In Trigonometry Nest, the functions sin(cos(x)) and cos(sin(x)) were introduced, and it was proven that one of these functions is always greater than the other.

Taking this concept to infinity, which of these functions is greater: C(x)=cos(sin(cos(......(sin(x))))...))), or S(x)=sin(cos(sin(......(cos(x))))...)))?

How does the ratio C(x)/S(x) depend on x?

C(x) =.768169156736796, S(x)= .6948196907307875, for all x. Therefore C(x)/S(x) doesn't depend on x.

Using the Inverse Symbolic Calculator, there is no match for either of these, or for their ratio.  So it doesn't seem to be a simple algebraic combination of well-known constants.

DEFDBL A-Z
DO
  INPUT x
  c = x: s = x
  a$ = "": ct = 0
  DO
   c = COS(SIN(c)): s = SIN(COS(s))
   PRINT x, c, s
   ct = ct + 1
   IF ct MOD 40 = 0 THEN
     DO: a$ = INKEY$: LOOP UNTIL a$ > ""
   END IF
  LOOP UNTIL a$ = CHR$(27)
LOOP

 

Posted by Charlie on 2005-11-07 10:37:33