Question

The complex number$z=-1+i$in polar form isz = __________.

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

c) The complex number graphed below can be expressed in rectangular form as __________ or in polar form as _________.

Short answer

a) The complex number $z=-1+i$in polar form is$z=\sqrt{2}(\cos \frac{3\pi }{4}+i\sin \frac{3\pi }{4})$.

b) The rectangular form of$z=2(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6})$ is$z=\sqrt{3}+i$ .

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

Step-by-step solution

Step 1. Given Information.

The complex number$z=-1+i$ .

Step 2. Applying the formula.

Consider the complex number$z=-1+i$.

Comparing the complex number with$z=a+ib$

We get a = -1 and b = 1

To convert the complex number into polar form, we compute the modulus r.

As we know that$r=\sqrt{a^2+b^2}$

$r=\sqrt{-1^2+1^2}$

$r=\sqrt{2}$

Now to compute$\theta$ argument as we can see that since the complex number is not unique and the complex number$z=-1+i$ lies in the second quadrant where tangent takes the negative values.

So,$\tan \theta =\frac{a}{b}$

$\tan \theta =\frac{-1}{1}$

$\theta ={\tan }^{-1}\left(\frac{-1}{1}\right)$

$\theta =\frac{3\pi }{4}$

Step 3. Converting the complex number into polar coordinate.

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

By substituting the value we get,

$a=\sqrt{2}\cos \frac{3\pi }{4}$

$b=\sqrt{2}\sin \frac{3\pi }{4}$

No substituting the values of a and b in the polar form of complex number$z=r(\cos \theta +i\sin \theta )$

$z=\sqrt{2}(\cos \frac{3\pi }{4}+i\sin \frac{3\pi }{4})$

Step 1. Given Information.

The complex number in polar form $z=2(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6})$.

Step 2. Applying the formula.

Consider the complex number$z=2(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6})$.

Comparing the complex number with$z=r(\cos \theta +i\sin \theta )$

We get r = 2 and $\theta =\frac{\pi }{6}$

To convert the complex number into rectangular form, we compute.

For a,

a$=r\cos \theta$

a $=2\cos \frac{\pi }{6}$

a$=2\times \frac{\sqrt{3}}{2}$

a$=\sqrt{3}$

For b,

b$=r\sin \theta$

b $=2\sin \frac{\pi }{6}$

b $=1$

Now substituting the values of a, b we get,

$z=\sqrt{3}+i$

Step 1. Given Information.

The complex number$z=1+i$ .

Step 2. Applying the formula.

Consider the complex number$z=1+i$.

Comparing the complex number with$z=a+ib$

We get a = 1 and b = 1

To convert the complex number into polar form, we compute.

For $\tan \theta =\frac{a}{b}$

$\tan \theta =\frac{1}{1}$

$\theta ={\tan }^{-1}\left(\frac{1}{1}\right)$

$\theta =\frac{\pi }{4}$

Now to determine the modulus we use the formula$r=\sqrt{a^2+b^2}$

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

$r=\sqrt{1^2+1^2}$

$r=\sqrt{2}$

Now substituting the values of r, $\theta$in the polar form of a complex number$z=r(\cos \theta +i\sin \theta )$

We get,

$z=\sqrt{2}(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4})$