Question

Write the expression in terms of sin x and cos x only. sin(2x)+cos(3x)

Answer 1

sin2x=2sinxcosx, cos2x=1-2sin^2x
sin(2x)+cos(3x)=2sinxcosx+cos(x+2x)
cos(x+2x)=cosx(1-2sin^2(x))-sinx2sinxcosx
sin(2x)+cos(3x)=2sinxcosx(1-sinx)+cosx(1-2sin^2(x))

Answer 2

Use the following identities:

`sin(2x)=2sin(x)cos(x)\\ cos(2x)=\cos^2{x}-\sin^2{x} \\ cos((x+y))=cos(x)cos(y)-sin(x)sin(y)`


`sin(2x)+cos(3x) \\=2sin(x)cos(x)+cos((2x+x)) \\=2sin(x)cos(x)+cos(2x)cos(x)-sin(2x)sin(x) \\=2sin(x)cos(x)+(\cos^2{x}-\sin^2{x})cos(x)-2sin(x)cos(x)sin(x) \\=2sin(x)cos(x)+\cos^3{x}-\sin^2{x}cos(x)-2\sin^2{x}cos(x) \\=2sin(x)cos(x)+\cos^3{x}-3\sin^2{x}cos(x) \\=sin(x)cos(x)(2+\cos^2{x}-3sin(x))`