SOLUTION: Rewrite as a sum, then simplify if possible: cos3xsin11x

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Question 770556: Rewrite as a sum, then simplify if possible: cos3xsin11x
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
cos(3x)sin(11x)

That's a product of a cosine and a sine.  Usually sines
come before cosines, so let's reverse the two factors:

Let's rewrite it as

sin(11x)cos(3x)

We think through the formulas for AħB

sin(AħB) = sin(A)cos(B) ħ cos(A)sin(B)
cos(AħB) = cos(A)cos(B) ∓ sin(A)sin(B)

and see that the first formula contains such a product, 
sin(A)cos(B),

so we let A=11x and B=3x, then 

sin(AħB) = sin(A)cos(B) ħ cos(A)sin(B)

becomes the two equations:

sin(11x+3x) = sin(11x)cos(3x) + cos(11x)sin(3x)
sin(11x-3x) = sin(11x)cos(3x) - cos(11x)sin(3x)

or

sin(14x) = sin(11x)cos(3x) + cos(11x)sin(3x)
 sin(8x) = sin(11x)cos(3x) - cos(11x)sin(3x)

Addinng those equations term by term:

sin(14x) + sin(8x) = 2sin(11x)cos(3x)

Solving for sin(11x)cos(3x) by multiplying both sides by 1%2F2

1%2F2[sin(14x) + sin(8x)] = sin(11x)cos(3x)

1%2F2sin(14x) + 1%2F2sin(8x)] = sin(11x)cos(3x)

Edwin