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There are some literature to say the solutions for all trigonometric ratios. But the following problem is given in particular interval. Kindly answer this question.

In the interval $x$ in $[0, 25\pi]$ how many solutions are there to $\sin x = -1/3$?

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Go back to the unit circle and remember that $\sin(x)$ also means the $y$ position of the point on the unit circle. So by going around the circle $12$ and a half times $(25\pi/2\pi)$, how many times does $y = -1/3$? Well it only seems to hit $-1/3$ when it is in the bottom half of the unit circle (twice), so by going around the entire circle $12$ times, you hit $\sin x = (-1/3) 12\cdot2=24$ times. You do not hit that value again by going around another half of the circle, so your answer is $24$.

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  • $\begingroup$ !Thanks a lot for your excellent explanation. $\endgroup$ – gama Nov 12 '12 at 10:11

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