Question

If $\theta$ lies in the II ${}^{\text{nd }}$ quadrant or $90<\theta <180^{\circ }$ or $\theta$ is an obtuse angle and $3\tan \theta +4=0$ then the value of $12\cot \theta -20\cos \theta -5\sin \theta$ is ___

Answer 1

Answer: 7

Given: $3\tan \theta +4=0$
$\Rightarrow 3\tan \theta =-4$
$\Rightarrow \tan \theta =-\frac{4}{3}(\theta$ lies in the IInd quadrant $)$
$\{$ By using Pythagoras theorem, we get $=AC$
$=5\because AC=\sqrt{(AB{)}^2+(BC{)}^2}\}$
$\sin \theta =\frac{4}{5}(+\mathrm{ve})\Rightarrow$ in second quadrant
$\cos \theta =-\frac{3}{5}$
$\cot \theta =-\frac{3}{4}$
Putting values of $\sin \theta ,\cos \theta$ and $\cot \theta$, we get
$12\left(\frac{-3}{4}\right)-20\left(-\frac{3}{5}\right)+5\left(\frac{4}{5}\right)$
$=-9+12+4=7$