Is high school wave peak equation accurate enough for real world?

Question

After preparing for my physics test I came across the equation for calculating peaks and valleys of light in double slit experiment - $N\lambda = dsin\theta$. But, looking at how it is proven. I'm wondering if it's accurate.

Because the proof is, as far as I understand, using the fact that on large distances(compared to distance between slits(d)), 2 lines connecting slits will look perpendicular. So, by drawing a line perpendicular to one, the residual distance should be wavelength.

This picture should help.

Source: https://www.khanacademy.org/science/physics/light-waves/interference-of-light-waves/v/youngs-double-slit-part-2

Here we see the right triangle on the left. But, shouldn't that triangle be the approximation?

I'm concerned about this because, as we move toward more slits, we see something called diffraction grating. Shouldn't this diffraction grating come into effect here too, if our results aren't accurate? Or are our results just slightly innacurate, such that while diffraction grating happens there the real result is actually very close?

Thanks for help. I hope you understand what I'm curious about!

Answer 1

As you have spotted there is an approximation involved in the theory. Almost every physical theory will involve some kind of approximation and/or is designed to only cover a restricted range of parameters. Many theories have no means of getting a useful form of solution (like a formula) and need numerical simulations to produce practical results.

The mathematics of a more detailed analysis of this problem would be a significant burden for high-school students and even first or second year undergraduate students, who would also be getting more advanced mathematical training as they studied physics.

The important issue is what the experiment shows - the wave behavior of light. The basic theory is "good enough" for this purpose. There are much more advanced versions of this experiment which are used in demonstrating aspects of quantum theory and these, unsurprisingly, require more advanced theories underpinning them.