Finding Angles without Using Trigonometry

Date: 08/27/2001 at 22:01:03
From: Marc Foster
Subject: Re: Given 1 side of an isosceles triangle ABC, solve for 
angle A

I know I learned this at one point in time or another, but I can't 
seem to figure it out.

          A
          /\
         /  \
        /    \
       /      \
      /        \
    B/__________\C

Where BC = 150

The height of the triangle = ((BC/2)^2)-40

AB Congruent to AC
BC does not equal AB
Solve for angle A

Is this possible?

Maybe I should just cut to the chase and ask the real question...

Given the lengths of three sides of a triangle, can you determine the
measures of the three angles without using cos or sine or even looking 
at a trig table?  In other words, can one solve the angles using only 
geometry and algebra?  

Even if the difficulty outweighs the practicality of the method, I 
would enjoy the peace of mind gained from knowing it.

Thanks,
Marc Foster

Date: 08/28/2001 at 13:05:04
From: Doctor Rob
Subject: Re: Given 1 side of an isosceles triangle ABC, solve for 
angle A

Thanks for writing, Marc.

In answer to your "cut-to-the-chase" question, the answer is a 
qualified yes.

If one knows the three sides a, b, and c, and wants angle A, one can
compute the quantity

   x = (b^2+c^2-a^2)/(2*b*c).

Then one can find

   A = Pi/2 - x - (1/2)*x^3/3 - (1/2)*(3/4)*x^5/5 -
          (1/2)*(3/4)*(5/6)*x^7/7 - ... .

Take as many terms as needed to get the accuracy you want. This will
give you the angle measure in radians. To convert to degrees, multiply
by 180/Pi.

(Of course trigonometry is present here, just disguised.  x = cos(A),
and A = Arccos(x) = Pi/2 - Arcsin(x).)

Example: Suppose the sides of a triangle are a = 7, b = 5, and c = 4.
Then

   x = (25+16-49)/(2*4*5) = -1/5,
   A = Pi/2 + 1/5 + 1/750 + 3/125000 + 1/1750000 + 7/450000000 + ...,
     = 1.5707963267 + 0.2000000000 + 0.0013333333 + 0.0000240000 +
         0.0000005714 + 0.0000000156 + ...,
     = 1.77215425 radians,

rounded to 8 decimal places.  In degrees, that is

   A = 1.77215425*180/3.1415926536 = 101.53696 degrees.

Similarly,

   B = 44.41531 degrees (x = 5/7),
   C = 34.04773 degrees (x = 29/35).

Aside from this kind of approach, you cannot find the angles without
using trigonometry.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/