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Prove that the formula for getting the angle between two lines is
Solution:
To derive the formula for getting the angle between two lines, let's draw the two intersecting lines in Rectangular Coordinate System as follows:
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| Photo by Math Principles in Everyday Life |
L1 is the first line that is inclined to the left and θ1 is the angle of inclination of a line in a counterclockwise direction from x-axis.
L2 is the second line that is inclined to the right and θ2 is the angle of inclination of a line in a counterclockwise direction from x-axis.
As you notice in the figure that L1, L2, and x-axis are the sides of a triangle. Let's label further the figure above as follows
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| Photo by Math Principles in Everyday Life |
The sum of the interior angles of a triangle is calculated as follows
Take tangent on both sides of the equation, we have
Substitute the right side of the equation with the slope of two lines as follows
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| Photo by Math Principles in Everyday Life |
The slope of L1 is
The slope of L2 is
Therefore, the formula for getting the angle between two lines is
