Using the Law of Cosines

Date: 05/21/98 at 05:15:17
From: Dan
Subject: Trigonometry

In triangle ABC, side a = 30m, side b = 36m, and side c = 10m.

The question asks to find the size of the angle between sides b and c. 
I think the question has something to do with advanced trigonometry, 
but I am unsure of how to go about it.

Your help would be appreciated!

Thanks,
Dan

Date: 05/21/98 at 07:34:11
From: Doctor Anthony
Subject: Re: Trigonometry

The convention is to use lower case letters a, b, c, to represent the 
sides of the triangle, and to use upper case letters A, B, C, to 
represent the angles, with side a opposite angle A, side b opposite 
angle B, and side c opposite angle C.

There are two principal formulae for solving triangles of any shape 
(not just right-angled triangles).

Which one you use depends on what information you are given. You will 
find these formulae proved in any textbook on elementary trigonometry.

Cosine Formula
---------------

   a^2 = b^2 + c^2 - 2bc*cos(A)     or

   b^2 = c^2 + a^2 - 2ca*cos(B)     or
 
   c^2 = a^2 + b^2 - 2ab*cos(C)

This is used if you are given three sides or if you are given two 
sides and the included angle. ("Included" means the angle betweem the 
two given sides).

Sine Formula
-------------

      a            b            c          
   --------  =  --------  =  --------
    sin(A)       sin(B)       sin(C)

This is used if you are given two angles and a side, or two sides and 
a non-included angle.

In the problem you stated, we are required to find the angle between 
sides b and c. This is angle A. Since we are given three sides, we use 
the cosine formula.

   a^2 = b^2 + c^2 - 2bc*cos(A)

   2bc*cos(A) = b^2 + c^2 - a^2

             b^2 + c^2 - a^2
   cos(A) = ------------------
                   2bc

Putting a = 30, b = 36, c = 10  we have:

            1296 + 100 - 900
   cos(A) = ----------------
                   720

             496     31
   cos(A) = ----- = ----- = 0.68888
             720     45

   A = 46.458 degrees

     = 46 deg, 27 min, 28 secs

If you require the other two angles, it is quicker now to use the sine 
formula. Also, remember that the three angles add up to 180 degrees, 
so if you have two angles, you can find the third by subtraction from 
180 degrees.

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