1. convergent equation

Can be a convergent equation (Term that I invent) like this example:

Find x in the equation:
X(cos(X)) = C
when C [Constant] = 3.

I use the calculator and choose x (x is in degree not Radians) the value 4.
4(cos(4)) = 3.99...
I need to decrease the value to find X.
Let x be 3:
3(cos(3)) = 2.99...
So the x is between 3 to 4.
You can continue to find x in that way by increasing and decreasing the value of X by calculator.
And Also:
You can choose any number of the equation:
X(cos(x)) = C
with different value of C.

Can you give me more examples of this phenomenon?
Is there a reason to this phenomenon?
Why it happen?

2. Re: convergent equation

Here's the plot for the equation x * cos(x * pi/180) from -5 to 5 and from -1000 to 1000.

Wolfram|Alpha: Computational Intelligence
Wolfram|Alpha: Computational Intelligence

Why is the curve of the equation close to the equation y = x between -5 and 5? It's because cos(0) = 1 so any number x close to 0 in radians will make cos(x) a value close to 1.
Hence x * cos(x) will be approximately x * cos(0) = x

Conclude two things:
1) The function is bounded above and below by curves y = x and y = -x. This is because cos(x) is bounded between -1 and 1
2) In the neighborhood when cos(x * pi/180) is close to 1, the function x * cos(x * pi/180) will be approximately x.