On page 431 of $ Physics: Calculus, $ 2d 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^o." $

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

$ \sin x \approx x $

(b) Use a graphing device to determine the value of $ x $ for which $ \sin x $ and $ x $ differ by converting from radians to degrees.