Question

The maximum value of the expression $sin\theta co{s}^2\theta \left(\forall \theta \in \left[0,\pi \right]\right)$ is

• A. $\frac{2}{3}$
• B. $\frac{2}{\sqrt{3}}$
• C. $\frac{2}{3\sqrt{3}}$
• D. $\frac{1}{\sqrt{3}}$

Answer 1

Answer: (C)

As $si{n}^2\theta +co{s}^2\theta =1,$
$\Rightarrow si{n}^2\theta +\frac{co{s}^2\theta }{2}+\frac{co{s}^2\theta }{2}=1$
Applying $AM\ge GM,$ we have
$\frac{si{n}^2\theta +\frac{co{s}^2\theta }{2}+\frac{co{s}^2\theta }{2}}{3}\ge {\left(\frac{si{n}^2\theta co{s}^4\theta }{4}\right)}^{\frac{1}{3}}$
$\Rightarrow {\left(\frac{1}{3}\right)}^3\ge \frac{si{n}^2\theta co{s}^4\theta }{4}$
$\Rightarrow {\left(sin\theta \cdot co{s}^2\theta \right)}^2\le \frac{4}{27}$
$\Rightarrow sin\theta \cdot co{s}^2\theta \le \frac{2}{3\sqrt{3}}$