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

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 $=A C$
$\left.=5 \because A C=\sqrt{(A B)^2+(B C)^2}\right\}$
$\sin \theta=\frac{4}{5}(+\mathrm{ve}) \Rightarrow$ 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$