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.

Sine and Cosine Rule
Simple derivation of sine and cosine rule.

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

 \frac{\sin A}{a} = \frac{\sin C}{c}.

The cosine rule only requires a little more work. Applying Pythagoras to the right hand part of the triangle gives

 (BC)^2 = (OB)^2 + (OC)^2.

Substituting BC = a, OB = c\sin A and OC = b-c\cos A gives

a^2 = c^2\sin^2A + b^2 + c^2\cos^2A - 2bc\cos A

which simplifies to the standard cosine rule

a^2 = b^2 + c^2 - 2bc\cos A

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

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