10.5 Polar Form of Complex Numbers

Find the absolute value of the complex number $z = 12 - 5 i .$

Given $z = 1 - 7 i ,$ find $\left|z\right| .$

Express $z = 3 i$ as $r \text{cis} \theta$ in polar form.

Find the polar form of $- 4 + 4 i .$

Write $z = \sqrt{3} + i$ in polar form.

Convert the polar form of the given complex number to rectangular form:

Find the rectangular form of the complex number given $r = 13$ and $\tan \theta = \frac{5}{12} .$ Assume the number is in the first quadrant.

Convert the complex number to rectangular form:

Find the product of $z_{1} z_{2} ,$ given $z_{1} = 4 \left(\cos \left(80°\right) + i \sin \left(80°\right)\right)$ and $z_{2} = 2 \left(\cos \left(145°\right) + i \sin \left(145°\right)\right) .$

Find the quotient of $z_{1} = 2 \left(\cos \left(213°\right) + i \sin \left(213°\right)\right)$ and $z_{2} = 4 \left(\cos \left(33°\right) + i \sin \left(33°\right)\right) .$

Find the product and the quotient of $z_{1} = 2 \sqrt{3} \left(\cos \left(150°\right) + i \sin \left(150°\right)\right)$ and $z_{2} = 2 \left(\cos \left(30°\right) + i \sin \left(30°\right)\right) .$

Evaluate the expression $\left(1 + i\right)^{5}$ using De Moivre’s Theorem.

Evaluate the cube roots of $z = 8 \left(\cos \left(\frac{2 \pi}{3}\right) + i \sin \left(\frac{2 \pi}{3}\right)\right) .$

Find the four fourth roots of $16 \left(\cos \left(120°\right) + i \sin \left(120°\right)\right) .$

A complex number is $a + b i .$ Explain each part.

How is a complex number converted to polar form?

What is De Moivre’s Theorem and what is it used for?

Find the absolute value of the given complex number.

$- 7 + ​ i$

Find the absolute value of the given complex number.

$\sqrt{2} - 6 i$

Find the absolute value of the given complex number.

$2.2 - 3.1 i$

Write the complex number in polar form.

$8 - 4 i$

Write the complex number in polar form.

$\sqrt{3} + i$

Convert the complex number from polar to rectangular form.

$z = 7 cis \left(\frac{\pi}{6}\right)$

Convert the complex number from polar to rectangular form.

$z = 4 cis \left(\frac{7 \pi}{6}\right)$

Convert the complex number from polar to rectangular form.

$z = 3 cis \left(240°\right)$

Find $z_{1} z_{2}$ in polar form.

$z_{1} = 2 \sqrt{3} cis \left(116°\right) ; z_{2} = 2 cis \left(82°\right)$

Find $z_{1} z_{2}$ in polar form.

$z_{1} = 3 cis \left(120°\right) ; z_{2} = \frac{1}{4} cis \left(60°\right)$

Find $z_{1} z_{2}$ in polar form.

$z_{1} = \sqrt{5} cis \left(\frac{5 \pi}{8}\right) ; z_{2} = \sqrt{15} cis \left(\frac{\pi}{12}\right)$

Find $\frac{z_{1}}{z_{2}}$ in polar form.

$z_{1} = 21 cis \left(135°\right) ; z_{2} = 3 cis \left(65°\right)$

Find $\frac{z_{1}}{z_{2}}$ in polar form.

$z_{1} = 15 cis \left(120°\right) ; z_{2} = 3 cis \left(40°\right)$

Find $\frac{z_{1}}{z_{2}}$ in polar form.

$z_{1} = 5 \sqrt{2} cis \left(\pi\right) ; z_{2} = \sqrt{2} cis \left(\frac{2 \pi}{3}\right)$

Find the powers of each complex number in polar form.

Find $z^{3}$ when $z = 5 cis \left(45°\right) .$

Find the powers of each complex number in polar form.

Find $z^{2}$ when $z = 3 cis \left(120°\right) .$

Find the powers of each complex number in polar form.

Find $z^{4}$ when $z = cis \left(\frac{3 \pi}{16}\right) .$

Evaluate each root.

Evaluate the cube root of $z$ when $z = 27 cis \left(240°\right) .$

Evaluate each root.

Evaluate the cube root of $z$ when $z = 32 cis \left(\frac{2 \pi}{3}\right) .$

Evaluate each root.

Evaluate the square root of $z$ when $z = 8 cis \left(\frac{7 \pi}{4}\right) .$

Find all answers rounded to the nearest hundredth.

Use the rectangular to polar feature on the graphing calculator to change $3 - 2 i$ to polar form.

Find all answers rounded to the nearest hundredth.

Use the polar to rectangular feature on the graphing calculator to change $4 cis \left(120°\right)$ to rectangular form.

Find all answers rounded to the nearest hundredth.

Use the polar to rectangular feature on the graphing calculator to change $5 cis \left(210°\right)$ to rectangular form.