find angle between the lines x+3y-8=0 &2x-3y+6=0

Question

how many terms of the series 148+146+144+142+...............must be taken to have teir sum amount to 3000? explain the double answer..

Answer 1

find angle between the lines x+3y-8=0 & 2x-3y+6=0

First we find the angle each line forms relative to the y axis.
We need the slope of each line.

x+3y-8=0
3y = -x + 8
y = (-1/3)x + 8
The slope is -1/3. That is the tangent of
the angle, so angle = arctan (-1/3)
The angle is -18.435 degrees, which means
it runs from the upper left to the lower right.

2x-3y+6=0
-3y = -2x - 6
y = (2/3)x - 6
The slope is 2/3. The angle is arctan (2/3).
This angle is 33.69 degrees. It runs from
the lower left to the upper right.

The angle between the two lines is measured
down 33.69 degrees from the second line to
the x axis, then down 18.435 degrees from
the x axis to the second line.
33.69 degrees + 18.435 degrees = 52.125 degrees