Trig problems (Interference/diffraction and slits))

[AznBoi]

Interference and diffraction problems involving slits includes a lot of geometry and trig right? I'm having a hard time deriving the equtions: $$d sin \theta=m \lambda$$ and $$y=\frac{m\lambda L}{d}$$ because there are some many geometry stuff you need to do to connect the angles.

Can someone explain to me why $$tan\theta=sin\theta$$ if theta is small? thanks

 

[Integral]

Can someone explain to me why $sin \theta = tan \theta$ if theta is small? thanks

Look at a right triangle. Now consider the definition of Sin and Tan.

$$Sin \theta = \frac o h$$

$$Tan \theta = \frac o a$$

if $\theta$ is small the adjacent side is very nearly the same length as the hypotenuse. So Sin and Tan are very nearly equal. Another way to see this is to look at the Taylor series expansion of Tan and Sin both have the same first term $\theta$ when you drop all but the linear term, you have $sin \theta = tan \theta = \theta$

 

[AznBoi]

Okay I get it now, thanks. How does one develop the ability to see these kinds of connections without drawing triangles etc.? Is it just practice with these type of trig problems?

Anyways, is this equation used to determine the distance bewteen bright/dark fringes in single slit diffractions of light? $$y=\frac{m\lambda L}{d}$$ Except that (d) is replaced with (w), the width of the slit right? This equation applies to both double slit and single slit? So the only difference is that (d) is the distance between the slits in the two slit diffractions while (d) is the width of the slit in single slit diffractions? Thanks.

 

[Office_Shredder]

Okay I get it now, thanks. How does one develop the ability to see these kinds of connections without drawing triangles etc.? Is it just practice with these type of trig problems?

sinx=x for small x is a pretty standard approximation that crops up time and again... and of course, tanx = sinx/cosx, and cosx is about 1 for small x