Is there an intuitive and useful explanation for the fact that the fixed point of natural cosine is 0 point 7390851332?

Answer 1

It's not. A fixed point is where the function crosses the axis and so the (first) fixed point of cosine is pi/2 in radians, 90 in degrees and 100 in grad. I say first fixed point because it has another fixed point at 3pi/4 in rad, 270 in deg and 300 in grad.

To show it by iteration with rad, use the N-R equation xn+1=xn- f(xn)/f'(xn) which gives us xn+1=xn+cos(xn)/sin(xn) if we start with an initial point x0=pi/4=0.785398163 then the iterations give us:

x1=1.785398163

x2=1.567440065

x3=1.570796339

x4=1.570796327=pi/2 to 10 s.f.

There's some confusion here about the phrase "fixed point". 0.7390851332 is a fixed point in the sense that cos(0.7390851332)=0.7390851332 . This is not to be confused with stationary point.