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Question: If \(A\) lies in the second quadrant and \(3\tan A + 4 = 0,\) the value of \(2\cot A - 5\cos A + \si...

If AA lies in the second quadrant and 3tan⁡A+4=0,3\tan A + 4 = 0, the value of 2cot⁡A−5cos⁡A+sin⁡A2\cot A - 5\cos A + \sin Ais t

A

−5310\frac{- 53}{10}

B

−710\frac{- 7}{10}

C

710\frac{7}{10}

D

2310\frac{23}{10}

Answer

2310\frac{23}{10}

Explanation

Solution

3tan⁡A+4=0⇒tan⁡A=−433\tan A + 4 = 0 \Rightarrow \tan A = - \frac{4}{3}

⇒sin⁡A=±tan⁡A1+tan⁡2A=±−4/31+16/9=45\Rightarrow \sin A = \pm \frac{\tan A}{\sqrt{1 + \tan^{2}A}} = \pm \frac{- 4/3}{\sqrt{1 + 16/9}} = \frac{4}{5}

(∵A(\because A is in 2nd quadrant) and cos⁡A=−35\cos A = - \frac{3}{5}. Thus,

2cot⁡A−5cos⁡A+sin⁡A=2(−34)−5(−35)+45=23102\cot A - 5\cos A + \sin A = 2\left( - \frac{3}{4} \right) - 5\left( - \frac{3}{5} \right) + \frac{4}{5} = \frac{23}{10}