Question

The smallest positive angle which satisfies the equation ​

$$2{\sin }^{2}x+\sqrt{3}\cos x+1=0$$ is

Options

• $$\frac{5\pi }{6}$$

   

• $$\frac{2\pi }{3}$$

   

• $$\frac{\pi }{3}$$

   

• $$\frac{\pi }{6}$$

   

Answer 1

$$\frac{5\pi }{6}$$
Given:
$$2{\sin }^{2}x+\sqrt{3}\cos x+1=0$$
$$\Rightarrow 2(1-{\cos }^{2}x)+\sqrt{3}\cos x+1=0$$
$$\Rightarrow 2-2{\cos }^{2}x+\sqrt{3}\cos x+1=0$$
$$\Rightarrow 2{\cos }^{2}x-\sqrt{3}\cos x-3=0$$
$$\Rightarrow 2{\cos }^{2}x-2\sqrt{3}\cos x+\sqrt{3}\cos x-3=0$$
$$\Rightarrow 2\cos x(\cos x-\sqrt{3})+\sqrt{3}(\cos x-\sqrt{3})=0$$
$$\Rightarrow (2\cos x+\sqrt{3})(\cos x-\sqrt{3})=0$$

$$\Rightarrow 2\cos x+\sqrt{3}=0$$ or,

$$\cos x-\sqrt{3}=0$$

∴ $$\cos x=-\frac{\sqrt{3}}{2}$$ or,

$$\cos x=\sqrt{3}$$ is not possible.

$$\Rightarrow \cos x=\cos \left(\frac{5\pi }{6}\right)$$
$$\Rightarrow x=2n\pi \pm \frac{5\pi }{6},n\in Z$$

For n = 0, the value of $$xis\pm \frac{5\pi }{6}$$.
Hence, the smallest positive angle is $$\frac{5\pi }{6}$$.