+0  
 
0
521
1
avatar

 The expression\(\sin^3 2x \cos 6x + \cos^3 2x \sin 6x\)
can be written in the equivalent form asin bx for some positive constants a and b  Find a+b.

 Apr 16, 2020
 #1
avatar+23245 
+1

Formulas:  sin3(x)   =  ¾·sin(x) - ¼·sin(3x)      --->   sin3(2x)  =  ¾·sin(2x) - ¼·sin(6x)

                  cos3(x)  =  ¼·cos(3x) + ¾·cos(x)   --->   cos3(2x)  =  ¼·cos(6x) + ¾·cos(2x)

 

[ sin3(2x) ] · cos(6x)  + [ cos3(2x) ] · sin(6x)

 

=  [ ¾·sin(2x) - ¼·sin(6x) ] · cos(6x)  +  [ ¼·cos(6x) + ¾·cos(2x) ] · sin(6x)

 

=  ¾·sin(2x) · cos(6x) - ¼·sin(6x) · cos(6x)  +  ¼·cos(6x) · sin(6x) + ¾·cos(2x) · sin(6x)

 

=  ¾·sin(2x) · cos(6x) + ¾·cos(2x) · sin(6x)

 

=  ¾ [ sin(2x) · cos(6x) + cos(2x) · sin(6x) ]

 

But:  sin(A + B)  =  sin(A)·cos(B) + cos(A)·sin(B)

 

=  ¾ · sin( 2x + 6x )

 

=  ¾ · sin( 8x )

 

Your turn ...

 Apr 17, 2020

1 Online Users

avatar