Derivation of Young's Double Slit Formula...

Bright fringes occur at points of constructive interference. This happens when the path difference between the waves is nλ (where n = 0, 1, 2...)

Dark fringes occur at points of destructive interference; this happens when the path difference is nλ/2 (where n = 1, 2, 3...)

From the diagram, for the first bright fringe, the path difference λ is given by the distance s₁Z.  Using trigonometry, we can show that:

sin θ = = , and

tan θ =

Now, for small values of θ, sinθ >> tanθ, so:

= , and so