Difference between revisions of "Sine theorem"
From Encyclopedia of Mathematics
(TeX) |
|||
| Line 2: | Line 2: | ||
For any triangle in the Euclidean plane with sides $a,b,c$ and opposite angles $A,B,C$, respectively, the equalities | For any triangle in the Euclidean plane with sides $a,b,c$ and opposite angles $A,B,C$, respectively, the equalities | ||
| − | $$\frac{a}{\sin A}=frac{b}{\sin B}=frac{c}{\sin C}=2R$$ | + | $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$$ |
hold, where $R$ is the radius of the circumscribed circle. | hold, where $R$ is the radius of the circumscribed circle. | ||
| Line 11: | Line 11: | ||
In spherical geometry the sine theorem reads | In spherical geometry the sine theorem reads | ||
| − | $$\frac{\sin a}{\sin A}=frac{\sin b}{\sin B}=frac{\sin c}{\sin C},$$ | + | $$\frac{\sin a}{\sin A}=\frac{\sin b}{\sin B}=\frac{\sin c}{\sin C},$$ |
and in Lobachevskii geometry: | and in Lobachevskii geometry: | ||
| − | $$\frac{\sinh a}{\sin A}=frac{\sinh b}{\sin B}=frac{\sinh c}{\sin C}.$$ | + | $$\frac{\sinh a}{\sin A}=\frac{\sinh b}{\sin B}=\frac{\sinh c}{\sin C}.$$ |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H.S.M. Coxeter, S.L. Greitzer, "Geometry revisited" , Math. Assoc. Amer. (1975)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H.S.M. Coxeter, S.L. Greitzer, "Geometry revisited" , Math. Assoc. Amer. (1975)</TD></TR></table> | ||
Latest revision as of 14:29, 19 March 2014
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.
Comments
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://encyclopediaofmath.org/index.php?title=Sine_theorem&oldid=31397
Sine theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sine_theorem&oldid=31397
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