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Instead, count the number of times $ \left| \sin(3\theta) + \cos(4\theta) \right| = 1 $, i.e., when $ \sin(3\theta) + \cos(4\theta) = \pm1 $.
The function $ f(\theta) = \sin(3\theta) + \cos(4\theta) $ is analytic and not periodic with simple period, but we can consider how many times it attains values $ \pm1 $.
But better: The function $ f(\theta) $ is of the form $ \sin a\theta + \cos b\theta $. The number of solutions to $ |f(\theta)| = 1 $ over a long interval can be approximated by analyzing how many times the envelope allows crossings.

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