Question

If $3\sin \theta +5\cos \theta =5$, then $5\sin \theta -3\cos \theta$is equal to

• A. $4$
• B. $\pm 3$
• C. $5$
• D. None of these

Answer 1

The correct option is B

$\pm 3$

Step 1. Finding the value of $\mathbf{5}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{\theta }\mathbf{}\mathbf{-}\mathbf{}\mathbf{3}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{\theta }\mathbf{}$:
Given,

$3\sin \theta +5\cos \theta =5...\left(i\right)$

Let $5\sin \theta –3\cos \theta =p...\left(ii\right)$

Step 2. Squaring and adding (i) and (ii), we get

$\begin{array}{rcl}9{\sin }^2\theta +25{\cos }^2\theta +30\sin \theta \cos \theta +25{\sin }^2\theta +9{\cos }^2\theta –30\sin \theta \cos \theta & =& 25+p^2\end{array}$

$\Rightarrow$ $\begin{array}{rcl}9({\sin }^2\theta +{\cos }^2\theta )+25({\cos }^2\theta +{\sin }^2\theta )& =& 25+p^2\end{array}$

$\Rightarrow$ $\begin{array}{rcl}9+25& =& 25+p^2\end{array}$

$\Rightarrow$ $\begin{array}{rcl}{p}^2& =& 9\end{array}$

$\begin{array}{rcl}\mathbf{\therefore }\mathit{p}\mathbf{}& \mathbf{=}& \mathbf{}\mathbf{\pm }\mathbf{3}\end{array}$

Hence, the correct answer is option (B).