Chapter 3, Problem 14PS

Radians and Degrees The fundamental limit

$\underset{x\to \infty }{\lim }\frac{\sin x}{x}=1$

assumes that x is measured in radians. Suppose you assume that x is measured in degrees instead of radians.

(a) Set your calculator to degree mode and complete the table.

z (in degrees)	0.1	0.01	0.0001
$\frac{\sin z}{z}$

(b) Use the table to estimate

$\underset{z\to 0}{\lim }\frac{\sin z}{z}$

for z in degrees. What is the exact value of this limit?

(Hint: ${180}^{\circ }=\pi$ )

(c) Use the limit definition of the derivative to find $D_z$ for z in degrees.

(d) Define the new functions $S(z)=\sin cz$ and $C(z)=\cos cz$, where $c=\pi /180$. Find S (90) and C(180). Use the Chain Rule to calculate D, [s (z)].

(e) Explain why calculus is made easier by using radians instead of degrees.