Understanding Tangents in Analytic Geometry

[ObsoleteBacon]

Okay so I am reading a book on analytic geometry because my algebra class is starting to really bore me, and I ran across this:

slope = m = -1/2 (easy enouph)

but then it states "since the slope is the tangent of the angle of inclination θ, we have: tan θ = -1/2

okay so I guess my first question is: What is a tangent in analytic geometry?

continuing with my story: the very next thing it gives my is the angle of inclination

So for this it goes: θ = tan^-1 (-1/2) = 153°26'.

What the heck? Can someone explain this problem above in greater detail. I know that tan^-1 is the inverse of tan but without understanding what tangent actually is that doesn't help me much. Also, how do you calculate this to get the answer? I have looked at a ton of online sources but can't seem to find any answers to my questions.

 

[DiracRules]

Have you done trigonometry?

I don't understand if your problem is with the tangent -trigonometric function- or the tangent -the line on a graph. I'll try explaining both.

The latter:
take a curve. Fix one point on it, let it be P. Then, fix another point, let's say S. Draw the straight line connecting P to S (called secant). Then, move S until it overlaps P. The line you drew will move along with S. When S overlaps P, the line (now called tangent) will touch the curve in one and only point (P) - at least, the intersection will be one looking near P, but it can intersect the curve in other points.
http://en.wikipedia.org/wiki/Tangent" on wiki.

As regards the trigonometric function, in a right triangle, the tangent of an angle (not the right angle, however) is the ratio between the opposite and the adjacent sides.

So, when it says that the slope is the tangent of the angle of inclination, it means that is the tangent of the angle of inclination of the tangent on the curve. The slope is the tangent of θ because the evaluation of the slope (\frac{y_2-y_1}{x_2-x_1}) can be seen as the ratio between the two catheti of the triangle made by the x axis, the tangent and the height of the tangent in the point you are considering.

Sorry for my bad english, but I've never studied geometry in english, so I don't know the specific language :D Hope you understand and that it is ok!

 

[ObsoleteBacon]

Thank you for the answer it helped a lot. I wasnt sure whether it was telling me to use the first or second way, but now i know its the trigonometric way (i think i spelled that right)