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Question

Mathematics Question on Complex Numbers and Quadratic Equations

The value of The value of 2(cos⁡ 75∘+i sin⁡ 75∘)0.2(cos⁡ 30∘+i sin⁡ 30∘)\frac{2(\cos \, 75^{\circ} + i \, \sin \, 75^{\circ})}{0.2(\cos \, 30^{\circ} + i \, \sin \, 30^{\circ})} is

A

52(1+i)\frac{5}{\sqrt{2}} (1 + i)

B

102(1+i)\frac{10}{\sqrt{2}} (1 + i)

C

102(1−i)\frac{10}{\sqrt{2}} ( 1 - i )

D

52(1−i)\frac{5}{\sqrt{2}} (1 - i)

Answer

102(1+i)\frac{10}{\sqrt{2}} (1 + i)

Explanation

Solution

2(cos⁡75∘+isin⁡75∘)0.2(cos⁡30∘isin⁡30∘)=2⋅ei75∘0.2⋅ei30∘\frac{2\left(\cos 75^{\circ}+i \sin 75^{\circ}\right)}{0.2\left(\cos 30^{\circ} i \sin 30^{\circ}\right)}= \frac{2 \cdot e^{i 75^{\circ}}}{0.2 \cdot e^{i 30^{\circ}}}
(∵cos⁡θ+isin⁡θ=eiθ)\left(\because \cos \theta +i \sin \theta=e^{i \theta}\right)
=10⋅ei75∘⋅e−i30∘=10 \cdot e^{i 75^{\circ}} \cdot e^{-i 30^{\circ}}
=10⋅ei45∘=10 \cdot e^{i 45^{\circ}}
=10(cos⁡45∘+isin⁡45∘)=10\left(\cos 45^{\circ}+i \sin 45^{\circ}\right)
(eiθ=cos⁡θ+isin⁡θ)\left(e^{i \theta}=\cos \theta +i \sin \theta\right)
=10(12+i12)= 10\left(\frac{1}{\sqrt{2}}+i \frac{1}{\sqrt{2}}\right)
=102(1+i)=\frac{10}{\sqrt{2}}(1+i)