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?
 A: 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.
A: Have a look at this drawing from Wikipedia:
Unit Circle Definitions of Trigonometric Functions.
When viewed this way, the tangent function actually represents the slope of a line perpendicular to the tangent line of that point (i.e. the slope of the radius that touches the angle point).
However, you can actually see that the "tangent line", consisting the values of the tangents, is the actual tangent line of the circle at the point from which the angles are measured, and I would guess that this is the source of the name.
A: Yes and no, resp.: yes, any line in the plane that forms an angle of $\,45^\circ\,$ with the positive direction of the $\,x-$axis has a slope of $\,\tan 45^\circ=1\,$, and no: it isn't something different.
It is not completely clear though what you mean by "tangent line"...perhaps you meant "tangent line at some point on the graph of a (derivable) function"?
