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

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Answer 1 · 2018-07-11T16:37:32

#1/37(34+19i)#

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)#