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Question

Rewrite 2sin(4x)cos(3x). as the sum of sine or/and cosine funtions.

Answer 1

Using various trigonometric identities...

sin(A+B) = sin(A)cos(B) + sin(B)cos(A)

cos(A+B) = cos(A)cos(B) - sin(A)sin(B)

f(x) = 2*sin(4x)*cos(3x) = 2*sin(2x+2x)*cos(x+2x)

sin(4x) = sin(2x+2x) = 2*sin(2x)*cos(2x)

sin(2x) = 2*sin(x)*cos(x)

cos(2x) = cos²(x) - sin²(x)

Now substitute what we have...

f(x) = 2 * (2 * sin(2x)*cos(2x)) * (cos(x)*cos(2x) - sin(x)*sin(2x))

f(x) = 4*sin(2x)*cos(2x)*cos(x)*cos(2x) - 4*sin(2x)*cos(2x)*sin(x)*sin(2x)

f(x) = U + V

U = 4*sin(2x)*cos(2x)*cos(x)*cos(2x)

U = 8*sin(x)*cos²(x)*cos(2x)

U = 8*sin(x)*cos²(x)*[ cos²(x) - sin²(x) ]

U = 8*sin(x)*cos⁴(x) - 8*sin³(x)*cos²(x)

V = 4*sin(2x)*cos(2x)*sin(x)*sin(2x)

V = 8*sin²(x)*cos(x)*cos(2x)*sin(2x)

V = 8*sin²(x)*cos(x)*(2*sin(x)*cos(x))*cos(2x)

V = 16*sin³(x)*cos²(x)*cos(2x)

V = 16*sin³(x)*cos²(x)*[ cos²(x) - sin²(x) ]

V = 16*sin³(x)*cos⁴(x) - 16*sin⁵(x)*cos²(x)

f(x) = U + V --> 8*sin(x)*cos⁴(x) - 8*sin³(x)*cos²(x) + 16*sin³(x)*cos⁴(x) - 16*sin⁵(x)*cos²(x)

f(x) = 8*sin(x)*cos⁴(x) * [ 1 + 2sin²(x) ] - 8*sin³(x)*cos²(x) * [ 1 + 2sin²(x) ]

f(x) = [ 1 + 2sin²(x) ] * [ 8*sin(x)*cos⁴(x) - 8*sin³(x)*cos²(x) ]

f(x) = [ 1 + 2sin²(x) ] * [ 8*sin(x)*cos²(x) * (cos²(x) - sin²(x)) ]