Question

If $3 \,sin\theta + 5\, cos\theta = 4$, then find the value of $5 \,sin\theta - 3\, cos\theta$.

• A. $\pm 3\sqrt{2}$
• B. $\pm 5\sqrt{2}$
• C. $\pm \sqrt{2}$
• D. $\pm 7\sqrt{2}$

Answer 1

Answer: (A)

Let $x = 5\, sin\theta - 3cos\theta$, then
$x^{2}=\left(5sin\theta-3cos\theta\right)^{2}=25sin^{2}\theta+9cos^{2}\theta-30sin\theta\,cos\theta$
$=25\left(1-cos^{2}\theta\right)+9\left(1-sin^{2}\theta\right)-30sin\theta\,cos\theta$
$= 25 + 9 - \left(25\,cos^{2}\theta+9\,sin^{2}\theta+30sin\theta\,cos\theta\right)$
$=34-\left(3sin\theta+5cos\theta\right)^{2}$
$=34-4^{2}=18$
$\Rightarrow x=\pm\sqrt{18}$
$=\pm3\sqrt{2}$.