Let there be two vectors \(\vec A\) and \(\vec B\) with D between them.

Using parallelogram law of vector addition resultant.

\(\vec R=\vec A+\vec B\)

Raw DF ⊥ OC extended as OF

Considering right angled ∆OFD

OD^{2} = OF^{2} + DF^{2}

= (OC + CF)^{2} + DF^{2}

= (A + Bcos θ)^{2} + (Bsin θ)^{2}

R^{2} = A^{2} + B^{2} + 2ABcos θ (∵ cos^{2}θ + sin^{2} θ = 1)

R = \(\sqrt{A^2+B^2+2ABcos\,\theta}\) …(i)

tan α = \(\frac{DF}{OF}=\frac{DF}{OC+CF}\)

= \(\frac{B\,sin \,\theta}{A+B\,cos\,\theta}\)

α = \(tan^{-1}\frac{B\,sin \,\theta}{A+B\,cos\,\theta}\) ...(ii)

Eqn. (i) is called law of cosines.