Two waves $y_1 = A_1\sin (ωt - β_1)$ and $y_2 = A_2 \sin (ωt - β_2)$ superimpose to form a resultant wave. The amplitude of this resultant wave is given by

$A_1 + A_2$ $|A_1 - A_2|$ $\sqrt{A_1^2+A_2^2+2A_1A_2\cos(β_1-β_2)}$ $\sqrt{A_1^2+A_2^2-2A_1A_2\cos(β_1-β_2)}$

$\sqrt{A_1^2+A_2^2+2A_1A_2\cos(β_1-β_2)}$

Phase diff = $ωt - β_1-ωt - β_2=β_2-β_1$ ∴ Amplitude of resultant wave $=\sqrt{A_1^2+A_2^2+2A_1A_2\cos(β_2-β_1)}$ $=\sqrt{A_1^2+A_2^2+2A_1A_2\cos(β_1-β_2)}$  [As $\cos(-θ)=+\cos θ$]