How can we calculate the value of $\cos 1$ where the angle is in radians (and not degrees). If this isn't possible, can we somehow find whether this value would be rational or irrational?
P.S: I know how to determine the irrationality of $\cos 1$ when angle is in degrees, and also am aware of its explicit formula. But those methods cannot be used here.
Best Answer
You can find the value of any $ \cos x $ via a power series:
$ \cos x = x-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}... $
This can be rewritten as:
$ \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!} $
And solved by painstaking summation to calculate that $ \cos1 \approx 0.540 $.