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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.

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    What makes you think that an exact value can be obtained? – 2017-01-31
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    @imranfat: did he say that ? – 2017-01-31
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    @YvesDaoust In the first line?? – 2017-01-31
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    @imranfat: I don't see the word *exact* nowhere. – 2017-01-31
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    @YvesDaoust He doesn't talk about rounding or decimals, if someone asks "Calculate $arccos0.5$ without calculator, do you expect $1.04$ or $\pi/3$? – 2017-01-31
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    @imranfat: it depends if the quantity has a closed-form expression or not. – 2017-01-31
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    That's true, but if the OP assumes that there is a closed form, whereas in fact there may not be (as far as I know), then.... – 2017-01-31
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    @imranfat: I don't see any assumption about a closed form in the post. – 2017-01-31
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    You can lookup this amazing table: http://intmstat.com/blog/2011/06/exact-values-sin-degrees.pdf; you can find more by using the angle bisection formula. But in most cases, (like your example I guess) there is no closed-form formula. – 2017-01-31

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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.

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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

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    There are more acute angles $x$ such that $x/\pi$ and $\cos x$ are radical numbers. ($x$ is in radians). – 2017-01-31
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    How do you think that the first trigonometric tables were established ? – 2017-01-31
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    @Yves the first trigonometric table were made by calulating the values of sin and cos from right angled triangles with one angle 0,30,45,.......respectively. A different triangle was drawn for each angle and the values of sin and cos were measured physically (since, sinx = altitude/hypo. and cosx = base/hypo) – 2017-02-03
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    @mikealise: I assume this was done at a time the calculators were still made of stones... Anyway, very early tables was obtained numerically, by the half-angle fomulas. – 2017-02-03
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    @Yves They didn't had calculators at that time (around 500bc), instead they used abacus. – 2017-02-03
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    @mikealise: I meant electronic calculators (by the way still made of silicon). – 2017-02-03
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$\theta = \arccos(2/\sqrt{5}) = \arctan(1/2)$ 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 $z^{2n}-1$).