Question

By writing sin 3x = sin(2x+x), show that sin 3x = 3 sin x - 4 sin^3 x.

Answer 1

Given the trigonometry expression:

sin 3x = sin(2x+x),

In trigonometry identity;

$Sin(A+B) = sinAcosB + cosAsinB$

The expression above will then become;

$sin(2x+x) =sin2xcosx+cos2xsinx\\sin(2x+x) =2sinxcosxcosx + (1-2sin^2x)sinx\\sin(2x+x) =2sinxcos^2x + (1-2sin^2x)sinx$

Recall that sin^2x + cos^x = 1, hence;

$sin(2x+x) =2sinx(1-sin^2x)+ (1-2sin^2x)sinx\\Expand\\sin(2x+x) =2sinx-2sin^3x+ sinx-2sin^3x\\sin(2x+x) =2sinx + sinx-2sin^3x-2sin^3x\\sisn(2x+x) = 3sinx-4sin^3x$

This shows that $sin3x =3sinx-4sin^3x$ (Proved!)

Learn more here: brainly.com/question/7331447