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Angle Inclination of a Line - Problem 3

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.

Let’s doa tougher problem that involves angle of inclination. Consider the lines L1 which is 5x plus 3y equals 30 and L2 which is 5x minus 2y equals -10. Find the measure of the acute angle alpha that they form at their interception.

So let me show you a graph of these two lines, here is L1, here’s is L2 and here’s where they intersect. I want to find the measure of the acute angle that they form because they form two. They form an obtuse angle here and an acute angle alpha there, this is also alpha by the vertical angle still.

Now if you noticed this little triangle, looking at this triangle I realize that this angle here, the lower left angle is actually the angle of inclination of line 2. This angle down here is not the angle of inclination of line 1 its angle of inclination is going to be negative but I’ll be able to figure out what this angle is by finding the angle of inclination of L1. So let’s find both angles of inclination that will allows us to find alpha.

First L1. So to find the angle of inclination I have to identify the slope of L1 so I’ll solve it for y and put it in slope intercept form. And I get 3y equals -5x plus 50 and I divide by 3, y equals -5/3x plus 50/3 so the slope is -5/3 and that means the angle of inclination is inverse tangent of -5/3 and I’ll round that to the nearest degree. Inverse tangent of -5 over 3 is -59 degrees.

Now let’s go back to the picture really quickly. If this angle is -59 degrees, well this angle of inclination then the magnitude of that is 59 that means this angle is 59 degrees so I just need to find this guy here.

Alright L2. So I need to solve this equation for y and I get -2y equals -5x minus 10 and I divide everything by -2. y equals -2 sorry positive 5/2x plus 5 so the slope is 5/2 and that means that its angle of inclination maybe I should call this theta 2 and this one theta 1. This is going to be inverse tangent of 5/2, so I'll use my calculator for that, inverse tangent 5 divided by 2 and I get 68.2 degrees, so approximately 68 degrees. So that’s the measure of this angle here, 68 degrees.

So remember that the angles of a triangle have to add up to 180 so all I have to do is take 180 degrees minus 68 degrees minus 59 degrees and I get 53 degrees. Alpha is 53 degrees. And that’s the acute angle between the two lines.

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