Trig Functions in MatricesDate: 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 Check out our web site! http://mathforum.org/dr.math/ |
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