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 $