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

Step-by-step explanation:

z1 = 2cis120

`\begin{gathered} =2(cos120+i\sin 120) \\ =2(-(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(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}*=\frac{2\sqrt[]{3}-2i}{2\sqrt[]{3}-2i} \\ \\ =\frac{-2\sqrt[]{3}+2i+2i(3)+2\sqrt[]{3}}{4(3)+4} \\ \\ =(8i)/(16) \\ \\ =(1)/(2)i \end{gathered}`

Therefore

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

a = 0

b = 1/2