Question

For what and only what value of $\alpha$ lying between 0 and $\pi$ is the inequality. $\sin \alpha . \cos^3 \alpha > \sin^3\alpha \cos \alpha$ valid ?

• A. $\alpha\in\left(0,\frac{\pi}{4}\right)or\left(\frac{3\pi}{4},\pi\right)$
• B. $\alpha\in\left(0,\frac{\pi}{2}\right)or\left(\frac{3\pi}{2},\pi\right)$
• C. $\alpha\in\left(\frac{\pi}{4},\frac{\pi}{2}\right)or\left(\frac{3\pi}{2},\frac{3\pi}{4}\right)$
• D. none of these.

Answer 1

Answer: (A)

$\alpha\in\left(0,\frac{\pi}{4}\right)or\left(\frac{3\pi}{4},\pi\right)$

Answer 2

We have $\sin\, \alpha \cos^2\,\alpha > \sin^3\alpha \, \cos\, \alpha$ $\Rightarrow \, \sin\, \alpha \, \cos\, \alpha [\cos^2\alpha - \sin^2\alpha ] > 0$ $\Rightarrow$ $\cos^3 \alpha \, \sin \, \alpha (1 =- \tan^2 \, \alpha) > 0$ $[ \because \, \sin \, \alpha > 0 \, for \, 0 < \alpha < \pi ]$ $\Rightarrow$ $\cos \, \alpha (1 - \tan^2 \alpha) 0$ $\Rightarrow$ $\cos \,\alpha > 0$ and $1 - \tan^2 \alpha > 0 \, i.e., \, \tan^2 \alpha < 1$ or $\cos \, \alpha < 0$ and $1 - \tan^2 \alpha < 0 \, i.e., \, \tan^2 \alpha >1$ $\Rightarrow$ $\alpha \, \in \left( 0, \frac{\pi}{4} \right)$ or $\alpha \, \in \left( \frac{3 \pi}{4} , \pi \right)$