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 Rotating slopes (Posted on 2006-03-21)
Given a line with slope y/x, find a simple formula for the slope of a second line that forms a 45 degree angle with this line (find slopes for both 45 degrees more and 45 degrees less.)
This can be done without trigonometry.

Find a general formula for any angle.
This requires trigonometry.

 See The Solution Submitted by Jer Rating: 3.2000 (5 votes)

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 Solution | Comment 2 of 5 |
`Let (0,0) and (x,y) be the coordinates of points O and Arespectively. The slope of OB (where OAB is an isoscelesright triangle, CCW) gives the slope for 45 degrees more`
`  B = (x,y) + (-y,x) = (x-y,x+y)`
`               x+y  slope(OB) = -----               x-y`
`The slope of OC (where OAC is an isosceles right triangle, CW)gives the slope for 45 degrees less`
`  C = (x,y) - (-y,x) = (y+x,y-x)`
`               y-x  slope(OC) = -----               y+x`
` `
`Let p/q be the slope of the general angle V and y/x the slopeof angle U.                                                                          y     p                                     --- + ---               tan(U) + tan(V)        x     q       px + qy  tan(U+V) = ------------------- = ------------- = ---------              1 - tan(U)*tan(V)          y   p      qx - py                                    1 - ---*---                                         x   q`
`                                                                          y     p                                     --- - ---               tan(U) - tan(V)        x     q       qy - px  tan(U-V) = ------------------- = ------------- = ---------              1 + tan(U)*tan(V)          y   p      py + qx                                    1 + ---*---                                         x   q`
`Note: If V = 45, then p = q and we get the slopes of the first      part of the problem.`
` `

Edited on March 21, 2006, 4:40 pm
 Posted by Bractals on 2006-03-21 16:39:26

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