Derivation of the Law of Sines

Introduction

The law of sines relates the sides and angles of a triangle. It states that in any triangle, the ratio of the side to the sine of the opposite angle of that side is the same for all three sides. The sine law states that  a triangle ABC with side lengths a, b, and c,

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

The derivation of this law is quite easy. It only require basic knowledge in trigonometric functions.

Theorem

For any triangle ABC,

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

Proof

Let ABC be a triangle and let h be its altitude. Recall that the sine of an angle in a triangle is the ratio of the length of its opposite leg to the length of its hypotenuse.

law of sines

In triangle ACD,

\displaystyle \sin A = \frac{h}{b}

This means that h = b \sin A.

In triangle BCD,

\displaystyle\sin B = \frac{h}{a}

which means that h = a \sin B.

Therefore, b \sin A = a \sin B.

Dividing both sides by \sin A \sin B gives us

\displaystyle \frac{b}{\sin B} = \frac{a}{\sin A}.

which is what we want to show. Now, it remains to be shown that

\frac{b}{\sin B} = \frac{c}{\sin C}.

Although this directly follows from the proof above, it is left as an exercise.

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