# GCSE Trigonometry - Super Simple Sine and Cosine Rule

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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 $OB$. By looking at the left part of the triangle, the definition of the sine function implies $OB=c\sin A$. Similarly, looking at the right part of the triangle implies $OB=a\sin C$. 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 $BC = a$, $OB = c\sin A$ and $OC = b-c\cos A$ gives

which simplifies to the standard cosine rule

by applying the trigonometric identity $\sin^2A + \cos^2A = 1$.