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Fringe pattern from interference, waves of different lengths

Fringe Patterns. See attached file for full problem description.

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The solution is in the attached file 70476.pdf

There is some kind of a confusion in that there is only one $theta$ in the picture, but you mention $theta_1$ and $theta_2$ further down in the text.
I shall assume that what is meant in the text is that $theta_1= theta$ and
$theta_2=-theta$.

On how to find a period:

Suppose we have a function of x ...

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