Problem 40E

40. On page 431 of Physics: Calculus, $2 \mathrm{~d}$ ed., by Eugene Hecht (Pacific Grove, CA: Brooks/Cole, 2000 ), in the course of deriving the formula $T=2 \pi \sqrt{L / g}$ for the period of a pendulum of length $L$, the author obtains the equation $a_{T}=-g \sin \theta$ for the tangential acceleration of the bob of the pendulum. He then says, „for small angles, the value of $\theta$ in radians is very nearly the value of $\sin \theta$; they differ by less than $2 \%$ out to about $20^{\circ} . „$

(a) Verify the linear approximation at 0 for the sine function:

$$ \sin x \approx x $$

(b) Use a graphing device to determine the values of $x$ for which $\sin x$ and $x$ differ by less than $2 \%$. Then verify Hecht’s statement by converting from radians to degrees.

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