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Finding Coordinates


Date: 12/31/96 at 15:57:27
From: Joe Rounceville
Subject: Finding the coordinates of the endpoint of a line 
given its angle from another line

Hi,

I want to tell you folks thanks a lot for your help.  Such speedy
responses, too!  What a great service you are providing!  Anyway, I've 
got a bit of a mind bender that falls somewhere in the trig arena.  
I've spent all evening pondering this one, and I've wasted several 
sheets of scratch paper on it.  It doesn't seem like it should be this 
tough, but I'm still sitting here with no answer, so here goes:

Given this:

               |
               |  B
               | /\
     __________|/__\________
              A|    \
               |     \
               |      \
               |       C

B is at (3,2), A is at the origin, the line BC is 4.123 units long, 
and angle ABC is 68.616 degrees.  How do I determine the coordinates 
of C?

Thanks again for all your help.

Joe Rounceville


Date: 01/01/97 at 13:54:18
From: Doctor Pete
Subject: Re: Finding the coordinates of the endpoint of a line 
given its angle from another line
 
Hi,

The first solution that pops into my head is to use the Law of Cosines 
to get the distance AC (since we know AB, BC, and angle ABC), then use 
the Law of Sines to obtain angle BAC.  This gives the angle AC makes 
with the x-axis and hence the polar form of point C.

I'll start you off with some of the calculations:

      (AC)^2 = (AB)^2 + (BC)^2 - 2(AB)(BC)Cos[ABC]
             = 3^2 + 2^2 + (4.123)^2 - 2*Sqrt[13]*4.213*Cos[68.616]
             = 19.15857

    Sin[BAC] = BC*Sin[ABC]/AC
        BAC  = 61.2959 deg

The angle AC makes with the (positive) x-axis is:

    ArcTan[2/3] - BAC = -27.6058 deg

Finally, use the polar-to-rectangular transformation:

      x = r*Cos[theta]
      y = r*Sin[theta]
 
to obtain the rectangular coordinates of C.

 -Doctor Pete,  The Math Forum
  Check out our web site!  http://mathforum.org/dr.math/   


Date: 01/01/97 at 21:00:46
From: Joe Rounceville
Subject: Re: Finding the coordinates of the endpoint of a line 
given its angle from another line

Thanks again for your help, Doctor Pete!  Using what you told me I was
able to create (for a school project) a realistic animation of someone 
running.  

Joe
    
Associated Topics:
High School Coordinate Plane Geometry
High School Geometry
High School Trigonometry

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