Sine rule: Difference between revisions

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imported>Paul Wormer
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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''' (also known as '''[[Law of sines|Law of Sines]]''') relates in a [[triangle]] the [[sine]]s of the three angles and the lengths of their opposite sides,
:<math>
\frac{a}{\sin\alpha} = \frac{b}{\sin\beta}= \frac{c}{\sin{\gamma}} = d,
</math>
where ''d'' is the diameter of the circle circumscribing the triangle and the angles and the lengths of the sides are defined in Fig. 1. From this follows  that the ratio of the sines of the angles of a triangle is equal to the ratio of the lengths of the opposite sides.
==Proof==
The easiest proof is purely geometric, not algebraic.
===Lemma===
[[Image:Proof sine rule.png|left|thumb|200px|Fig. 2. The angles &alpha; and &alpha;' share the chord ''a''. The center of the circle is at ''C'' and its diameter is ''d''.]]
In Fig. 2 the arbitrary angle &alpha; satisfies,
:<math>
\sin\alpha = \frac{a}{d},
</math>
where ''d'' is the diameter of the circle and ''a'' is the chord opposite &alpha;. To prove this we consider the angle &alpha;' that has the diameter of the circle as one of its sides, see Fig. 2.  The two angles, &alpha; and &alpha;'  share a segment of the circle (have the chord ''a'' in common).  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''.  A  well-known theorem of plane geometry states that &alpha; = &alpha;'  and it follows that the angle &alpha; 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.
 
==External link==
[http://madmath.madslideruling.com/precalculus/sinerule.html Life lecture on Sine rule]

Latest revision as of 05:16, 21 October 2008

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