Question

For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly find $\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{r}\mathbf{,}\mathit{\theta }$in your head for these simple problems. Then plot the number and label it in five ways as in Figure 3.3. Also plot the complex conjugate of the number.

$\mathbf{-}\sqrt{\mathbf{3}}\mathbf{+}\mathit{i}$

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Short answer

The required values are mentioned below.

$\begin{array}{rcl}x& =& -\sqrt{3}\\ y& =& 1\\ r& =& 2\\ \theta & =& \frac{5\pi }{6}\\ & & \end{array}$

Step-by-step solution

Given Information

The complex number is $-\sqrt{3}+i$.

Definition of the Complex number.

A complex number is a number that is made up of both real and imaginary numbers.

Find the value.

The formula is mentioned below.

$\begin{array}{rcl}x+iy& =& r(\cos \theta +i\sin \theta )\\ & =& r{e}^{i\theta }\\ & & \end{array}$

The complex number$-\sqrt{3}+i=x+iy$ .

$$\begin{gathered} x=-\sqrt{3} \\ y=1 \end{gathered}$$

Find the value of$r$ $r$.

The formula is $r=\sqrt{x^2+y^2}$

$$\begin{gathered} r=\sqrt{{(-\sqrt{3})}^2+1^2} \\ =2 \end{gathered}$$

Find the value of$\theta$ $\theta$.

The formula is $\theta ={\cos }^{-1}\frac{x}{r}$

$$\begin{gathered} \theta ={\cos }^{-1}\frac{-\sqrt{3}}{2} \\ =\frac{5\pi }{6} \end{gathered}$$

The value of the number becomes as follows.

$\begin{array}{rcl}-\sqrt{3}+i& =& 2\left(\cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6}\right)\\ & =& 2e^{i\frac{4\pi }{6}}\end{array}$

The required values are mentioned below.

$$\begin{gathered} x=-\sqrt{3} \\ y=1 \\ r=2 \\ \theta =\frac{5\pi }{6} \end{gathered}$$

The graph is mentioned below where the red line represnts function and black line represents function’s conjugate