Question

How do you divide `( -i+8) / (2i +7 )` in trigonometric form?

Answer 1

`1/53(54-23i)`

Explanation:

We have

`\frac{-i+8}{2i+7}`

`=\frac{8-i}{7+2i}`

`=\frac{\sqrt{65}(\cos(-\tan^{-1}(1/8))+i\sin(-\tan^{-1}(1/8)))}{\sqrt{53}(\cos(\tan^{-1}(2/7))+i\sin(\tan^{-1}(2/7)))}`

`=\sqrt{\frac{65}{53}}(\cos(-\tan^{-1}(1/8)-\tan^{-1}(2/7))+i\sin(-\tan^{-1}(1/8)-\tan^{-1}(2/7)))`

`=\sqrt{\frac{65}{53}}(\cos(-\tan^{-1}(23/54))+i\sin(-\tan^{-1}(23/54)))`

`=\sqrt{\frac{65}{53}}(54/\sqrt3445-i23/\sqrt{3445})`

`=\sqrt{\frac{65}{53\times 3445}}(54-23i)`

`=1/53(54-23i)`