Chapter 8.3, Problem 3E

(a)

To determine

To find: the polar form of the given complex number.

Answer to Problem 3E

The complex number $z=2\left(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}\right)$is $z=\sqrt{3}+i$in rectangular form.

  $z=-1+i$is equivalent to $z=\sqrt{2}\left(\cos \frac{3\pi }{4}+i\sin \frac{3\pi }{4}\right)$

Explanation of Solution

Given: The complex numbers $z=-1+i$

Calculation:

Convert a complex number $z=a+bi$ using the formula

  $z=r\left(\cos \theta +i\sin \theta \right)$

Where, $r$ is the complex number of $z$ and $\theta$is an argument of $z$.

To convert the complex number $z=-1+i$ to polar form, first compute the modulus:

  $\begin{array}{l}r=\sqrt{a^2+b^2}\\ \text{ =}\sqrt{{\left(-1\right)}^2+1^2}\\ \text{ =}\sqrt{2}\end{array}$

Now compute an argument $\theta$ . Since the argument of any complex number is not unique, first note that $z=-1+i$lies in Quadrant || where tangent takes negative values.

  $\begin{array}{l}\tan \theta =\frac{b}{a}\\ \theta ={\tan }^{-1}\left(-\frac{1}{1}\right)\\ \theta =\frac{3\pi }{4}\end{array}$

So $z=-1+i$is equivalent to $z=\sqrt{2}\left(\cos \frac{3\pi }{4}+i\sin \frac{3\pi }{4}\right)$

To convert a complex number from polar form to regular coordinates, use the formulas :

  $\begin{array}{l}a=r\cos \theta \\ b=r\sin \theta \end{array}$

In order to convert the number $z=2\left(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}\right)$ , solve for $a$ and $b$ . Note that the modulus $r=2$and the argument $\theta =\frac{\pi }{6}$ .

  $\begin{array}{l}a=r\cos \theta \\ \text{ =2cos}\frac{\pi }{6}\\ \text{ =2}\cdot \frac{\sqrt{3}}{2}\\ \text{ =}\sqrt{3}\end{array}$

For $b$:

  $\begin{array}{l}b=r\sin \theta \\ \text{ =2cos}\frac{\pi }{6}\\ \text{ =2}\cdot \frac{1}{2}\\ \text{ =1}\end{array}$

So, the complex number $z=2\left(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}\right)$is $z=\sqrt{3}+i$in rectangular form.

Conclusion:

The complex number $z=2\left(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}\right)$is $z=\sqrt{3}+i$in rectangular form.

  $z=-1+i$is equivalent to $z=\sqrt{2}\left(\cos \frac{3\pi }{4}+i\sin \frac{3\pi }{4}\right)$

(b)

To determine

To find: the rectangular and polar form of the given complex number.

Answer to Problem 3E

Here, the complex number graphed can be expressed in rectangular form as $z=1+i$ or in polar form as $z=\sqrt{2}\left(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\right)$

Explanation of Solution

Given:

  

Calculation:

Consider the graph on the complex plane:

  

The complex number is $z=1+i$ in rectangular form. To convert to polar form, compute the modulus and argument. Note that $z$ lies in Quadrant |, where tangent takes positive values.

For the modules $r$ :

  $\begin{array}{l}r=\sqrt{a^2+b^2}\\ =\sqrt{1^2+1^2}\\ =\sqrt{2}\end{array}$

For the argument $\theta$ :

  $\begin{array}{l}\tan \theta =\frac{b}{a}\\ \theta ={\tan }^{-1}\frac{1}{1}\\ \theta =\frac{\pi }{4}\end{array}$

So the polar form is given by the equation $z=\sqrt{2}\left(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\right)$ .

Hence :

The complex number graphed below can be expressed in rectangular form as $z=1+i$ or in polar form as $z=\sqrt{2}\left(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\right)$

Conclusion:

Therefore, the complex number graphed can be expressed in rectangular form as $z=1+i$ or in polar form as $z=\sqrt{2}\left(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\right)$