Trig Functions in Matrices

Date: 7/22/96 at 9:39:16
From: Michael W.S. Morton
Subject: Trig Functions in Matrices

Show that

[   1         1         1    ]          #-B       B-r       r-#
[sin(@+#)  sin(@+B)  sin(@+r)] = 4 sin ----- sin ------ sin -----
[cos(@+#)  cos(@+B)  cos(@+r)]           2         2          2

Date: 8/3/96 at 19:43:11
From: Doctor Anthony
Subject: Re: Trig Functions in Matrices

This works out quite quickly if you expand the determinant by the top 
row.  As a first step you then obtain three lots of 

   sinX*cosY - cosX*sinY = sin(X-Y).

The three terms are:

   sin(B-r) + sin(r-#) + sin(#-B).

You now combine the first two sines using:

   sinX + sinY = 2sin{(X+Y)/2}cos{(X-Y)/2}.

The third term, sin(#-B) can be written 
   
   2sin{(#-B)/2}cos{(#-B)/2}

and you get a common term to take out with the other terms:

   2sin{(B-#)/2}[cos{B+#-2r)/2} - cos{(B-#)/2}].

The cosX - cosY in the bracket can be factored into

   cosX - cosY = 2sin{(X+Y)/2}sin{(Y-X)/2}.

This leads to:

   4sin{(#-B)/2}sin{(B-r)/2}sin{(r-#)/2}.

-Doctor Anthony,  The Math Forum
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