Question

How do you divide `(-4+5i) / (-1+6i)` in trigonometric form?

Answer 1

`1/37(34+19i)`

Explanation:

We have

`\frac{-4+5i}{-1+6i}`

`=\frac{\sqrt{41}e^{i(\pi-\tan^{-1}(5/4))}}{\sqrt{37}e^{i(\pi-\tan^{-1}(6))}}`

`=\sqrt{\frac{41}{37}}(e^{i(\pi-\tan^{-1}(5/4))}e^{-i(\pi-\tan^{-1}(6))})`

`=\sqrt{\frac{41}{37}}(e^{i(\tan^{-1}(6)-\tan^{-1}(5/4))})`

`=\sqrt{\frac{41}{37}} e^{i\tan^{-1}(19/34)}`

`=\sqrt{\frac{41}{37}}(\cos(\tan^{-1}(19/34))+i\sin(\tan^{-1}(19/34)))`

`=\sqrt{\frac{41}{37}}(34/\sqrt1517+i19/\sqrt{1517})`

`=\sqrt{\frac{41}{37\times 1517}}(34+19i)`

`=1/37(34+19i)`