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 )#?

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Answer 1 · 2016-08-02T01:31:16

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

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)#