# General solutions of Trigonometric problem.

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 $24$.