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Rotating slopes (Posted on 2006-03-21) Difficulty: 3 of 5
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 Solution | Comment 2 of 5 |

Let (0,0) and (x,y) be the coordinates of points O and A
respectively. The slope of OB (where OAB is an isosceles
right 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 slope
of 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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