Chapter 4: Applications of Derivatives - Section 4.2 - The Mean Value Theorem - Exercises 4.2 - Page 199: 64

See the explanation below.

Step 1. Let $f(x)=sin(x)$ and $a\lt b$; we can see that the function satisfies the conditions to use the Mean Value Theorem, which gives $f'(c)=\frac{f(b)-f(a)}{b-a}=\frac{sin(b)-sin(a)}{b-a}$ where $c$ is in the interval $(a,b)$ Step 2. Since $|f'(c)|=|cos(c)|\leq1$, we have $|\frac{sin(b)-sin(a)}{b-a}|\leq1$ and $|sin(b)-sin(a)|\leq|b-a|$