Question

The number of values of $\theta$ satisfying $4\cos \theta +3\sin \theta =5$ as well as $3\cos \theta +4\sin \theta =5$

• A. one
• B. two
• C. zero
• D. none of these

Answer 1

The correct option is C zero

Squaring the first equation,

$16{\cos }^{{}^2}\theta =25+9{\sin }^2\theta -30\sin \theta$

$25{\sin }^2\theta -30\sin \theta +9=0$

$\sin \theta =\frac{3}{5}$

Squaring the other equation,

$9-9{\sin }^2\theta =25+16{\sin }^2\theta -40\sin \theta$

$25{\sin }^2\theta -40\sin \theta +16=0$

$\sin \theta =\frac{4}{5}$

$\sin \theta$ cannot have two different values for same $\theta$ ,

So common solutions $=0$