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Difference between revisions of "Law of Sines"

(Added proof for the non-extended law of sines.)
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Given a [[triangle]] with sides of length a, b and c, opposite [[angle]]s of measure A, B and C, respectively, and a [[circumcircle]] with radius R,  <math>\frac{a}{\sin{A}}=\frac{b}{\sin{B}}=\frac{c}{\sin{C}}=2R</math>.
 
Given a [[triangle]] with sides of length a, b and c, opposite [[angle]]s of measure A, B and C, respectively, and a [[circumcircle]] with radius R,  <math>\frac{a}{\sin{A}}=\frac{b}{\sin{B}}=\frac{c}{\sin{C}}=2R</math>.
  
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==Proof of the Law of Sines==
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==Proof ==
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=== Method 1 ===
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In the diagram below, circle <math> O </math> [[circumscribe]]s triangle <math> ABC </math>.  <math> OD </math> is perpendicular to <math> BC </math>.  Since <math> \triangle ODB \cong \triangle ODC </math>, <math> BD = CD = \frac a2 </math> and <math> \angle BOD = \angle COD </math>.  But <math> \angle BAC = 2\angle BOC </math> making <math> \angle BOD = \angle COD = \theta </math>.  Therefore, we can use simple trig in right triangle <math> BOD </math> to find that
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<center><math> \sin \theta = \frac{\frac a2}R \Leftrightarrow \frac a{\sin\theta} = 2R. </math> </center>
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The same holds for b and c thus establishing the identity.
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<center>[[Image:Lawofsines.PNG]]</center>
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=== Method 2 ===
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This method only works to prove the regular (and not extended) Law of Sines.
 +
 
 
The formula for the area of a triangle is:  
 
The formula for the area of a triangle is:  
 
<math> \displaystyle [ABC] = \frac{1}{2}ab\sin C </math>
 
<math> \displaystyle [ABC] = \frac{1}{2}ab\sin C </math>

Revision as of 21:11, 29 June 2006

Given a triangle with sides of length a, b and c, opposite angles of measure A, B and C, respectively, and a circumcircle with radius R, $\frac{a}{\sin{A}}=\frac{b}{\sin{B}}=\frac{c}{\sin{C}}=2R$.

Proof

Method 1

In the diagram below, circle $O$ circumscribes triangle $ABC$. $OD$ is perpendicular to $BC$. Since $\triangle ODB \cong \triangle ODC$, $BD = CD = \frac a2$ and $\angle BOD = \angle COD$. But $\angle BAC = 2\angle BOC$ making $\angle BOD = \angle COD = \theta$. Therefore, we can use simple trig in right triangle $BOD$ to find that

$\sin \theta = \frac{\frac a2}R \Leftrightarrow \frac a{\sin\theta} = 2R.$

The same holds for b and c thus establishing the identity.

Lawofsines.PNG

Method 2

This method only works to prove the regular (and not extended) Law of Sines.

The formula for the area of a triangle is: $\displaystyle [ABC] = \frac{1}{2}ab\sin C$

Since it doesn't matter which sides are chosen as $a$, $b$, and $c$, the following equality holds:

$\displaystyle \frac{1}{2}bc\sin A = \frac{1}{2}ac\sin B = \frac{1}{2}ab\sin C$

Multiplying the equation by $\displaystyle \frac{2}{abc}$ yeilds:

$\displaystyle \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

See also

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