It is known that sin⁡β=45 and 0<β<πthen the value of 3sin⁡(α+β)−2cos⁡11π6cos⁡(α+β)sin⁡αis

E=3sin⁡(α+β)−4cos⁡(α+β)3sin⁡α=3(sin⁡αcos⁡β+cos⁡αsin⁡β)−4(cos⁡αcos⁡β-sin⁡αsin⁡β)3sin⁡α=53 for 0<β<π2 and E=3(7+24cot⁡α)15 for π2<β<π

It is known that sin⁡β=45 and 0<β<πthen the value of 3sin⁡(α+β)−2cos⁡11π6cos⁡(α+β)sin⁡αis

E=3sin⁡(α+β)−4cos⁡(α+β)3sin⁡α=3(sin⁡αcos⁡β+cos⁡αsin⁡β)−4(cos⁡αcos⁡β-sin⁡αsin⁡β)3sin⁡α=53 for 0<β<π2 and E=3(7+24cot⁡α)15 for π2<β<π

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It is known that $\sin β = \frac{4}{5} and 0 < β < π$ then the value of $[\frac{\sqrt{3} \sin (α + β) - \frac{2}{\cos \frac{11 π}{6}} \cos (α + β)}{\sin α}]$ is

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Independent of α for all ß in (0, π)

$\frac{5}{\sqrt{3}} for \tan β > 0$

$\frac{\sqrt{3} (7 + 24 \cot α)}{15} for \tan β < 0$

$zero for \tan β > 0$

answer is B, C.

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Detailed Solution

$\begin{array}{l} E = \frac{3 \sin (α + β) - 4 \cos (α + β)}{\sqrt{3} \sin α} \\ = \frac{3 (\sin αcos β + \cos αsin β) - 4 (\cos αcos β - \sin αsin β)}{\sqrt{3} \sin α} \\ = \frac{5}{\sqrt{3}} for 0 < β < \frac{π}{2} \\ and E = \frac{\sqrt{3} (7 + 24 \cot α)}{15} for \frac{π}{2} < β < π \end{array}$