Question

Type the correct answer in each box. If necessary, use / for the fraction bar.if 24 = 2cis120° 22=4cis30°and =a+bithena=+and b=ResetNext

Answer 1

Answer:

a = 0

b = 1/2

Explanation:

z1 = 2cis120

$\begin{gathered} =2(cos120+i\sin 120) \\ =2(-\frac{1}{2}+i\frac{\sqrt[]{3}}{2}) \\ \\ =-1+i\sqrt[]{3} \end{gathered}$

z2 = 4cis30

$\begin{gathered} =4(\cos 30+i\sin 30) \\ \\ =4(\frac{\sqrt[]{3}}{2}+i\frac{1}{2}) \\ \\ =2\sqrt[]{3}+2i \end{gathered}$

z1/z2

$=\frac{-1+i\sqrt[]{3}}{2\sqrt[]{3}+2i}$

Rationalize the surd above, we have:

$\begin{gathered} \frac{-1+i\sqrt[]{3}}{2\sqrt[]{3}+2i}\times=\frac{2\sqrt[]{3}-2i}{2\sqrt[]{3}-2i} \\ \\ =\frac{-2\sqrt[]{3}+2i+2i(3)+2\sqrt[]{3}}{4(3)+4} \\ \\ =\frac{8i}{16} \\ \\ =\frac{1}{2}i \end{gathered}$

Therefore

a + ib = 0 + i(1/2)

a = 0

b = 1/2