# Sine theorem

For any triangle in the Euclidean plane with sides $a,b,c$ and opposite angles $A,B,C$, respectively, the equalities

$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$$

hold, where $R$ is the radius of the circumscribed circle.

In spherical geometry the sine theorem reads

$$\frac{\sin a}{\sin A}=\frac{\sin b}{\sin B}=\frac{\sin c}{\sin C},$$

and in Lobachevskii geometry:

$$\frac{\sinh a}{\sin A}=\frac{\sinh b}{\sin B}=\frac{\sinh c}{\sin C}.$$

#### References

 [a1] H.S.M. Coxeter, S.L. Greitzer, "Geometry revisited" , Math. Assoc. Amer. (1975)
How to Cite This Entry:
Sine theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Sine_theorem&oldid=31398
This article was adapted from an original article by Yu.A. Gor'kov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article