Difference between revisions of "Sine theorem"
From Encyclopedia of Mathematics
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| − | For any triangle in the Euclidean plane with sides | + | {{TEX|done}} |
| + | 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 | + | hold, where $R$ is the radius of the circumscribed circle. |
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In spherical geometry the sine theorem reads | In spherical geometry the sine theorem reads | ||
| − | + | $$\frac{\sin a}{\sin A}=frac{\sin b}{\sin B}=frac{\sin c}{\sin C},$$ | |
and in Lobachevskii geometry: | and in Lobachevskii geometry: | ||
| − | + | $$\frac{\sinh a}{\sin A}=frac{\sinh b}{\sin B}=frac{\sinh c}{\sin C}.$$ | |
====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> | ||
Revision as of 14:28, 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