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21.3: Exercises

  • Page ID
    54470
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    Exercise \(\PageIndex{1}\)

    Plot the complex numbers in the complex plane.

    1. \(4+2i\)
    2. \(-3-5i\)
    3. \(6-2i\)
    4. \(-5+i\)
    5. \(-2i\)
    6. \(\sqrt{2}-\sqrt{2}i\)
    7. \(7\)
    8. \(i\)
    9. \(0\)
    10. \(2i-\sqrt{3}\)
    Answer

    clipboard_e1477883f6fdca92e59ee54217c6094d5.png

    Exercise \(\PageIndex{2}\)

    Add, subtract, multiply, and divide, as indicated.

    1. \((5-2i)+(-2+6i)\)
    2. \((-9-i)-(5-3i)\)
    3. \((3+2i)\cdot (4+3i)\)
    4. \((-2-i)\cdot (-1+4i)\)
    5. \(\dfrac{2+3i}{2+i}\)
    6. \((5+5i)\div (2-4i)\)
    Answer
    1. \(3+4 i\)
    2. \(-14+2 i\)
    3. \(6+17 i\)
    4. \(6-7 i\)
    5. \(\dfrac{7}{5}+\dfrac{4}{5} i\)
    6. \(-\dfrac{1}{2}+\dfrac{3}{2} i\)

    Exercise \(\PageIndex{3}\)

    Find the absolute value \(|a+bi|\) of the given complex number, and simplify your answer as much as possible.

    1. \(|4+3i|\)
    2. \(|1-2i|\)
    3. \(|-3i|\)
    4. \(|-2-6i|\)
    5. \(|\sqrt{8}-i|\)
    6. \(|-2\sqrt{3}-2i|\)
    7. \(|-5|\)
    8. \(|-\sqrt{17}+4\sqrt{2}i|\)
    Answer
    1. \(5\)
    2. \(\sqrt{5}\)
    3. \(3\)
    4. \(2 \sqrt{10}\)
    5. \(3\)
    6. \(4\)
    7. \(5\)
    8. \(7\)

    Exercise \(\PageIndex{4}\)

    Convert the complex number into polar form \(r(\cos(\theta)+i\sin(\theta))\).

    1. \(2+2i\)
    2. \(4\sqrt{3}+4i\)
    3. \(3-2i\)
    4. \(-5+5i\)
    5. \(4-3i\)
    6. \(-4+3i\)
    7. \(-\sqrt{5}-\sqrt{15}i\)
    8. \(\sqrt{7}-\sqrt{21}i\)
    9. \(-5-12i\)
    10. \(6i\)
    11. \(-10\)
    12. \(-\sqrt{3}+3i\)
    Answer
    1. \(2 \sqrt{2}\left(\cos \left(\dfrac{\pi}{4}\right)+i \sin \left(\dfrac{\pi}{4}\right)\right)\)
    2. \(8\left(\cos \left(\dfrac{\pi}{6}\right)+i \sin \left(\dfrac{\pi}{6}\right)\right)\)
    3. approximately \(\sqrt{13}(\cos (-.588)+i \sin (-.588))\) or \(\sqrt{13}(\cos (-.187 \pi)+i \sin (-.187 \pi))\)
    4. \(5 \sqrt{2}\left(\cos \left(\dfrac{3 \pi}{4}\right)+i \sin \left(\dfrac{3 \pi}{4}\right)\right)\)
    5. approximately \(5(\cos (-.644)+i \sin (-.644))\) or \(5(\cos (-.205 \pi)+i \sin (-.205 \pi))\)
    6. approximately \(5(\cos (2.498)+i \sin (2.498))\) or \(5(\cos (.795 \pi)+i \sin (.795 \pi))\)
    7. \(2 \sqrt{5}\left(\cos \left(\dfrac{4 \pi}{3}\right)+i \sin \left(\dfrac{4 \pi}{3}\right)\right)\)
    8. \(2 \sqrt{7}\left(\cos \left(-\dfrac{\pi}{3}\right)+i \sin \left(-\dfrac{\pi}{3}\right)\right)\)
    9. approximately \(13(\cos (4.318)+i \sin (4.318))\) or \(13(\cos (1.374 \pi)+i \sin (1.374 \pi))\)
    10. \(6\left(\cos \left(\dfrac{\pi}{2}\right)+i \sin \left(\dfrac{\pi}{2}\right)\right)\)
    11. \(10(\cos (\pi)+i \sin (\pi))\)
    12. \(2 \sqrt{3}\left(\cos \left(\dfrac{2 \pi}{3}\right)+ i \sin \left(\dfrac{2 \pi}{3}\right)\right)\)

    Exercise \(\PageIndex{5}\)

    Convert the complex number into the standard form \(a+bi\).

