# How would you calculate the Tangent without a calculator? [duplicate]

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.

• Learn some values by heart and interpolate. Jul 30, 2013 at 19:45
• @metacompactness: that's what calculators do... Jul 30, 2013 at 19:48
• @User58220 And he wants to be a human calculator. Jul 30, 2013 at 19:51
• 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. Jul 30, 2013 at 19:53
• There are nice continued fraction approximations. Jul 30, 2013 at 20:40

Use Taylor series for sin in radians:$$\sin(x) = x-\frac{x^3}{3!}+\frac{x^5}{5!}-...$$ Then calculate tan:$$\tan(x)=\frac{\sin(x)}{ \sqrt{1-\sin^2(x)}}$$
• @DavisDude $\sin^2 (x)$ means $[\sin(x)]^2$. Jul 30, 2013 at 21:15
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.