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Question

Mathematics Question on Complex Numbers and Quadratic Equations

The value of cos⁡30∘+isin⁡30∘cos⁡60∘−isin⁡60∘\frac{\cos 30{}^\circ +i\sin 30{}^\circ }{\cos 60{}^\circ -i\sin 60{}^\circ } is equal to

A

ii

B

−i-i

C

1+3i2\frac{1+\sqrt{3}i}{2}

D

1−3i2\frac{1-\sqrt{3}i}{2}

Answer

ii

Explanation

Solution

LHS=cos⁡30∘+isin⁡30∘cos⁡60∘−isin⁡60∘LHS=\frac{\cos 30{}^\circ +i\sin 30{}^\circ }{\cos 60{}^\circ -i\sin 60{}^\circ }
=32+i1212−i32=3+i21−i32=\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}}
=3+i1−i3×1+i31+i3=3+3i+i−31+3=\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}
=4i4=i=\frac{4i}{4}=i