Question

The value of $\frac{\cos 30{}^{\circ }+i\sin 30{}^{\circ }}{\cos 60{}^{\circ }-i\sin 60{}^{\circ }}$ is equal to

• A. $i$
• B. $-i$
• C. $\frac{1+\sqrt{3}i}{2}$
• D. $\frac{1-\sqrt{3}i}{2}$
• E. $1+i$

Answer 1

Answer: (A)

$LHS=\frac{\cos 30{}^{\circ }+i\sin 30{}^{\circ }}{\cos 60{}^{\circ }-i\sin 60{}^{\circ }}$
$=\frac{\frac{\sqrt{3}}{2}+i\frac{1}{2}}{\frac{1}{2}-i\frac{\sqrt{3}}{2}}=\frac{\frac{\sqrt{3}+i}{2}}{\frac{1-i\sqrt{3}}{2}}$
$=\frac{\sqrt{3}+i}{1-i\sqrt{3}}\times \frac{1+i\sqrt{3}}{1+i\sqrt{3}}=\frac{\sqrt{3}+3i+i-\sqrt{3}}{1+3}$
$=\frac{4i}{4}=i$