Question

If ${\sin }^3\theta +\sin \theta \cos \theta +{\cos }^3\theta =1$ , then $\theta$ is equal to $\left(nϵz\right)$

• A. $2n\pi +\frac{\pi }{4}$
• B. $2n\pi +\frac{\pi }{2}$
• C. $2n\pi -\frac{\pi }{2}$
• D. $2n\pi$

Answer 1

Answer: (B), (D)

The correct options are
B $2n\pi$
D $2n\pi +\frac{\pi }{2}$

${\sin }^3\theta +\sin \theta \cos \theta +{\cos }^3\theta =1$
$(\sin \theta +\cos \theta )(1-\sin \theta \cos \theta )+\sin \theta \cos \theta -1=0$
$(1-\sin \theta \cos \theta )(\sin \theta +\cos \theta -1)=0$
Either $1-\sin \theta \cos \theta =0$
$1-\frac{1}{2}\sin 2\theta =0$
$\Rightarrow \sin 2\theta =2$ (not possible )
or $\sin \theta +\cos \theta -1=0$
$(\frac{1}{\sqrt{2}}\sin \theta +\frac{1}{\sqrt{2}}\cos \theta )=\frac{1}{\sqrt{2}}$
$\Rightarrow \cos (\theta -\frac{\pi }{4})=\cos \frac{\pi }{4}$
$\Rightarrow \theta -\frac{\pi }{4}=2n\pi \pm \frac{\pi }{4}$
$\Rightarrow \theta =2m\pi$ or $\theta =2n\pi +\frac{\pi }{2}m,n\in I$