# How can I prove the sine theorem?

1. Oct 30, 2009

1. The problem statement, all variables and given/known data

How can I prove this theorem in the triangle ABC? sin(A)/a=sin(B)/b=sin(C)/c

2. Relevant equations

A*B and ...

3. The attempt at a solution

I have drawn a triangle and tried to prove it, but i couldn't. (I don't know how to send a picture to my post!) I know how to prove the cosine theorem but i can't prove the sine theorem.

2. Oct 30, 2009

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Last edited: Oct 30, 2009
3. Oct 30, 2009

### Pagan Harpoon

Consider the three vectors A to B, B to C and C to A. Now consider geometrically their cross products. The magnitude of each is equal to twice the area of the triangle, so the three are equal.

4. Oct 30, 2009

### Gib Z

I particularly like this one:

For any triangle, inscribe it in a circle. So the triangle touches the circle at 3 points, call them A, B, and C, and their corresponding angles a, b, and c, and the lengths of the sides opposite them L(a), L(b), L(c).

Move point A along the circumference so that AB passes the center of the circle. Angles subtended by equal chords onto the circumference are equal, so angle BAC is still a.

Now we have a triangle in a semi circle, and so angle ACB is equal to 90 degrees. By trigonometry, sin a = L(a)/Diameter, and so Diameter = sin a/L(a)

Making similar transformations for the other sides shows sin b/L(b) and sin c/L(c) are also equal to the diameter, and hence equal to each other.

5. Oct 31, 2009