How to calculate $\arccos (x)$?

Question

how do i calculate for example $\arccos (2/\sqrt [2] 5)$ without calculator? I had some exercises with calculator and i could not find any good explanation for calculating per hand.

Answer 1

When there is no closed-formula and no calculator is allowed, about the only resort is the Taylor formula.

In your case, you are lucky, as

$$\arccos\frac2{\sqrt 5}=\arctan\frac12,$$ and you can use the Gregory's series

$$\arctan\frac12=\frac12-\frac1{3\cdot2^3}+\frac1{5\cdot2^5}-\frac1{7\cdot2^7}+\cdots$$ which gives you at least two more bits of accuracy on every term with not too painful by-hand computation.

Answer 2

You cannot calculate it's value without calculator.

You can only calculate values of inverse trigonometric functions for common angles viz. 0, 30, 45, 60, 90, 180, etc

Answer 3

$\theta = \arccos(2/\sqrt{5}) = \arctan(1/2)$ is not a rational multiple of $\pi$. This can be seen from the fact that $\exp(i\theta) = (2+i)/\sqrt{5}$ is not an algebraic integer (its minimal polynomial is $z^4-(6/5) z^2+1$), but $\exp(i \pi m/n)$ is (since it's a root of $z^{2n}-1$).