Question

Obtain the geometric, polar and exponential representationsof (i5 – 1)
–1
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Answer 1

Answer on Question #45757 – Engineering – Other

Obtain the geometric, polar and exponential representations of $\left(\mathrm{i}5 - 1\right)^{-1}$.

Solution:

$$z = \frac{1}{5i - 1} = \frac{5i + 1}{(5i - 1)(5i + 1)} = \frac{5i + 1}{-25 - 1} = \frac{5i + 1}{-26} = -\frac{1}{26} - \frac{5}{26}i$$

Geometric representation = x + yi

Polar representation:

$$\begin{aligned} & z = r(\cos \varphi + i \sin \varphi), r = |z| \\ r = \sqrt{x^2 + y^2} = \sqrt{\left(-\frac{1}{26}\right)^2 + \left(-\frac{5}{26}\right)^2} = \sqrt{\frac{1}{676} + \frac{25}{676}} = \sqrt{\frac{26}{676}} = \sqrt{\frac{1}{26}} \\ & z = \sqrt{\frac{1}{26}} \left(-\sqrt{\frac{1}{26}} + i \left(\sqrt{\frac{25}{26}}\right)\right) \\ \varphi = \operatorname{atan} \left(\frac{-\frac{5}{26}}{-\frac{1}{26}}\right) - \pi = \operatorname{atan} 5 - \pi \\ \end{aligned}$$ $$z = \sqrt {\frac {1}{26}} (\cos (\operatorname {atan} 5 - \pi) + i \sin (\operatorname {atan} 5 - \pi))$$

Exponential representation:

$$z = \frac {z = \mathrm {re} ^ {\mathrm {i} \varphi}}{\sqrt {\frac {1}{26}} \mathrm {e} ^ {\mathrm {i} (\mathrm {atan} 5 - \pi)}}$$

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