# Is it possible to calculate $\sin(\alpha)$ (and other trigonometric functions) as a rational number? [duplicate]

I am creating a computer library for arbitrary-precision calculations, by expressing numbers as rationals (with an arbitrary-precision numerator and denominator).

Now, I am exploring the possibility to add trigonometric functions to this library. I know from college that certain values like $\sin(\frac{1}{2}) = \frac{\pi}{2}$ and $\sin($ are defined and easy to remember.

Is there a way to find rational solutions of $\sin(\alpha)$ for all possible angles $\alpha$ ?

## marked as duplicate by Dietrich Burde, zz20s, MathOverview, Ben Sheller, Antonios-Alexandros RobotisMar 25 '16 at 17:26

• How is $\sin(\frac{1}{2})=\frac{\pi}{2}$? Do you instead mean, that there are specific values e.g. $\sin(\frac{\pi}{6})=\frac{1}{2},~\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$ that are easy to remember? – Hirshy Mar 25 '16 at 16:03
• What do you mean by "rational solutions of $\sin(\alpha)$"? Do you mean rational approximations to $\sin(\alpha)$? $\qquad$ – Michael Hardy Mar 25 '16 at 16:58