The sine of an angle

The sine of an angle (specified in radians) can be computed by making use of the approximation sin x ≈ x if x is sufficiently small, and the trigonometric identity

sin(x) = 3sin(x / 3) − 4sin³(x / 3)

to reduce the size of the argument of sin . (For purposes of this exercise an angle is considered ''sufficiently small'' if its magnitude is not greater than 0.1 radians.) These ideas are incorporated in the following procedures:

(define (cube x) (* x x x))

(define (p x) (- (* 3 x) (* 4 (cube x))))

(define (sine angle)
   (if (not (> (abs angle) 0.1))
       angle
       (p (sine (/ angle 3.0)))))

a. How many times is the procedure p applied when (sine 12.15) is evaluated?

b. What is the order of growth in space and number of steps (as a function of a ) used by the process generated by the sine procedure when (sine a) is evaluated?

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