Question

Find z_1 x z_2 for z_1 = 9(cos225° + isin225°) and z_2 = 3(cos45° + isin45°).The 'z' is all subscript.

Answer 1

Using Euler's Formula:

`re^(i\theta)=r(\cos (\theta)+i\sin (\theta))`

Since:

`\begin{gathered} z1=9(\cos (225)+i\sin (225)) \\ z2=3(\cos (45)+i\sin (45)) \\ \end{gathered}`

We can rewrite them as:

`\begin{gathered} z1=9e^(225i) \\ z2=3e^(45i) \end{gathered}`

So:

`\begin{gathered} z1* z2=(9e^(225i))(3e^(45i))=27e^(225i+45i)=27e^(270i) \\ so\colon \\ z1* z2=27(\cos (270)+i\sin (270)) \end{gathered}`
`\begin{gathered} a=r\cos (\theta) \\ b=r\sin (\theta) \\ where \\ r=27 \\ \theta=270 \end{gathered}`

So: