Question

General solution of ${\sin }^{3}x+{\cos }^{3}x+\frac{3}{2}\sin 2x=1$

• A. $x=n\pi$ when n is even integer
• B. $x=2n\pi$ when n is odd integer
• C. $x=n\pi +\frac{\pi }{2}$ when n is odd integer
• D. $x=n\pi -\frac{\pi }{2}$

Answer 1

The correct option is A $x=n\pi$ when n is even integer

${\sin }^{3}x+{\cos }^{3}x+\frac{3}{2}\sin 2x=1$

$\Rightarrow {\sin }^{3}x+{\cos }^{3}x+\frac{3}{2}\times 2\sin x\cos x=1$

$\Rightarrow {\sin }^{3}x+{\cos }^{3}x+3\sin x\cos x\left(1\right)$ since ${\sin }^{2}x+{\cos }^{2}x=1$

$\Rightarrow {\sin }^{3}x+{\cos }^{3}x+3\sin x\cos x({\sin }^{2}x+{\cos }^{2}x)=1$

$\Rightarrow {(\sin x+\cos x)}^3=1$

$\Rightarrow \sin x+\cos x=1$

$\Rightarrow \frac{1}{\sqrt{2}}\sin x+\frac{1}{\sqrt{2}}\cos x=\frac{1}{\sqrt{2}}$

$\Rightarrow \cos \frac{\pi }{4}\sin x+\sin \frac{\pi }{4}\cos x=\frac{1}{\sqrt{2}}$

$\Rightarrow \sin (x+\frac{\pi }{4})=\sin \frac{\pi }{4}$

$\Rightarrow x+\frac{\pi }{4}=n\pi +\frac{\pi }{4}$

$\Rightarrow x=n\pi$ where $n$ is an even integer.