SICP Exercise 1.39

Question

A continued fraction representation of the tangent function was published in 1770 by the German mathematician J.H. Lambert:

$$\tan x=\frac{x}{1-\frac{x^2}{3-\frac{x^2}{5-\ddots }}}$$

where $x$ is in radians. Define a procedure (tan-cf x k) that computes an approximation to the tangent function based on Lambert’s formula. k specifies the number of terms to compute, as in exercise 1.37.

Answer

Our expected result for $\tan (1)$ with 4 decimal places accuracy is $1.5574$. Amazingly, we only need 4 iterations to get close enough (but as always, more is better).

copy(define (square-unless-one x)
  (if (= x 1) x
      (* x x)))

(define (get-n x) (- (* 2 x) 1))

(define (tan-cf x k)
  (define (iterate counter)
    (if (= k counter) (/ (square-unless-one x) (get-n counter))
        (/ (square-unless-one x) (- (get-n counter) (iterate (+ counter 1))))))
  (iterate 1))

(tan-cf 1.0 4)

Results:

copy1.5573770491803278