Play with scientific calculator

I posted an entry where I showed a very simple method to solve x = cos(x) by using a scientific calculator. Now, let's continue playing with a scientific calculator.

1. Bring your scientific calculator (you can use your iphone or a web calculator).
2. Change the mode to "rad" (radian).
3. Push "1" (one) key.
4. Push "sin" key.
5. Push "1/x" key.
6. Go back to step 4, unless you want out.
   

You will see two numbers:

• 0.8975395... (after step 4.)
• 1.1141571... (after step 5.)

The second number is a solution of the equation x = 1/sin(x)(this equation has infinite number of solutions), and the first number is its inverse.


Then, change "sin" to "cos" in step 4. Does it converge?

No.

I simulated the procedure by using SCILAB, and obtained a sequence shown below.

This looks like a chaotic sequence.