Heiseinberg's Microscope - Trigonometric and Interpretation questions

[s3a]

Homework Statement

The problem along with its solution is attached as ProblemSolution.jpg.

Homework Equations

Δx = λ/sinθ (Eq. 1)
Δp_x = (h/λ)(sinε) (Eq. 2)

The Attempt at a Solution

In Wikipedia, I found this article: http://en.wikipedia.org/wiki/Heisenberg's_microscope which uses (Eq. 2) from the “Relevant equations” section above but, comparing it to my attached jpg file, ε = 2θ so I feel that the equation should become Δp_x = (h/λ)(sin(2θ)) but my jpg file states that it is Δp_x ≈ 2(h/λ)(sin(θ)) instead. Why is this the case? I even confirmed with Wolfram Alpha that sin(2θ) ≠ 2sin(θ). sin(2θ) = 2sinθcosθ but I don't see any cosθ anywhere.

Also, in the solution in the jpg file, there is the Δp_x ≈ 2h(v/c)sinθ equation; is it this equation that “[shows] that if we minimize Δx by reducing λ, this will result in a loss of information about the x-component of the elctron momentum?” If so, then what's the purpose of the Δx Δp_x product? Is the value of Δx Δp_x ≈ 4πħ = 2h supposed to be an approximation to the Heisenber Uncertainty Principle equation: Δx Δp_x ≥ ħ/2? Lastly, the last part of the solution says “We can attempt to overcome this difficulty by [ . . .].” What's the difficulty we are attempting to overcome? Is it the difficulty of measuring both the position and momentum of an electron simultaneously? Could someone please explain, confirm and/or deny these things to me even if it seems obvious to you?

If more information is needed or you want me to rephrase something, just ask.

Any input would be GREATLY appreciated!
Thanks in advance!

 

[vela]

You didn't attach the file.

 

[s3a]

Sorry lol but, I just did now. :)

 

[harts]

All of these equations, on your pdf and on wikipedia, are all approximations. Maybe they all roughly explain the same thing?

 

[harts]

http://spiff.rit.edu/classes/phys314/lectures/heis/heis.html

This lecture thing I found when I googled Heisenberg's microscope kinda defends my angle approximation idea.

Now that I look at this lecture, I'm pretty sure they are getting that 2 from the fact that the uncertainty in p ranges from -hθ/λ to +hθ/λ?

so Δp_x=2hθ/λ (or sinθ if youd like)

As far as I can tell, wikipedia and your pdf are giving different but adequate explanations of the same thing.

 

[s3a]

Sorry for the late response and thank you! Those notes were also aesthetically pleasing to my eyes. :)