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Unformatted text preview: 105. The key trigonometric identity used in this proof is sin(2θ) = 2sinθ cosθ. Now, we
wish to show that Eq. 3619 becomes (when d = a) the pattern for a single slit of width 2a
(see Eq. 365 and Eq. 366):
2
⎛sin(2πasinθ/λ)⎞
I(θ) = Im ⎜
⎟.
⎝ 2πasinθ/λ ⎠
We note from Eq. 3620 and Eq. 3621, that the parameters β and α are identical in this
case (when d = a), so that Eq. 3619 becomes
⎛cos(πasinθ/λ)sin(πasinθ/λ)⎞
I(θ) = Im ⎜
⎟.
πasinθ/λ
⎠
⎝
2 Multiplying numerator and denominator by 2 and using the trig identity mentioned above,
we obtain
2
2
⎛2cos(πasinθ/λ)sin(πasinθ/λ)⎞
⎛sin(2πasinθ/λ)⎞
I(θ) = Im ⎜
⎟ = Im ⎜
⎟
2πasinθ/λ
⎠
⎝
⎝ 2πasinθ/λ ⎠
which is what we set out to show. ...
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 Spring '08
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 Physics

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