Question

The greatest value of $\sin \mathrm{x}\cos \mathrm{x}$ is​

• A. $1$
• B. $\frac{1}{2}$
• C. $\frac{1}{4}$
• D. $2$

Answer 1

The correct option is B

$\frac{1}{2}$

Given:

$\begin{array}{rcl}& & \mathrm{sinx}.\mathrm{cosx}\end{array}$

Multiply and divide the expression by $2$

$\begin{array}{rcl}\mathrm{sinx}.\mathrm{cosx}& =& \frac{2\mathrm{sinx}.\mathrm{cosx}}{2}\\ & =& 2\sin 2\mathrm{x}\left[\mathrm{Note}:\mathrm{Sin}2\mathrm{x}=2\mathrm{sinx}\mathrm{cosx}\right]\end{array}$

We know that,

$\begin{array}{rcl}\sin 2x& \in & [-1,1]\\ & \Rightarrow & \frac{\sin 2x}{2}\in \left[\frac{-1}{2},\frac{1}{2}\right]\end{array}$

Hence, the greatest value of $\sin x.\cos x$ is $\frac{1}{2}$​.