Question

If $12 \,\cot^{2} \,\theta-31 \,\text{cosec} \,\theta + 32 = 0$, then the value of $sin \,\theta$ is

• A. $\frac{3}{5}$ or $1$
• B. $\frac{2}{3}$ or $\frac{-2}{3}$
• C. $\frac{4}{5} or \frac{3}{4}$
• D. $\pm\frac{1}{2}$

Answer 1

Answer: (C)

$12 \,cot^{2} \,\theta-31 \,cosec \,\theta + 32 = 0$
$\Rightarrow 12 \,\left(cosec^{2} \,\theta -1\right)-31 \,cosec \,\theta + 32 = 0$
$\Rightarrow 12 \,cosec^{2} \,\theta -31 \,cosec \,\theta + 20 = 0$
$\Rightarrow 12 \,cosec^{2} \,\theta -16 \,cosec \,\theta - 15 \,cosec \,\theta + 20 = 0$
$\Rightarrow \left(4\,cosec \,\theta -5\right)\left(3\,cosec \,\theta -4\right) = 0$
$\Rightarrow \,cosec \,\theta = \frac{5}{4}, \frac{4}{3} \,\therefore sin \,\theta = \frac{4}{5}, \frac{3}{4}.$