r = 7.2
z = 7.2 cis 73π/90
See explanation below
Explanation:r is given as |z|
z = -6 + 4i
[tex]\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}[/tex]tan α = 4/-6 = 2/-3
[tex]\begin{gathered} \tan \alpha\text{= }\frac{-2}{3} \\ \alpha=tan^{-1}(\frac{-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}[/tex]convert to radians:
1π rad = 180°
[tex]\begin{gathered} \text{146.3}\degree\text{ }\times\text{ }\frac{\pi}{180\degree}\text{ = }\frac{\text{146.3}\degree\text{ }\times\pi}{180\degree} \\ \text{146.3}\degree\text{ }\times\text{ }\frac{\pi}{180\degree}\text{ = }0.81\pi \\ OR \\ \text{ 146.3}\degree\text{ }\times\text{ }\frac{\pi}{180\degree}\text{ = }\frac{73\pi}{90}\text{ (approx i}mately) \end{gathered}[/tex][tex]\begin{gathered} \text{Polar form:} \\ z\text{ = r cis }\theta \\ |z|=\text{ r }\approx\text{ }7.2 \\ \\ \theta\text{ = }\frac{73\pi}{90} \\ z\text{ = 7.2 cis }\frac{73\pi}{90} \end{gathered}[/tex]