Question

Greatest value of $\sin 2\theta +{\cos }^2\theta +4$ is

• A. $\frac{9+\sqrt{5}}{2}$
• B. $\frac{9-\sqrt{5}}{2}$
• C. $4+\sqrt{2}$
• D. $4-\sqrt{2}$

Answer 1

The correct option is A $\frac{9+\sqrt{5}}{2}$

Let $f\left(x\right)=\sin 2\theta +{\cos }^2\theta +4$
$=\sin 2\theta +\frac{\cos 2\theta +1}{2}+4[\because \cos 2\theta =2{\cos }^2\theta -1]$
$f\left(x\right)=\frac{2\sin 2\theta +\cos 2\theta }{2}+\frac{9}{2}$
as we know, range of $a\sin x\pm b\cos x$ is $[-\sqrt{a^1+b^2},\sqrt{a^2+b^2}]$
$\therefore g\left(\theta \right)=2\sin 2\theta +\cos 2\theta$
range of $g\left(x\right)ϵ[-\sqrt{2^2+1},\sqrt{2^2+1}]$
$ϵ[-\sqrt{5},\sqrt{5}]$
$\therefore$ range of $f\left(x\right)ϵ[\frac{9-\sqrt{5}}{2},\frac{9+\sqrt{5}}{2}]$