Trigonometry/Law of Sines

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Consider this triangle:

It has three sides

• A, length A, opposite angle a at vertex a
• B, length B, opposite angle b at vertex b
• C, length C, opposite angle c at vertex c

The perpendicular, oc, from line ab to vertex c has length h

The Law of Sines states that:

The law can also be written as the reciprocal:

[edit] Proof

The perpendicular, oc, splits this triangle into two right-angled triangles. This lets us calculate h in two different ways

• Using the triangle cao gives

• Using the triangle cbo gives

• Eliminate h from these two equations

• Rearrange

By using the other two perpendiculars the full law of sines can be proved. QED.