Is it valid to use the small angle approximation in this problem?

I'm trying to solve this question.

I use the equation $$\boxed{d\sin(ฮธ)=mฮป}$$

I keep finding $$575$$ $$\mathrm{nm}$$, the answer key says the result is $$485$$ $$\mathrm{nm}$$ but I think it's only possible to get that result by assuming $$\tan(\theta) = \theta$$ which gives the angle to be used in the equation above.

I don't think we can use the small-angle approximation here as $$\theta$$ is not too small (not $$\theta\ll 1$$).

What I'm missing?

• Yes the small angle approximation is valid here, have you tried calculating $\tan{\theta}$ here? Because the difference between that and $\theta$ will be in the thousandths place. Aug 24, 2021 at 23:49
• All right. My big question is why we find a different result when we don't use the small-angle approximation. For example, if tan๐ is equal to ๐, it doesn't matter which one I use. Taking tan๐ gives 575 nm as the final result, while taking ๐ gives 485 nm as the final result. Aug 25, 2021 at 1:19
• It's hard to say as you haven't shown any of your work, I'd have to guess at why your getting a error which I believe to be arithmetic in nature. Aug 25, 2021 at 1:43
• Could it be a radians-degrees thing? Aug 25, 2021 at 1:47
• @Triatticus, the distance between the centre of the central bright spot and the second-order dot is 1.46 and the distance between the grating and the screen is 1.98. Doing arctan = (1.46/1.98) would give the angle theta which is 0.63537 rad, whereas tan(0.63537) is 0.7373, so it's not in the thousandths and I don't think the small-angle approximation is valid here as using tan๐ and ๐ give quite different results in the wavelength calculation like 575 nm and 485 nm. Aug 25, 2021 at 2:14

$$0.635$$ rad is too large for the approximation $$\tan\theta\approx\theta$$ to be valid, since the error percentage has exceeded $$1\%$$. To keep it within $$1\%$$, that is to the thousandths place, you need at least small as $$0.2441$$ rad, as give by https://en.wikipedia.org/wiki/Small-angle_approximation. In practice (experiment), if we encountered such problem, in order to make the small-angle approximation be valid, we can increase the distance between the grating and the screen for a fixed wavelength.