- Demonstrate the ability to find the product of two complex numbers
- Demonstrate the ability to find the quotient of two complex numbers
#1:
Instructions: Simplify, write the answer in polar form.
$$a)\hspace{.1em}\sqrt{31}(\text{cos}\hspace{.1em}135° + i \hspace{.1em}\text{sin}\hspace{.1em}135°) \cdot 5(\text{cos}\hspace{.1em}90° + i \hspace{.1em}\text{sin}90°)$$
$$b)\hspace{.1em}4\left(\text{cos}\frac{11π}{6}+ i \hspace{.1em}\text{sin}\frac{11π}{6}\right) \cdot 4\left(\text{cos}\frac{5π}{6}+ i \hspace{.1em}\text{sin}\frac{5π}{6}\right)$$
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#2:
Instructions: Simplify, write the answer in rectangular form.
$$a)\hspace{.1em}4\left(\text{cos}\frac{5π}{6}+ i \hspace{.1em}\text{sin}\frac{5π}{6}\right) \cdot 5\left(\text{cos}\frac{π}{6}+ i \hspace{.1em}\text{sin}\frac{π}{6}\right)$$
$$b)\hspace{.1em}2(\text{cos}\hspace{.1em}300° + i \hspace{.1em}\text{sin}\hspace{.1em}300°) \cdot 6(\text{cos}\hspace{.1em}30° + i \hspace{.1em}\text{sin}30°)$$
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#3:
Instructions: Simplify, write the answer in polar form.
$$a)\hspace{.1em}\left(\frac{3\sqrt{2}}{2}- \frac{3\sqrt{2}}{2}i \right)\left(-\frac{5\sqrt{3}}{2}- \frac{5}{2}i\right)$$
$$b)\hspace{.1em}\frac{4(\text{cos}\hspace{.1em}60° + i \hspace{.1em}\text{sin}60°)}{2(\text{cos}\hspace{.1em}225° + i \hspace{.1em}\text{sin}225°)}$$
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#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.
$$b)\hspace{.1em}\frac{6\left(\text{cos}\hspace{.1em}\large{\frac{5π}{3}}+ i \hspace{.1em}\text{sin}\large{\frac{5π}{3}}\right)}{3\left(\text{cos}\hspace{.1em}\large{\frac{π}{2}}+ i \hspace{.1em}\text{sin}\large{\frac{π}{2}}\right)}$$
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#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.
$$b)\hspace{.1em}\frac{-\sqrt{2}+ i\sqrt{2}}{-\large{\frac{5\sqrt{2}}{2}}- \large{\frac{5\sqrt{2}}{2}}i}$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.1em}5\sqrt{31}(\text{cos}\hspace{.1em}225° + i \hspace{.1em}\text{sin}225°)$$
$$b)\hspace{.1em}16\left(\text{cos}\frac{2π}{3}+ i \hspace{.1em}\text{sin}\frac{2π}{3}\right)$$
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#2:
Solutions:
$$a)\hspace{.1em}{-}20$$
$$b)\hspace{.1em}6\sqrt{3}- 6i$$
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#3:
Solutions:
$$a)\hspace{.1em}15\left(\text{cos}\frac{11π}{12}+ i \hspace{.1em}\text{sin}\frac{11π}{12}\right)$$
$$b)\hspace{.1em}2(\text{cos}\hspace{.1em}195° + i \hspace{.1em}\text{sin}195°)$$
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#4:
Solutions:
$$a)\hspace{.1em}\frac{2}{5}(\text{cos}\hspace{.1em}165° + i \hspace{.1em}\text{sin}\hspace{.1em}165°)$$
$$b)\hspace{.1em}{-}\sqrt{3}- i$$
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#5:
Solutions:
$$a)\hspace{.1em}{-}1 - i\sqrt{3}$$
$$b)\hspace{.1em}\frac{2}{5}\left(\text{cos}\frac{3π}{2}+ i \hspace{.1em}\text{sin}\frac{3π}{2}\right)$$
