Value of cos 1 (angle is in radians)?

Question

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.

Answer 1

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 $.

Answer 2

We may approximate as follows:

$1 Rad ≈ \frac{\pi}{3.15}=\frac{\pi}{3}\times \frac{1}{1,05}$

$1.05=\frac{100}{105}=\frac{105-5}{105}=1-\frac{1}{21}$

$1 Rad= \frac {\pi}{3}-\frac{\pi}{63}$

$cos(\frac {\pi}{3}-\frac{\pi}{63})=cos\frac {\pi}{3}cos\frac{\pi}{63}+sin\frac {\pi}{3}sin\frac{\pi}{63} ≈0.499+0.043=0.542$