Why Is the Small Angle Approximation Used in Optics Problems?

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Homework Help Overview

The discussion revolves around the use of the small angle approximation in optics problems, particularly in the context of understanding when and why this approximation is applicable. Participants explore the implications of this approximation and its relevance to specific problems without providing a definitive solution.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the rationale behind the small angle approximation, with one suggesting the use of specific angle values to evaluate the approximation's validity. Others express uncertainty about determining when the approximation is appropriate, especially in the absence of given angles.

Discussion Status

The discussion is ongoing, with participants sharing insights and suggestions for evaluating small angles. There is a recognition of the need to understand the conditions under which the approximation holds, but no consensus has been reached regarding its application in the specific problem at hand.

Contextual Notes

Some participants note the importance of using radians for the small angle approximation to be valid, while others highlight the lack of explicit angle information in the problem, which complicates the application of the approximation.

Gauss M.D.
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Homework Statement



Had the same problem as this threadstarter:

https://www.physicsforums.com/showthread.php?t=109059

Homework Equations





The Attempt at a Solution



I managed to find a ratio of tangents for the two angles. From there, it seems you're supposed to go "well tan(x) ≈ x for small angles so let's magically assume this is a small angle and go grab a donut".

Why is the small angle approximation appropriate for this problem and how do I avoid getting stuck on similar problems in the future?
 
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You could try a range of small angles and judge for yourself, e.g.,

x=1°: x=... radians, sin x=..., tan x=...

x=2°: x=... radians, sin x=..., tan x=...

x=3°:


By working this out for yourself, you'll be left with a better appreciation of the result. :smile:

▣[/color] Remember, the trig approximations expect x to be in radians.
 
Do what nascent oxygen recommends...you will be surprised how 'BIG' the angle can be yet still be considered 'SMALL'
 
Of course the small angle approximation only works if you use radians.
 
Integral said:
Of course the small angle approximation only works if you use radians.

As recommended !
 
No, I get the small angle approximation, I just don't get how I am supposed to know that it is applicable here. I mean, we're not given any angles. We're supposed to figure it out through trig/geometry trickery.Theoretically, the angles could be pi/2 for all I know.
 

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