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If I have $\arcsin(8/20)$, what angle is this? This is from Wikipedia and the answer is $21.8^{\circ}$. But I do not understand, by hand, how I can get the degrees.

This is from the Wikipedia page:

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$8/20 = .4$

$.4$ is the ratio of rise to run, but I don't know what they did after that.

How did that get to $21.8$ degrees (or $.38$ radians)?

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    In general you cannot solve for the angle "by hand" unless it is a "nice" angle for which you have memorized its sine or tangent or whatever. For instance, $\arctan(1)=45^\circ$ or $\arctan(\sqrt{3})=60^\circ$. – 2017-01-09
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    Arctangent rather than arcsine. They presumably used a calculator or some tables – 2017-01-09
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    @kccu So they just used a calculator? Is it simply a massive set of steps do get 21.8? – 2017-01-09
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    @Henry Thanks. I fixed it. – 2017-01-09
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    @johnny It's not that it's a "massive set of steps," but rather there is no set of steps which would get you to the answer. Unless it is a "nice" value, you just have to use a calculator or a table. – 2017-01-09
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    @kccu What would I be calculating? Perhaps that is the problem I do not understand. – 2017-01-09
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    @johnny I don't think I understand your question. Are you asking what you would type into the calculator, or what the meaning of the answer is? If you ask the calculator for $\arctan(0.4)$, the answer the calculator gives out is the angle $\theta$ such that $\tan(\theta)=0.4$. – 2017-01-09

2 Answers 2

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Generally, you don't. When $\sin x$ is algebraic $x$ is transcendental (with x represented in radians).

That means that the solution requires working with an infinite series.

One of the easier ones is.

$\arctan x = \sum_\limits{n=0}^{\infty} \frac {(-1)^n x^{2n+1})}{2n+1}$ (With the results in radians.)

You don't usually learn this until calculus. So, if you are in trig and this seems over your head, don't worry.

$\arcsin = \arctan \frac {0.4}{\sqrt {1-0.4^2}} = \arctan \frac {2}{\sqrt {21}}$

$\arctan \frac {2}{\sqrt {21}} \approx \frac {2}{\sqrt {21}} (1-\frac {4}{63}+\frac {16}{2205}\cdots)= 0.412$

$0.38$ radians is $\arctan 0.4 = \arcsin \frac 2{\sqrt{29}}$

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    As stated above, the power series for $\tan^{-1}\quad$ yields an angle in radians. My understanding of your question is that you want to know how to get degrees out of radians. If that is your question, just multiply the radians by 180, then divide by $\pi.$ – 2017-01-09
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$$ \arctan(x)=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+..... $$ The point to be noted here is that this series continues on without any end. This simply means that we cannot find the exact value of the $\arctan$ of any angle in radians. Hence, we approximate.

Calculators are able to find an approximated version of the $\arctan$ by using this series. They only consider a few hundred terms of this series.

If you consider more terms, you end up with an answer closer to the actual result.

Note that for small angles the second and the consecutive terms in the series become very small and can be neglected.

In the case of your question : $$ \arctan(0.4) \approx 0.4 - \frac{(0.4)^3}{3} + \frac{(0.4)^5}{5} \\ \implies \arctan(0.4) \approx 0.38 $$