can describe and each adds to a fuller understanding of the tan function.


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*First, the basics: the value of $\tan$ is equal to the value of $\sin$ over $\cos$.
$$\\tan(45^\circ)=\frac{\sin(45^\circ)}{\cos(45^\circ)}=\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}=1$$

*So, the $\tan$ function for a given angle does give the slope of the radius, but only on a unit circle or only when the radius is one.  For instance, when the radius is 2, then $2\tan(45^\circ)=2$, but the slope of the 45 degree angle is still 1.

*The value of the $\tan$ for a given angle is the length of the line, tangent to the circle at the point on the circle intersected by the angle, from the point of intersection (A) to the $x$-axis (E).
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*The value of the tangent line can also be described as the length of the line $x=r$ (which is a vertical line intersecting the $x$-axis where $x$ equals the radius of the circle) from $y=0$ to where the vertical line intersects the angle.


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The explanations in examples 3 and 4 might seem counter intuitive at first, but if you think about it, you can see that they are really just reflections across a line of half the specified angle. Image to follow.
The images included are both from Wikipedia.
