Question

Can you help me with number 3 and do it just as the paper says so I can understand it better.

Answer 1

r = 7.2

z = 7.2 cis 73π/90

See explanation below

Step-by-step explanation:

r is given as |z|

z = -6 + 4i

`\begin{gathered} r\text{ = }\sqrt[]{(-6)^2+(4)^2} \\ r\text{ = }\sqrt[]{36\text{ + 16}} \\ r\text{ = }\sqrt[]{52} \\ r\text{ }\approx\text{ }7.2\text{ (nearest tenth)} \end{gathered}`

tan α = 4/-6 = 2/-3

`\begin{gathered} \tan \alpha\text{= }(-2)/(3) \\ \alpha=tan^(-1)((-2)/(3))\text{ }\approx\text{ -33.69}\degree \\ \\ \sin ce\text{ z = -6 + 4i is in the second quadrant, add 180}\degree\colon \\ \text{-33.69}\degree\text{ + 180}\degree\text{ = 146.31}\degree\text{ }\approx\text{146.3}\degree \end{gathered}`

convert to radians:

1π rad = 180°

`\begin{gathered} \text{146.3}\degree\text{ }*\text{ }(\pi)/(180\degree)\text{ = }\frac{\text{146.3}\degree\text{ }*\pi}{180\degree} \\ \text{146.3}\degree\text{ }*\text{ }(\pi)/(180\degree)\text{ = }0.81\pi \\ OR \\ \text{ 146.3}\degree\text{ }*\text{ }(\pi)/(180\degree)\text{ = }(73\pi)/(90)\text{ (approx i}mately) \end{gathered}`
`\begin{gathered} \text{Polar form:} \\ z\text{ = r cis }\theta \\ |z|=\text{ r }\approx\text{ }7.2 \\ \\ \theta\text{ = }(73\pi)/(90) \\ z\text{ = 7.2 cis }(73\pi)/(90) \end{gathered}`