Question

What is the result of multiplication of complex numbers - V3+ i and (cos + i sin 57 ) ? 5п 3​

Answer 1

Given:-

$\text { Multiplication of complex numbers is carried out as under: }$

$(a+i b)(c+i d)=a c+i a d+i b c+i^{2} b d=(a c-b d)+i(a d+b c)$

$\text { First determine the values of the trigonometric ratios. }$

$\text { Then multiply the complex numbers as shown above. }$

$\begin{array}{l}\cos (5 \pi / 3)=\cos 300^{\circ}=\cos \left(360^{\circ}-60^{\circ}\right) \\=\cos 60^{\circ} \text { since } \cos \left(360^{\circ}-x\right)=\cos x \\=1 / 2 . \\\left.\sin (5 \pi / 3)=\sin 300^{\circ}=\sin 360^{\circ}-60^{\circ}\right) \\=-\sin 60^{\circ} \text { since } \sin \left(360^{\circ}-x\right)=-\sin x \\=-\sqrt{3} / 2\end{array}$

$\begin{array}{l}(1 / 8)(\cos (5 \pi / 3)+i \sin (5 \pi / 3)) \\=(1 / 8)(1 / 2-i \sqrt{3} / 2) \\=(1 / 16)(1-i \sqrt{3})\end{array}$$\text { Hence }-\sqrt{3}+i \text { and }(1 / 8)(\cos (5 \pi / 3)+i \sin (5 \pi / 3)) ?$

$\begin{array}{l}=(-\sqrt{3}+i)(1 / 16)(1-i \sqrt{3}) \\=(1 / 16)(-\sqrt{3}+i)(1-i \sqrt{3}) \\=(1 / 16)(-\sqrt{3}+3 i+i+\sqrt{3}) \\=(1 / 16) 4 i \\=i / 4\end{array}$

hence the solution for the given equation is = $\frac{i}{4}$