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(xy)+sin(x+y)
now we have
sin(49x)+sin(51x)=sin(50xx)+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)<=2cos(x) and
2sin(50x)cos(x)>=2cos(x) for all x
now if cos(x)>=0 we have
sin(50x)<=1
2sin(50x)<=2
2sin(50x)cos(x)<=2cos(x)
and
sin(50x)>=1
2sin(50x)cos(x)>=2cos(x)
if cos(x)<0 then we have
sin(50x)>=1
2sin(50x)>=2
2sin(50x)cos(x)<=2cos(x)
and
sin(50x)<=1
2sin(50x)<=2
2sin(50x)cos(x)<=2cos(x)
thus sin(49x)+sin(51x) is bounded by +2cos(x)

Posted by Daniel
on 20090623 12:36:42 