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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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