# Thread: Solve cos x = x with a calculator

1. ## 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

2. 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

3. Did you know that you can do the same with sine?

$\displaystyle sin(0)=0$

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

$\displaystyle sin(0)=0$
so I realised.

5. Originally Posted by chopet
so I realised.
Well, this is why the trig functions are so beautiful.

6. 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.

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

$\displaystyle sin(0)=0$
That amused me!

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# cos x=-0.54

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