Question

If A lies in second quadrant 3tan A + 4 = 0, then the value of 2cot A − 5cosA + sin A is equal to

विकल्प

• $$-\frac{53}{10}$$

   

• $$\frac{23}{10}$$

   

• $$\frac{37}{10}$$

   

• $$\frac{7}{10}$$

   

Answer 1

It is given that $$\frac{\pi }{2}<A<\pi$$.

$$3\tan A+4=0$$

$$\Rightarrow \tan A=-\frac{4}{3}$$

$$\Rightarrow \cot A=-\frac{3}{4}$$
Now,
$$\sec A=\pm \sqrt{1+{\tan }^{2}A}=\pm \sqrt{1+\frac{16}{9}}=\pm \sqrt{\frac{25}{9}}=\pm \frac{5}{3}$$
$$\therefore \sec A=-\frac{5}{3}\left(\text{ A lies in 2nd quadrant }\right)$$
$$\Rightarrow \cos A=-\frac{3}{5}$$
Also,

$$\sin A=\pm \sqrt{1-{\cos }^{2}A}=\pm \sqrt{1-\frac{9}{25}}=\pm \sqrt{\frac{16}{25}}=\pm \frac{4}{5}$$

$$\therefore \sin A=\frac{4}{5}\left(\text{ A lies in 2nd quadrant }\right)$$

So,
$$2\cot A-5\cos A+\sin A$$
$$=2\times (-\frac{3}{4})-5\times (-\frac{3}{5})+\frac{4}{5}$$
$$=-\frac{3}{2}+3+\frac{4}{5}$$
$$=\frac{-15+30+8}{10}$$
$$=\frac{23}{10}$$

Hence, the correct answer is option B.