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If I know sin/cos/tan... of an angle x, how do I know the x without look at the table?

How do computers calculate the angle? How the table of sin cos tan was constructed? How the old mathematicians knew this angles without computers?

Blue
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thiago
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  • Ah, so you're asking about the arc-functions (arcsine, ...). – Sean Roberson Nov 27 '20 at 01:49
  • yeah but how calculate it by hand? – thiago Nov 27 '20 at 01:55
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    Related: "How does a calculator calculate the sine, cosine, tangent using just a number?" . It's about calculating trig values instead of inverse-trig values, but the ideas are similar. A site search should give you additional relevant results; this kind of question has been asked many times. – Blue Nov 27 '20 at 02:16
  • “Old mathematicians”. How old are we talking? Newton-Raphson and other numerical methods were developed in the 17th Century. Maybe the ancient Greeks around the 5th Century BC would have used something like bisection/binary search? Just speculating though – Adam Rubinson Nov 27 '20 at 02:24
  • I thought that was a simple formula for the angle, but apparently you uses some expressions to aproximate. Thanks. – thiago Nov 27 '20 at 02:45
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    @thiago: The lack of a "simple formula" in these cases is why we call the trig and inverse-trig functions transcendental: they transcend "simple" algebra in the sense that there's no polynomial relation between their arguments and their results. The discovery that power series (loosely speaking, "infinite polynomials") relate arguments and results was a key advancement in the understanding of the trig functions (and functions in general). – Blue Nov 27 '20 at 02:54
  • Oh, now I understand, thank you very much for explaining. – thiago Nov 27 '20 at 04:20

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