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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.

DavisDude
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    Learn some values by heart and interpolate. – user5402 Jul 30 '13 at 19:45
  • @metacompactness: that's what calculators do... – DJohnM Jul 30 '13 at 19:48
  • @User58220 And he wants to be a human calculator. – user5402 Jul 30 '13 at 19:51
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    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. – André Nicolas Jul 30 '13 at 19:53
  • There are nice [continued fraction approximations](http://math.stackexchange.com/questions/432771/continued-fraction-for-tannx). – ccorn Jul 30 '13 at 20:40

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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:$$\tan(x)=\frac{\sin(x)}{ \sqrt{1-\sin^2(x)}}$$

DJohnM
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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.