Question

If ${\sin }^3\theta \cos \theta -{\cos }^3\theta \sin \theta =\frac{1}{4},$ then the values of $\theta$ which satisfy the equation is

• A. $\frac{n\pi }{2}+\frac{\pi }{8},n\in Z$
• B. $\frac{n\pi }{2}-\frac{\pi }{8},n\in Z$
• C. $\frac{n\pi }{8}-\frac{\pi }{2},n\in Z$
• D. $\frac{n\pi }{8}+\frac{\pi }{2},n\in Z$

Answer 1

The correct option is B $\frac{n\pi }{2}-\frac{\pi }{8},n\in Z$

${\sin }^3\theta \cos \theta -{\cos }^3\theta \sin \theta =\frac{1}{4}\Rightarrow 4\sin \theta \cos \theta ({\sin }^2\theta -{\cos }^2\theta )=1\Rightarrow 2\sin 2\theta (-\cos 2\theta )=1\Rightarrow \sin 4\theta =-1\Rightarrow 4\theta =2n\pi -\frac{\pi }{2}\Rightarrow \theta =\frac{n\pi }{2}-\frac{\pi }{8},n\in Z$