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Angle Inclination of a Line - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

The angle inclination of a line is the angle formed by the intersection of the line and the x-axis. Using a horizontal "run" of 1 and m for slope, the angle of inclination, theta=tan-1(m), or m=tan(theta). Therefore, if the angle or the slope is known, the other can be found using one of the equations. If the angle of inclination is negative, then the slope of the line is also negative.

What is the angle of inclination of a line? Well I've got a line drawn here the equation y=mx+b it should be familiar to you. This is the angle of inclination theta it's the angle that a line makes with the horizontal. Now I want to figure out how I can calculate that line how it's related to the equation of a line and so I've drawn little triangle here and then I'll label the sides, let's say I label this one 1 what will this side be? Now I know that slope, the slope of the line equals rise over run so the slope is going to equal question mark over 1 so this is going to be the slope.
Now if I look at this right triangle and it is a right triangle, I can use right triangle trigonometry to find a relationship between theta and m right? m is the vertical length of this of this side so tangent of theta equals m over 1 tangent theta equals m so this is a relationship between the angle of inclination and slope the tangent of the angle of inclination is the slope and the angle of inclination is the inverse tangent of slope, so you can calculate slope from angle of inclination and you can calculate angle of inclination from slope.
Now let's take a look at another, at some special cases. Horizontal or vertical lines. Now horizontal line is not going to ever not necessarily going to cross the x axis of course for the line to y=0 it is the x axis but we define it's angle of inclination to be 0 and of course the tangent of 0 is 0 so the slope is going to be the tangent of 0 which is 0 and that's what we that's how we define the slope of a horizontal line it is 0.
What about a vertical one? Well I'd say that the angle of inclination here is pi over 2 so theta equals pi over 2, what's the slope? Tangent of theta right tangent of pi over 2 but of course that's undefined and this is exactly what we would want the slope of a vertical line is undefined there's no slope, so just to review, the angle of inclination of a line the angle that the line makes with the horizontal the relationship between that and slope is the tangent of theta the angle of inclination equals the slope and the angle of inclination equals the inverse tangent of slope.

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