How do you express #sin(2x)+sin(4x)# in terms of #sin(x)# and #cos(x)# ?

1 Answer
Nov 17, 2017

Answer:

In terms of #sin(x)# and #cos(x)# we find:

#sin(2x)+sin(4x) = 2 sin(x)cos(x)(1 + 2 cos^2(x) - 2 sin^2(x))#

Explanation:

Note that:

#sin(2x) = 2 sin(x)cos(x)#

#cos(2x) = cos^2(x) - sin^2(x)#

So:

#sin(4x) = sin(2(2x))#

#color(white)(sin(4x)) = 2 sin(2x)(cos(2x)#

#color(white)(sin(4x)) = 2(2sin(x)cos(x))(cos^2(x)-sin^2(x))#

#color(white)(sin(4x)) = 4sin(x)cos^3(x)-4sin^3(x)cos(x)#

So:

#sin(2x)+sin(4x) = 2 sin(x)cos(x) + 4sin(x)cos^3(x)-4sin^3(x)cos(x)#

#color(white)(sin(2x)+sin(4x)) = 2 sin(x)cos(x)(1 + 2 cos^2(x) - 2 sin^2(x))#