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If 3 tan $A + 4 = 0$ , then the value of $2 \cot A - 5 \cos A + \sin A$ is equal to

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$\frac{23}{10} if \frac{π}{2} < A < π$

$\frac{23}{10} if \frac{3 π}{2} < A < 2 π$

$\frac{- 53}{10} if \frac{π}{2} < A < π$

$\frac{- 53}{10} if \frac{3 π}{2} < A < 2 π$

answer is A, D.

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Detailed Solution

We have $\tan A = - \frac{4}{3}$

If A lies in 2nd quadrant, then

$\sin A = \frac{4}{5}, \cos A = \frac{- 3}{5}$ , and $\cot A = \frac{- 3}{4}$

If A lies in 4th quadrant, then

$\sin A = \frac{- 4}{5}$ , $\cos A = \frac{3}{5}$ and $\cot A = \frac{- 3}{4}$

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