In physics textbooks, the period T of a pendulum of length L is often given as T ≈ 2π√L/g , provided that the pendulum swings through a relatively small arc. In the course of deriving this formula, the equation a_T − gsinθ for the tangential acceleration of the bob of the pendulum is obtained, and then sinθ is replaced by θ the remark that for small angles, θ (in radians) is very close to sin θ.

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

sin θ ≈ θ

(b) If θ = π/18 (equivalent to 10°) and we approximate sin θ by θ, what is the percentage error?

(c) Use a graph to determine the values of θ for which sin θ and θ differ by less than 2%. What are the values in degrees?