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# If (sin 3theta)/(cos 2theta)lt 0, then theta lies in

Question from  Class 12  Chapter Trigonometric Equations

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`((3pi),(8),(23pi)/(48))``((7pi)/(24),(3pi)/(8))``((13pi)/(48),(7pi)/(24))``((2pi)/(4),(7pi)/(12))`

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Solution :
Let `(sin 3theta)/(cos 2theta)lt 0` <br> Case I. `sin theta lt 0` and `cos 2theta gt 0` <br> `rArr 3theta in (pi, 2pi)` and `2theta in (-(pi)/(2),(-pi)/(2))` <br> `rArr theta in((pi)/(3),(2pi)/(3))` and `theta in (-(pi)/(4), (pi)/(4))` <br> Case II. `sin 3theta gt 0` and `cos 2 theta lt 0` <br> `therefore 3theta in(0, pi)` and `2theta in((pi)/(2),(3pi)/(2))` <br> `rArr theta in(0,(pi)/(3))` and `theta in((pi)/(4),(3pi)/(4))` <br> From case I and II, we have <br> `theta in((pi)/(4),(pi)/(3))` <br> Since `((13pi)/(48),(7pi)/(24))in((pi)/(4),(pi)/(3))` <br> `therefore theta in((13pi)/(48),(7pi)/(24))`

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