Sine Law Û Cosine Law

Sine Law and Cosine Law are two important laws in solving triangles in trigonometry. Can we prove Sine Law using Cosine Law? and vice versa?

Prove Sine Law by Cosine Law

(1)

which is cyclic in a, b, c, that is, when we replace a by b, b by c and c by a, we can get the same function.

Therefore, when we work out we get the same expression in (1).

We then get

and the Sine Law

Prove Cosine Law by Sine Law

In the proof, some knowledge of compound angle formulas is needed.

Since by Sine Law, sin A = ka, sin B = kb, sin C = kc, where k is a constant.

Substitute in the above expression, we get

which is the familiar Cosine Law.