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I am trying to be able to be able to do this on paper without help from a calculator and I’m a complete beginner. If you can, please explain in the simplest terms without excluding any math or detail.

user11937382
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  • I'm not sure there's a good way to calculate it - you might do best by "calculate a table of values of $\sin$ and look for where the desired value occurs" - and then perhaps using root finding methods to improve your approximation. I'm sure they made tables of this back in the day, but I bet it was a lot of work - but maybe someone else knows a better answer! – Milo Brandt Jan 19 '20 at 15:36
  • https://math.stackexchange.com/questions/197874/maclaurin-expansion-of-arcsin-x – saulspatz Jan 19 '20 at 15:57
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    When you say you want to calculate arcsin, could you tell us how you calculate the sine? – Martin Argerami Jan 19 '20 at 16:50
  • I don’t know how to do that either. – user11937382 Jan 19 '20 at 16:55

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There is an infinite expansion for $\sin^{-1}x$ that you could do by hand which goes as- $$\sin^{-1}x= x + \frac 1 2 \frac {x^3} 3 + \frac {1\cdot 3} {2 \cdot 4} \frac {x^5} 5 + \frac {1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6} \frac {x^7} 7 + \cdots$$ Ofcourse, the answer will not be exact but calculating the first three terms should be sufficient.

Sam
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