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Equations with trigonometry in them can look scary, but that's nothing that a quick little substitution can't fix.
In the domain \( [0^\circ , 1000^\circ ] \), how many solutions are there to \( \sin \theta \cos \theta = \frac {1} {2} \)?
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In the interval \( [0, 2 \pi ] \), how many solutions are there to
\[ \sin \theta = 2 \sin 2 \theta ? \]
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How many \(x\)'s in the interval \(0 \leq x \leq 29\pi\) satisfy \[\sin \left(2x-\frac{\pi}{5}\right)=\sin \frac{\pi}{8}?\]
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How many \(x\)'s in the interval \(0 \leq x \leq 60\pi\) satisfy \[6\cos 2x+23\sin x-4=0?\]
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In the interval \( [0, 2 \pi ] \), how many solutions are there to
\[ \cos^2 \theta = \cos 2 \theta ? \]
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