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,

 

[tacman]

To solve for the intersecting point of these two waves, we can set them equal to each other and solve for x or y. In this case, we can use the inverse cosine function to find the intersecting point. By taking the inverse cosine of both sides of the equation, we can eliminate the cosine function and solve for the intersecting point. The solution will give us the value of x and y at the point where the two waves intersect.