Question

I do not understand how I can prove this:

Use the racetrack principle and the fact that sin0=0 to show that sinx is less than or equal to x for all of x greater than or equal to 0.

Answer 1

The racetrack principle stipulates that:
If \(f'(x)\le g'(x) \) for all \(x > 0\) and if f(0) = g(0) then \(f(x) \le g(x)\) for all \(x>0\)
For this case we consider f(x)=sin(x) and g(x)=x
Then f'(x)=cos(x) and g'(x)=1.
We have that: \(cos(x) \le 1\) for all \(x>0\) and sin(0)=0 qed