Question

$\left(2+\sqrt{3}\right)\sin \left(\theta \right)+2\cos \left(\theta \right)$ lies between

• A. $-\sqrt{3}\mathrm{and}\sqrt{3}$
• B. $-\left(2+\sqrt{3}\right)\mathrm{and}\left(2+\sqrt{3}\right)$
• C. $-\left(2+\sqrt{5}\right)\mathrm{and}\left(2+\sqrt{5}\right)$
• D. none of these

Answer 1

The correct option is C

$-\left(2+\sqrt{5}\right)\mathrm{and}\left(2+\sqrt{5}\right)$

Explanation for correct option :

Given, $\left(2+\sqrt{3}\right)\sin \left(\theta \right)+2\cos \left(\theta \right)$

We know that,

$\mathbf{-}\sqrt{{\mathbf{a}}^{\mathbf{2}}\mathbf{+}{\mathbf{b}}^{\mathbf{2}}}\mathbf{⩽}\mathit{a}\mathit{c}\mathit{o}\mathit{s}\left(\theta \right)\mathbf{+}\mathit{b}\mathit{s}\mathit{i}\mathit{n}\left(\theta \right)\mathbf{⩽}\sqrt{{\mathbf{a}}^{\mathbf{2}}\mathbf{+}{\mathbf{b}}^{\mathbf{2}}}$

$$\begin{gathered} \Rightarrow -\sqrt{{\left(2\right)}^2+{\left(2+\sqrt{3}\right)}^2}⩽\left(2+\sqrt{3}\right)\sin \left(\theta \right)+2\cos \left(\theta \right)⩽\sqrt{{\left(2\right)}^2+{\left(2+\sqrt{3}\right)}^2} \\ \Rightarrow -\sqrt{4+4+3+4\sqrt{3}}⩽\left(2+\sqrt{3}\right)\sin \left(\theta \right)+2\cos \left(\theta \right)⩽\sqrt{4+4+3+4\sqrt{3}} \\ \Rightarrow -\sqrt{11+4\sqrt{3}}⩽\left(2+\sqrt{3}\right)\sin \left(\theta \right)+2\cos \left(\theta \right)⩽\sqrt{11+4\sqrt{3}} \\ \Rightarrow -\sqrt{17.928}⩽\left(2+\sqrt{3}\right)\sin \left(\theta \right)+2\cos \left(\theta \right)⩽\sqrt{17.928} \\ \Rightarrow -4.234⩽\left(2+\sqrt{3}\right)\sin \left(\theta \right)+2\cos \left(\theta \right)⩽4.234 \end{gathered}$$

Since, $\sqrt{5}=2.236\&\sqrt{3}=1.732$

From the option $2+\sqrt{5}=4.236$

Therefore, correct answer is option $\left(C\right)$