How could I turn 2sin5xcos3x into a sum of trig functions?

2 Answers
Aug 6, 2018

2sin5xcos3x=color(red)(sin(8x)+sin(2x)2sin5xcos3x=sin(8x)+sin(2x)

Explanation:

We know "the "color(blue)"Product Identity"the Product Identity

color(blue)(2sinAcosB=sin(A+B)+sin(A-B)2sinAcosB=sin(A+B)+sin(AB)

Substitute , A=5x and B=3xA=5xandB=3x

2sin5xcos3x=sin(5x+3x)+sin(5x-3x)2sin5xcos3x=sin(5x+3x)+sin(5x3x)

2sin5xcos3x=color(red)(sin(8x)+sin(2x)larr"sum of trig. functions."2sin5xcos3x=sin(8x)+sin(2x)sum of trig. functions.

Aug 6, 2018

sin8x+sin2xsin8x+sin2x

Explanation:

"using the "color(blue)"product to sum formula"using the product to sum formula

•color(white)(x)2sinxcosy=sin(x+y)+sin(x-y)x2sinxcosy=sin(x+y)+sin(xy)

"here "x=5x" and "y=3xhere x=5x and y=3x

2sin5xcos3x2sin5xcos3x

=sin(5x+3x)+sin(5x-3x)=sin8x+sin2x=sin(5x+3x)+sin(5x3x)=sin8x+sin2x