Sine rule: Difference between revisions

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imported>Paul Wormer
(Redirecting to Law of sines)
 
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#REDIRECT [[Law of sines]]
[[Image:Sine rule.png|right|thumb|300px|Fig. 1. '''Sine rule:''' sinα:sinβ:sinγ=a:b:c]]
In [[trigonometry]], the '''sine rule''' states that the ratio of the sines of the angles of a triangle is equal to the ratio of the lengths of the opposite sides, see Fig.1.
:<math>
\frac{\sin\alpha}{\sin{\beta}} = \frac{a}{b},\qquad \frac{\sin\beta}{\sin{\gamma}} = \frac{b}{c}.
</math>
==Proof==
The easiest proof is purely geometric.
===Lemma===
[[Image:Proof sine rule.png|left|thumb|200px|Fig. 2. The angles &alpha; and &beta; share the chord ''a''. The center of the circle is at ''C'' and its diameter is ''d''.]]
In Fig. 2 the angle &beta; satisfies,
:<math>
\sin\beta = \frac{a}{d}.
</math>
Indeed, in Fig. 2 we see two angles, &alpha; and &beta;, that share a segment of the circle (have the chord ''a'' in common). By a well-known theorem of plane geometry it follows that the angles are equal. The angle &alpha;, having the diameter of the circle ''d'' as one of its sides, has as opposite angle a right angle. Hence  sin(&alpha;) = ''a''/''d'',  the length of chord ''a'' divided by the diameter ''d''.  It follows that the angle &beta;, with a corner on the circumference of the same circle as &alpha;, but other than that arbitrary, has the same sine as &alpha;.
 
[[Image:Proof sine rule2.png|right|thumb|200px|Fig. 3]]
===Proof of sine rule===
From the lemma follows that the angles in Fig. 3 are
:<math>
\sin\alpha = \frac{a}{d}, \quad\sin\beta = \frac{b}{d},\quad\sin\alpha = \frac{c}{d},
</math>
where ''d'' is the diameter of the circle. From this result the sine rule follows.

Latest revision as of 05:16, 21 October 2008

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