Difference between revisions of "Law of Sines"
m (Law of sines moved to Law of Sines: consistency (I hope)) |
(Template) |
||
Line 10: | Line 10: | ||
<center>[[Image:Lawofsines.PNG]]</center> | <center>[[Image:Lawofsines.PNG]]</center> | ||
+ | {{asy replace}} | ||
=== Method 2 === | === Method 2 === | ||
Line 15: | Line 16: | ||
The formula for the area of a triangle is: | The formula for the area of a triangle is: | ||
− | <math> | + | <math>[ABC] = \frac{1}{2}ab\sin C </math> |
Since it doesn't matter which sides are chosen as <math>a</math>, <math>b</math>, and <math>c</math>, the following equality holds: | Since it doesn't matter which sides are chosen as <math>a</math>, <math>b</math>, and <math>c</math>, the following equality holds: | ||
− | <math> | + | <math>\frac{1}{2}bc\sin A = \frac{1}{2}ac\sin B = \frac{1}{2}ab\sin C </math> |
− | Multiplying the equation by <math> | + | Multiplying the equation by <math>\frac{2}{abc} </math> yeilds: |
− | <math> | + | <math>\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} </math> |
==See also== | ==See also== |
Revision as of 18:16, 24 September 2007
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, .
Contents
Proof
Method 1
In the diagram below, circle circumscribes triangle
.
is perpendicular to
. Since
,
and
. But
making
. Therefore, we can use simple trig in right triangle
to find that
![$\sin \theta = \frac{\frac a2}R \Leftrightarrow \frac a{\sin\theta} = 2R.$](http://latex.artofproblemsolving.com/0/3/f/03f858d898d88e6bb7665ea3514b180f2e8ebcf5.png)
The same holds for b and c thus establishing the identity.
This picture could be replaced by an asymptote drawing. It would be appreciated if you do this.
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:
Since it doesn't matter which sides are chosen as ,
, and
, the following equality holds:
Multiplying the equation by yeilds:
See also
This article is a stub. Help us out by expanding it.