ch36-p105 - 105. The key trigonometric identity used in...

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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. 36-19 becomes (when d = a) the pattern for a single slit of width 2a (see Eq. 36-5 and Eq. 36-6): 2 ⎛sin(2πasinθ/λ)⎞ I(θ) = Im ⎜ ⎟. ⎝ 2πasinθ/λ ⎠ We note from Eq. 36-20 and Eq. 36-21, that the parameters β and α are identical in this case (when d = a), so that Eq. 36-19 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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