Question

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

Answer 1

`1/25(11+27i)`

Explanation:

We have

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

`=\frac{\sqrt{68}(\cos(\tan^{-1}(4))+i\sin(\tan^{-1}(4)))}{\sqrt{50}(\cos(\tan^{-1}(1/7))+i\sin(\tan^{-1}(1/7)))}`

`=\sqrt{\frac{68}{50}}(\cos(\tan^{-1}(4)-\tan^{-1}(1/7))+i\sin(\tan^{-1}(4)-\tan^{-1}(1/7)))`

`=\sqrt{\frac{34}{25}}(\cos(\tan^{-1}(27/11))+i\sin(\tan^{-1}(27/11)))`

`=\sqrt{\frac{34}{25}}(11/\sqrt850+i27/\sqrt{850})`

`=\sqrt{\frac{34}{25\times 850}}(11+27i)`

`=1/25(11+27i)`