Question

The set of positive values of $n$ for which ${(\sqrt{\frac{1+cos{30}^{\circ }}{2}}+i\sqrt{\frac{1-cos{30}^{\circ }}{2}})}^n$ is real, is

• A. $4,8,12,....$
• B. $6,12,18,....$
• C. $8,16,24,....$
• D. $12,24,36,....$

Answer 1

The correct option is D $12,24,36,....$

${(\sqrt{\frac{1+\cos {30}^o}{2}}+i\sqrt{\frac{1-\cos 30}{2}})}^n$ is real

$\Rightarrow {(\sqrt{\frac{1+\cos 30}{2}}+i\sqrt{\frac{1-\cos 30}{2}})}^n=k+o\left(i\right)$

$\Rightarrow {(\sqrt{\frac{2{\cos }^{2}15}{2}}+i\sqrt{\frac{2{\sin }^{2}15}{2}})}^n=k$

$\Rightarrow (\cos 15+i\sin 15{)}^n=k$

$\Rightarrow \cos n\left({15}^o\right)=k$

$\Rightarrow \cos \left(165n\right)+i\sin \left(15n\right)=k$

on comparing

$\cos \left(15n\right)=k$

$\sin \left(15n\right)=o$

$\Rightarrow {15}^{o}n=\alpha \pi$

$\Rightarrow n=\frac{\alpha \pi }{{15}^o}$

$\Rightarrow n=\frac{\pi }{\pi /2}$

$\Rightarrow n=12\alpha$

$\therefore 12,24,\mathrm{36........}$