# Solve cos x = x with a calculator

• Apr 15th 2008, 06:29 PM
chopet
Solve cos x = x with a calculator
Why is it that we can start with a number (say, 1), and press cos x repeatedly, we can get the solution of cos x = x?

e.g. cos (1) = 0.54
cos (0.54) = 0.85
cos (0.85) = 0.65
cos (0.65) = 0.79
cos (0.79) = 0.71
cos (0.71) = 0.76
cos (0.76) = 0.72
.....

until it converges to 0.73
• Apr 15th 2008, 06:49 PM
mr fantastic
Quote:

Originally Posted by chopet
Why is it that we can start with a number (say, 1), and press cos x repeatedly, we can get the solution of cos x = x?

e.g. cos (1) = 0.54
cos (0.54) = 0.85
cos (0.85) = 0.65
cos (0.65) = 0.79
cos (0.79) = 0.71
cos (0.71) = 0.76
cos (0.76) = 0.72
.....

until it converges to 0.73

Reading this will get you started:

One Dimensional Dynamical Systems

Emergence of Chaos from Interactive Mathematics Miscellany and Puzzles
• Apr 16th 2008, 04:43 AM
colby2152
Did you know that you can do the same with sine?

$\displaystyle sin(0)=0$
• Apr 16th 2008, 04:49 AM
chopet
Quote:

Originally Posted by colby2152
Did you know that you can do the same with sine?

$\displaystyle sin(0)=0$

so I realised. :)
• Apr 16th 2008, 04:50 AM
colby2152
Quote:

Originally Posted by chopet
so I realised. :)

Well, this is why the trig functions are so beautiful.
• Apr 16th 2008, 07:43 AM
wingless
This is called Picard's Method (to find roots in R).

It can also be applied to other functions.

$\displaystyle x = e^x-1$

If you start with a positive x value, it'll diverge. But a negative initial value will diverge to zero. You can also change the function to $\displaystyle x = \ln (x+1)$ to make it diverge for positive x values.
• Apr 16th 2008, 07:53 AM
angel.white
Quote:

Originally Posted by colby2152
Did you know that you can do the same with sine?

$\displaystyle sin(0)=0$

That amused me!