Cosines of Three Angles

Date: 05/08/2003 at 22:02:14
From: Shannon
Subject: Angles of triangles

Show that the largest angle of the triangle whose sides are 4, 5, and 
6 is exactly twice the size of the smallest angle.

Date: 05/08/2003 at 22:54:00
From: Doctor Mitteldorf
Subject: Re: Angles of triangles

You can calculate the cosines of the three angles using the Law of
Cosines.  For example, the largest angle: 

   6^2 = 4^2 + 5^2 - 2(4)(5) cos(C)

   cos(C) = 1/8

Once you have the two cosines, you can test the statement that angle
C is twice angle A by using the cosine formula for double an angle:

   cos(2A) = cos^2(A) - sin^2(A) = 2cos^2(A) - 1

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