Test Objectives
  • Demonstrate the ability to find the product of two complex numbers
  • Demonstrate the ability to find the quotient of two complex numbers
Product and Quotient Theorems Practice Test:

#1:

Instructions: Simplify, write the answer in polar form.

$$a)\hspace{.1em}\sqrt{31}(\text{cos}\hspace{.1em}135° + i \hspace{.1em}\text{sin}\hspace{.1em}135°) \cdot 5(\text{cos}\hspace{.1em}90° + i \hspace{.1em}\text{sin}90°)$$

$$b)\hspace{.1em}4\left(\text{cos}\frac{11π}{6}+ i \hspace{.1em}\text{sin}\frac{11π}{6}\right) \cdot 4\left(\text{cos}\frac{5π}{6}+ i \hspace{.1em}\text{sin}\frac{5π}{6}\right)$$


#2:

Instructions: Simplify, write the answer in rectangular form.

$$a)\hspace{.1em}4\left(\text{cos}\frac{5π}{6}+ i \hspace{.1em}\text{sin}\frac{5π}{6}\right) \cdot 5\left(\text{cos}\frac{π}{6}+ i \hspace{.1em}\text{sin}\frac{π}{6}\right)$$

$$b)\hspace{.1em}2(\text{cos}\hspace{.1em}300° + i \hspace{.1em}\text{sin}\hspace{.1em}300°) \cdot 6(\text{cos}\hspace{.1em}30° + i \hspace{.1em}\text{sin}30°)$$


#3:

Instructions: Simplify, write the answer in polar form.

$$a)\hspace{.1em}\left(\frac{3\sqrt{2}}{2}- \frac{3\sqrt{2}}{2}i \right)\left(-\frac{5\sqrt{3}}{2}- \frac{5}{2}i\right)$$

$$b)\hspace{.1em}\frac{4(\text{cos}\hspace{.1em}60° + i \hspace{.1em}\text{sin}60°)}{2(\text{cos}\hspace{.1em}225° + i \hspace{.1em}\text{sin}225°)}$$


#4:

Instructions: Simplify, write the answer in polar form.

$$a)\hspace{.1em}\frac{2(\text{cos}\hspace{.1em}120° + i \hspace{.1em}\text{sin}120°)}{5(\text{cos}\hspace{.1em}315° + i \hspace{.1em}\text{sin}315°)}$$

Instructions: Simplify, write the answer in rectangular form.

$$b)\hspace{.1em}\frac{6\left(\text{cos}\hspace{.1em}\large{\frac{5π}{3}}+ i \hspace{.1em}\text{sin}\large{\frac{5π}{3}}\right)}{3\left(\text{cos}\hspace{.1em}\large{\frac{π}{2}}+ i \hspace{.1em}\text{sin}\large{\frac{π}{2}}\right)}$$


#5:

Instructions: Simplify, write the answer in rectangular form.

$$a)\hspace{.1em}\frac{6(\text{cos}\hspace{.1em}330° + i \hspace{.1em}\text{sin}330°)}{3(\text{cos}\hspace{.1em}90° + i \hspace{.1em}\text{sin}90°)}$$

Instructions: Simplify, write the answer in polar form.

$$b)\hspace{.1em}\frac{-\sqrt{2}+ i\sqrt{2}}{-\large{\frac{5\sqrt{2}}{2}}- \large{\frac{5\sqrt{2}}{2}}i}$$


Written Solutions:

#1:

Solutions:

$$a)\hspace{.1em}5\sqrt{31}(\text{cos}\hspace{.1em}225° + i \hspace{.1em}\text{sin}225°)$$

$$b)\hspace{.1em}16\left(\text{cos}\frac{2π}{3}+ i \hspace{.1em}\text{sin}\frac{2π}{3}\right)$$


#2:

Solutions:

$$a)\hspace{.1em}-20$$

$$b)\hspace{.1em}6\sqrt{3}- 6i$$


#3:

Solutions:

$$a)\hspace{.1em}15\left(\text{cos}\frac{11π}{12}+ i \hspace{.1em}\text{sin}\frac{11π}{12}\right)$$

$$b)\hspace{.1em}2(\text{cos}\hspace{.1em}195° + i \hspace{.1em}\text{sin}195°)$$


#4:

Solutions:

$$a)\hspace{.1em}\frac{2}{5}(\text{cos}\hspace{.1em}165° + i \hspace{.1em}\text{sin}\hspace{.1em}165°)$$

$$b)\hspace{.1em}{-}\sqrt{3}- i$$


#5:

Solutions:

$$a)\hspace{.1em}{-}1 - i\sqrt{3}$$

$$b)\hspace{.1em}\frac{2}{5}\left(\text{cos}\frac{3π}{2}+ i \hspace{.1em}\text{sin}\frac{3π}{2}\right)$$