Optics (Intensity of light): Which in degrees, which in rad?

[Const@ntine]

This might be a bit of a silly question, but it's been driving me nuts for a couple of hours now. Background first: I'm reading Serway's Physics for Scientists & Engineers, Vol 2 Ed.8, and I'm currently in the Optics Chapters (light and whatnot). In O3.3 (O3 is generally about light going through two openings and meeting up at some point at a surface) it has certain equations/formulas, but the problem is that it doesn't specify if the "phase difference" φ it introduces is measure in radians or degrees, or how the formulas work.

See, thus far I've been working with angles in degrees, so it's so far so good. Then I reached that Part, and it beared enough simalirites with the Wave Functions (it even points it out) from previous Chapters, so I figured φ would be measured in rad. Here are the given equations:

φ = 2π*d*sinθ/λ
φ is the Phase Difference, d is the distance between the two openings, and λ thus far has been the length of the light wave.

I = I_max*cos²(π*d*sinθ/λ)=I_max*cos²(φ/2)
I is the intensity that we can measure at the point the two waves meet.

My problem is that when I moved on to the exercises, I couldn't make any sense of it. I assumed that θ would be measured traditionaly in degrees, and when the time came to compute/measure φ, I'd turn my findings into radians. As we know, π = 3.14 rad = 180 degrees. But all of my results are out of whack. Sometimes I get the correct answer by not turning degrees into rads. Others I turn them, and I get slightly different results.

Can anyone who's read this explain to me what I'm missing? If you want I can post an exarcise as an example.

Any help is appreciated!

 

[Domullus]

In physic's formulas everything is in radians. Forget the degree. The π is 3.14. Period. So my suggestion is to always calculate everything in radians (and turn calculator mode to rad). Degrees are used only when you need to visualize angles of the triangle, because we are more comfortable to think about 90° degree angle and not some odd 1.57 rad angle which sounds scary (but is the same). Also note that all angles that came in physics formulas and are expressed in degrees always ends under trigonometric function (sin, cos, tan, etc.). Therefore it is does not matter if you enter degree or radians if you evaluate trigonometric function correctly it equalizes everything.

Note, that people tend to imagine the phase also like a triangle (45 degree phase, 180 degree phase end etc.), so most often they depict it in degrees. Simple rule of thumb: if you have the numbers and the phase is really small (lets say <10) - it is probably depicted in radians, otherwise it is degrees. And when you make calculations and see π in equation, just put 3.14 and calculate everything in radians.

 

[Const@ntine]

In physic's formulas everything is in radians. Forget the degree. The π is 3.14. Period. So my suggestion is to always calculate everything in radians (and turn calculator mode to rad). Degrees are used only when you need to visualize angles of the triangle, because we are more comfortable to think about 90° degree angle and not some odd 1.57 rad angle which sounds scary (but is the same). Also note that all angles that came in physics formulas and are expressed in degrees always ends under trigonometric function (sin, cos, tan, etc.). Therefore it is does not matter if you enter degree or radians if you evaluate trigonometric function correctly it equalizes everything.

Alright, I'll try that and report back. But to make it clear, you're basically saying that if I have my calculator turned to rad, even if I calculate θ in radians, there won't be a problem, right? So I should just go ahead and use radians for everything.

I was sure that trigonometric function = radians, but I've been getting some weird results in these exercises (they're like, 5, so it's not of huge importance, but I want to know). Anyway, I'll try again and come back.

Thanks!

 

[Domullus]

Just look at the first formula:

φ = 2π*d*sinθ/λ

θ - could be in radians or degrees - it does not matter because if it is radians, your calculate sinθ in (rad) mode, if it is degrees - then in deg mode. You will get the same result. Now what is φ. See the 2π in the formula. If you insert 3.14 here, you get φ in radians, if 180° - you get answer in degrees. Easy as that.

 

[Const@ntine]

Just look at the first formula:

φ = 2π*d*sinθ/λ

θ - could be in radians or degrees - it does not matter because if it is radians, your calculate sinθ in (rad) mode, if it is degrees - then in deg mode. You will get the same result. Now what is φ. See the 2π in the formula. If you insert 3.14 here, you get φ in radians, if 180° - you get answer in degrees. Easy as that.

That's what I figured, but maybe I got lost somewhere (twas getting pretty late last night). Thanks for the clear-up, I'll try it as soon as possible and see where I was wrong.

 

[Const@ntine]

I tried it all in rads (all 5 exercises) and yeah, it works. Some decimals are a bit of (I get 0.969 and he gets 968) but it's probably a Significant Digits issue.

Thanks again for the help!