Trigonometry/Law of Sines

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:

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

The law can also be written as the reciprocal:

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

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

$h=B\sin a$

• Using the triangle cbo gives

$h=A\sin b$

• Eliminate h from these two equations

$A\sin b=B\sin a$

• Rearrange

$\frac{A}{\sin a}=\frac{B}{\sin b}$

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