Finding the Intersecting Point of Two Simple Waves

[Werg22]

Two simple waves but on is a Y function and the other a X function, on do you find the intersecting point?

y=cos(x)
x=cos(y)

cos^-1(x)=cos(x)

 

[VietDao29]

You can use Newton's method to solve this:
$$x_n = \frac{f(x_{n - 1})}{f'(x_{n - 1})}$$
And the solution x is:
$$x = \lim_{n \rightarrow \infty} x_n$$
Newton's method
You can change the equation a bit so it's easier to take the dirivative of the function:
Since the graph of Arccos(x) is the reflection of the graph Cos(x) across the line y = x.
So the intersection of the two graph Arccos(x) and Cos(x) is right on the line y = x. So the equation can be changed to:
$$\cos(x) = x \Leftrightarrow \cos(x) - x = 0$$
Let f(x) = cos(x) - x.
So f'(x) = -sin(x) - 1.
Using the formula, you have:
$$x_n = \frac{\cos (x_{n - 1}) - x_{n - 1}}{-\sin (x_{n - 1}) - 1}$$
Since the two graph cos(x) and x will cut each other at some x lies between 0, and pi. So you just choose $$x_0 \in [0, \ \pi]$$, eg: x_0 = 0.5,...
Viet Dao,