y=sin(49x)+sin(51x) has an interesting shape.

Prove that it is bounded by y=±2cos(x)

for this proof I will first use the identity
2sin(x)cos(y)=sin(x-y)+sin(x+y)
now we have
sin(49x)+sin(51x)=sin(50x-x)+sin(50x+x)
and thus we have
sin(49x)+sin(51x)=2sin(50x)cos(x)
thus we want to show that both
2sin(50x)cos(x)<=2|cos(x)| and
2sin(50x)cos(x)>=-2|cos(x)| for all x

now if cos(x)>=0 we have
sin(50x)<=1
2sin(50x)<=2
2sin(50x)cos(x)<=2|cos(x)|
and
sin(50x)>=-1
2sin(50x)cos(x)>=-2|cos(x)|

if cos(x)<0 then we have
sin(50x)>=1
2sin(50x)>=2
2sin(50x)cos(x)<=2|cos(x)|
and
sin(50x)<=-1
2sin(50x)<=-2
2sin(50x)cos(x)<=-2|cos(x)|

thus sin(49x)+sin(51x) is bounded by +-2cos(x)

Posted by Daniel on 2009-06-23 12:36:42