The sine and cosine rule can be quickly derived by considering a triangle, divided as shown in the figure below. Angles are given by capital letters, and lengths are given by lowercase letters.
First work out the length . By looking at the left part of the triangle, the definition of the sine function implies . Similarly, looking at the right part of the triangle implies . Equating these two expressions instantly gives the sine rule
The cosine rule only requires a little more work. Applying Pythagoras to the right hand part of the triangle gives
Substituting , and gives
which simplifies to the standard cosine rule
by applying the trigonometric identity .