Answer
(a) $sin~x \approx x$
(b) Hecht's statement is a valid statement.
Work Step by Step
(a) Let $f(x) =sin~x$
$f'(x) = cos~x$
When $x = 0$:
$f(0) = sin~0 = 0$
$f'(0) = cos~0 = 1$
We can find the linear approximation at $a=0$:
$f(x) \approx f(a)+f'(a)(x-a)$
$f(x) \approx f(0)+f'(0)(x-0)$
$f(x) \approx 0+(1)(x)$
$f(x) \approx x$
$sin~x \approx x$
(b) Using the graph, we can calculate that the values of $x$ and $sin~x$ differ by less than 2% for $-0.34 \leq x \leq 0.34$
We can verify this:
$\frac{-0.34-sin(-0.34)}{-0.34}\times 100\% = 1.9\%$
$\frac{0.34-sin(0.34)}{0.34}\times 100\% = 1.9\%$
We can convert $0.34~rad$ to degrees:
$0.34\times \frac{180^{\circ}}{\pi} = 19.5^{\circ}$
Hecht's statement is a valid statement.
