Sine rule: Difference between revisions

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\sin\beta = \frac{a}{d}.
\sin\beta = \frac{a}{d}.
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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;.  
This follows because the two angles, &alpha; and &beta;, in Fig. 2  share a segment of the circle (have the chord ''a'' in common). By a well-known theorem of plane geometry it follows that the two 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]]
[[Image:Proof sine rule2.png|right|thumb|200px|Fig. 3]]


===Proof of sine rule===
===Proof of sine rule===

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Fig. 1. Sine rule: sinα:sinβ:sinγ=a:b:c

In trigonometry, the sine rule (also known as Law of Sines) 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. Equivalently,

Proof

The easiest proof is purely geometric.

Lemma

Fig. 2. The angles α and β share the chord a. The center of the circle is at C and its diameter is d.

In Fig. 2 the angle β satisfies,

This follows because the two angles, α and β, in Fig. 2 share a segment of the circle (have the chord a in common). By a well-known theorem of plane geometry it follows that the two angles are equal. The angle α, having the diameter of the circle d as one of its sides, has as opposite angle a right angle. Hence sin(α) = a/d, the length of chord a divided by the diameter d. It follows that the angle β, with a corner on the circumference of the same circle as α, but other than that arbitrary, has the same sine as α.

Fig. 3

Proof of sine rule

From the lemma follows that the angles in Fig. 3 are

where d is the diameter of the circle. From this result the sine rule follows.

External link

Life lecture on Sine Law