Question

If $\sin 2\theta \cos 2\theta =-\frac{1}{4},$ then the values of $\theta$ which satisfy the equation is

• A. $\theta =\frac{n\pi }{8}+(-1{)}^n(-\frac{\pi }{48}),n\in Z$
• B. $\theta =\frac{n\pi }{4}+(-1{)}^n(-\frac{\pi }{24}),n\in Z$
• C. $\theta =\frac{n\pi }{2}+(-1{)}^n(-\frac{\pi }{24}),n\in Z$
• D. $\theta =\frac{n\pi }{4}+(-1{)}^n(-\frac{\pi }{12}),n\in Z$

Answer 1

The correct option is B $\theta =\frac{n\pi }{4}+(-1{)}^n(-\frac{\pi }{24}),n\in Z$

$\sin 2\theta \cos 2\theta =-\frac{1}{4}$
$\Rightarrow 2\sin 2\theta \cos 2\theta =-\frac{1}{2}$
$\Rightarrow \sin 4\theta =-\frac{1}{2}$
$\Rightarrow 4\theta =n\pi +(-1{)}^n(-\frac{\pi }{6})$
$\Rightarrow \theta =\frac{n\pi }{4}+(-1{)}^n(-\frac{\pi }{24}),n\in Z$