Question

The range of $\frac{1}{3sin\theta +4cos\theta +2}$ is

• A. $\left[-\frac{1}{3},\frac{1}{7}\right]$
• B. $\left[\frac{1}{3},\frac{1}{7}\right]$
• C. $\left(-\infty ,-\frac{1}{3}\right]\cup \left[\frac{1}{7},\infty \right)$
• D. None of these

Answer 1

Answer: (C)

We know that range of the expression $asinx+bcosx$ is $\left[-\sqrt{a^2+b^2},\sqrt{a^2+b^2}\right]$ i.e.,
$-\sqrt{a^2+b^2}\le asinx+bcosx\le \sqrt{a^2+b^2}$
So, $-\sqrt{3^2+4^2}\le 3sinx+4cosx\le \sqrt{3^2+4^2}$
$\Rightarrow -5\le 3sinx+4cosx\le 5$
$\Rightarrow -5+2\le 3sinx+4cosx+2\le 5+2$
$\Rightarrow -3\le 3sinx+4cosx+2\le 7$
$\Rightarrow -\frac{1}{3}\ge \frac{1}{3sinx+4cosx+2}\ge \frac{1}{7}$
So, range of $\frac{1}{3\sin x+4\cos x+2}$ is $\left(-\infty ,-\frac{1}{3}\right]\cup \left[\frac{1}{7},\infty \right)$.