Chapter 6.5, Problem 25E

To determine

Tofind:the complex number in the form of trigonometric form without using calculator and represent the complex number graphically.

Answer to Problem 25E

  $\text{z=5}\left(\text{cos}\frac{3\pi }{2}\text{+isin}\frac{3\pi }{2}\right)$ .

Explanation of Solution

Given:

  $\text{z=5i}$ .

Concept used:

Complex form: $\text{z=a+ib}$

In trigonometric form: $\text{z=r}\left(\text{cosθ+isinθ}\right)$

Where, $\text{a=rcosθ}$ and $\text{b=rsinθ}$ .

Modulus: $\text{r=}\sqrt{{\text{a}}^{\text{2}}{\text{+b}}^{\text{2}}}$ .

Argument: ${\text{θ=tan}}^{\text{-1}}\frac{\text{b}}{\text{a}}$ .

And

To plot the graph:

Step 1. Determine the real and imaginary party of the complex number.

Step2. Move along the horizontal axis to show the real part of the number.

Step3. Move parallel to the vertical to show the imaginary part of the number.

Step4. Plot the point.

Calculation:

Consider the complex number $\text{z=5i}$ .

Representing it graphically it can get the following figure.

  

The trigonometric form of the complex number $\text{z=a+bi}$ is given by:

  $\text{z=r}\left(\text{cosθ+isinθ}\right)$

Here ${\text{θ=tan}}^{\text{-1}}\frac{\text{b}}{\text{a}}$ , $\text{b=rsinθ}$ and $\text{r=}\sqrt{{\text{a}}^{\text{2}}{\text{+b}}^{\text{2}}}$ .

Substituting $\text{a=0 and b=5}$

Hence,

  ${\text{θ=tan}}^{\text{-1}}\frac{\text{b}}{\text{a}}$

  $\Rightarrow {\text{tan}}^{-1}\frac{5}{0}$

As the complex lies in 3^rd quadrant

  $\text{θ=π+}\frac{\text{π}}{\text{2}}\text{=}\frac{\text{3π}}{\text{2}}$

And

  $\text{r=}\sqrt{{\text{a}}^{\text{2}}{\text{+b}}^{\text{2}}}$

  $\text{r=}\sqrt{0^{\text{2}}{\text{+5}}^{\text{2}}}$

  $\text{r=}\sqrt{25}$

  $\text{r=5}$ .

Hence the trigonometric form of the complex number $\text{z=5i}$ is given by:

  $\text{z=5}\left(\text{cos}\frac{3\pi }{2}\text{+isin}\frac{3\pi }{2}\right)$ .