Question

If $A={\sin }^2\theta +{\cos }^4\theta$, then for all real values of $\theta$

• A. $1\le A\le 2$
• B. $\frac{3}{4}\le A\le 1$
• C. $\frac{13}{16}\le A\le 1$
• D. $\frac{3}{4}\le A\le \frac{13}{16}$

Answer 1

The correct option is B $\frac{3}{4}\le A\le 1$

Let $A={\sin }^2\theta +{\cos }^4\theta$
$={\cos }^4\theta -{\cos }^2\theta +1$
$={\cos }^2\theta (1-{\sin }^2\theta )-{\cos }^2\theta +1$
$=1-\frac{(\sin 2\theta {)}^2}{4}$
$\therefore$ range of $(\sin 2\theta {)}^2\in [0,1]$
$\therefore$ range of $A\in [1-\frac{1}{4},1-\frac{0}{4}]$
$A\in [\frac{3}{4},1]$