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?

Both series seem to converge quickly, and no matter what the value of X...

C(X) = 0.99999995360882925082568429452716

and

S(X) = 0.017452405627728157369451750236583

Therefore, C(X) is greater.

Also C(X)/S(X) = 57.298688498280271407522980498822 (which, by the way, can be approximated as tangent of 89.000152273918042090339324201263 degrees), and doesn't seem to depend on X.  This is my point of view.

Just in case, I tried this series MANUALLY using Windows' Calculator program.  To check youself, write any number (by default, it will think in degrees), then type S for sin and O for cos.

For C(X), since the last operation is sine, type S first, then O S O S O S O S O S O S O and by now it should've converged no matter what the value of X.

For S(X), since the last operation is cosine, type O first, then S O S O S O S O S O S and by now it should've converged no matter what the value of X.  Try it yourself.

NOW, I've seen the previous comments and wondered if they were true, so I also tried it in Excel and found these values.

C(X) = 0.768169157
S(X) = 0.694819691

This puzzled me for quite a while, until I realized the ONLY thing that differs between them.

Calculator uses degrees as default.  By switching the calculator to read radians, the values converge to those found by Excel.  However, when switching back to degrees, it still returns the values I mentioned before.  This is quite puzzling.

Another strange detail is that, using Calculator, the series does NOT converge using radians.  The numbers keep varying after every iteration, although it stays the same to about 6 decimal places.  However, when using degrees and finding the values I found, the series converges quickly to the most decimal points it can handle (about 42).

From what I can see, this possibly means that the only thing that seems to matter is whether or not you are working in radians or degrees.  So X can be of any value, C(X) will be greater, but the ratio seems to depend on whether or not you're working with degrees or radians.

Any thoughts?

Edited on November 8, 2005, 12:33 am

Posted by Alexis on 2005-11-07 23:48:09