Show sin(x) is less than or equal to x

Question

Show that sin(x)≤x for all x≥0 (hint look at the derivative of x-sin(x))

Answer 1

if  y = x - sin(x) then y' = 1 - cos(x).

 

This says y'>=0 so y is a non-decreasing function.

 

Since y=0 when x=0 this says since y is non-decreasing it does not go below zero.

 

 

Any questions comment back.

Answer 2

sin(x) will always range between -1 and 1.  It is divergent.  It will never converge to a single limiting value as x goes to infinity.

 

Let sin(x)=c, where c is some value between -1 and 1

 

Then limit of sin(x)/x as x becomes greater than 0 is c/x

 

For the value x=1,

 

c=x when c=1,

 

c<x when c is between -1 and 1

 

when x>1

 

c will always be less than x.  -1 and 1 are both less than 2, 3, 4, and so on.

 

limit of c/x as x becomes greater than zero is c/∞, which will eventually be 0.

 

If the ratio of c/x becomes zero, then sin(x) is less than or equal to x for values of x greater than 0.