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I was just curious as to how you would calculate it without a calculator. I don't care if it's in radians or degrees, but I just would like it to be specified.

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    $\begingroup$ Learn some values by heart and interpolate. $\endgroup$
    – user5402
    Jul 30, 2013 at 19:45
  • $\begingroup$ @metacompactness: that's what calculators do... $\endgroup$
    – DJohnM
    Jul 30, 2013 at 19:48
  • $\begingroup$ @User58220 And he wants to be a human calculator. $\endgroup$
    – user5402
    Jul 30, 2013 at 19:51
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    $\begingroup$ Tables of the tangent function were made by Islamic mathematicians about a millenium ago. If my calculator dies, no problem, it is back to the tables. Any further multiplications needed to find a numerical answer can be done by slide rule. $\endgroup$
    – André Nicolas
    Jul 30, 2013 at 19:53
  • $\begingroup$ There are nice continued fraction approximations. $\endgroup$
    – ccorn
    Jul 30, 2013 at 20:40

2 Answers 2

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Use Taylor series for sin in radians:$$\sin(x) = x-\frac{x^3}{3!}+\frac{x^5}{5!}-...$$ Then calculate tan:$$\tan(x)=\frac{\sin(x)}{ \sqrt{1-\sin^2(x)}}$$

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  • $\begingroup$ @DavisDude $\sin^2 (x)$ means $[\sin(x)]^2$. $\endgroup$
    – Lord Soth
    Jul 30, 2013 at 21:15
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The Taylor series for $\sin$ and $\cos$ converge quickly enough that a few digits of accuracy is possible with relatively few computations by hand (generally, computing 2-3 terms will give about two digits of accuracy). Then the division can be carried out.

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