Question

3 - 4i equals

Answer 1

Step 1: Compute the value of the complex expression

Given $3-4\mathrm{i}$
Let $\mathrm{z}=3-4\mathrm{i}$

$\begin{array}{rl}& \left|\mathrm{z}\right|=\sqrt{3^2+4^2}=5\left\{\right|\mathrm{a}-\mathrm{bi}|=\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2}\}\\ & \mathrm{z}=3-4\mathrm{i}={\mathrm{re}}^{\mathrm{i\theta }}\\ & \mathrm{\theta }=\arg \mathrm{\&}\mathrm{r}=\left|\mathrm{z}\right|\\ & \mathrm{\theta }={\tan }^{-1}\frac{\mathrm{b}}{\mathrm{a}}={\tan }^{-1}\left(\frac{-4}{3}\right)=\mathrm{\pi }-{\tan }^{-1}\left(\frac{4}{3}\right)\cdots \{{\tan }^{-1}(-\mathrm{\theta })=\mathrm{\pi }-{\tan }^{-1}(\mathrm{\theta }\left)\right\}\\ & \mathrm{\theta }\text{lies in the fourth quadrant.}\\ & \text{Argument}=-\mathrm{a}=-|\mathrm{\pi }-{\tan }^{-1}\left(\frac{4}{3}\right)|\\ & \mathrm{z}={\mathrm{re}}^{\mathrm{i\theta }}\\ & \times \mathrm{z}=5{\mathrm{e}}^{-\mathrm{i}[\mathrm{\pi }-{\tan }^{-1}\left(\frac{4}{3}\right)]}\end{array}$
The value of the expression is $\mathbf{z}\mathbf{=}\mathbf{5}{\mathbf{e}}^{\mathbf{-}\mathbf{i}\mathbf{[}\mathbf{\pi }\mathbf{-}\mathbf{t}\mathbf{a}{\mathbf{n}}^{\mathbf{-}\mathbf{1}}\mathbf{}\mathbf{\left(}\frac{\mathbf{4}}{\mathbf{3}}\mathbf{\right)}\mathbf{]}}$