How do you find A, B and C, given that A sin ( B x + C ) = cos ( cos^(-1) sin x + sin^(-1) cos x ) + sin (cos^(-1) sin x + sin^(-1) cos x )?

1 Answer
Aug 2, 2016

sqrt 2 sin ( 2 x - pi/4)

Explanation:

Use sin x = cos (pi/2-x), cos x = sin ( pi/2 - x)

cos^(-1) cos a =a, sin^(-1) sin b = b and

sin 2A - sin 2 B = 2 cos ( A + B ) sin ( A - B ).

Let u =cos^(-1) sinx + sin ^(-1)cos x

=cos^(-1) cos (pi/2-x) + sin ^(-1) sin (pi/2-x)

=(pi/2-x) + (pi/2-x)

=pi - 2x

So, the given equation is

A sin ( B x + C )

=cos (pi-2x)+sin(pi-2x)

=-cos 2x+ sin 2x

=sin 2x - sin (pi/2-2x)

= 2 cos (pi/4) sin ( 2x - pi/4)

=sqrt 2 sin ( 2x - pi/4)