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 = 2π√L/g for the period of a pendulum of length L, the author obtains the equation aT = – g sin θ for the tangential acceleration of the bob of the pendulum. He then says, “for small angles, the value of in radians is very nearly the value of sin θ; they differ by less than 2% out to about 20°.”
(a) Verify the linear approximation at 0 for the sine function: sin x ≈ x
(b) Use a graphing device to determine the values of for which sin x and differ by less than 2%. Then verify Hecht’s statement by converting from radians to degrees.