    1. \(6(\cos(134^\circ)+i\sin(134^\circ))\)
    2. \(\dfrac 1 2 \left(\cos\left(\dfrac \pi {17}\right)+i\sin\left(\dfrac \pi {17}\right)\right)\)
    3. \(2(\cos(270^\circ)+i\sin(270^\circ))\)
    4. \(\cos\left(\dfrac{\pi} 6\right)+i\sin\left(\dfrac{\pi}6\right)\)
    5. \(10\left(\cos\left(\dfrac{7\pi}{6}\right)+i\sin\left(\dfrac{7\pi}{6}\right)\right)\)
    6. \(6 \left(\cos\left(-\dfrac{5\pi}{12}\right)+i\sin\left(-\dfrac{5\pi}{12}\right)\right)\)
    Answer
    1. approximately \(-4.168+4.316 i\)
    2. approximately \(.491+0.0919 i\)
    3. \(-2 i\)
    4. \(\dfrac{\sqrt{3}}{2}+\dfrac{1}{2} i\)
    5. \(-5 \sqrt{3}-5 i\)
    6. approximately \(1.553-5 / 796 i\)

    Exercise \(\PageIndex{6}\)

    Multiply the complex numbers and write the answer in standard form \(a+bi\).

    1. \(4(\cos(27^\circ)+i\sin(27^\circ)) \cdot 10(\cos(33^\circ)+i\sin(33^\circ))\)
    2. \(7\left(\cos\left(\dfrac{2\pi}{9}\right)+i\sin\left(\dfrac{2\pi}{9}\right)\right) \cdot 6\left(\cos\left(\dfrac{\pi}{9}\right)+i\sin\left(\dfrac{\pi}{9}\right)\right)\)
    3. \(\left(\cos\left(\dfrac{13\pi}{12}\right)+i\sin\left(\dfrac{13\pi}{12}\right)\right) \cdot \left(\cos\left(\dfrac{-11\pi}{12}\right)+i\sin\left(\dfrac{-11\pi}{12}\right)\right)\)
    4. \(8\left(\cos\left(\dfrac{3\pi}{7}\right)+i\sin\left(\dfrac{3\pi}{7}\right)\right) \cdot 1.5\left(\cos\left(\dfrac{4\pi}{7}\right)+i\sin\left(\dfrac{4\pi}{7}\right)\right)\)
    5. \(0.2(\cos(196^\circ)+i\sin(196^\circ)) \cdot 0.5(\cos(88^\circ)+i\sin(88^\circ))\)
    6. \(4\left(\cos\left(\dfrac{7\pi}{8}\right)+i\sin\left(\dfrac{7\pi}{8}\right)\right) \cdot 0.25\left(\cos\left(\dfrac{-5\pi}{24}\right)+i\sin\left(\dfrac{-5\pi}{24}\right)\right)\)
    Answer
    1. \(40\left(\cos \left(60^{\circ}\right)+i \sin \left(60^{\circ}\right)\right)=20+20 \sqrt{3} i\)
    2. \(42\left(\cos \left(\dfrac{\pi}{3}\right)+i \sin \left(\dfrac{\pi}{3}\right) \right)=21+21 \sqrt{3} i\)
    3. \(\cos \left(\dfrac{\pi}{6}\right)+i \sin \left(\dfrac{\pi}{6}\right)=\dfrac{\sqrt{3}}{2}+\dfrac{1}{2} i\)
    4. \(12(\cos (\pi)+i \sin (\pi))=-12\)
    5. \(.1\left(\cos \left(284^{\circ}\right)+i \sin \left(284^{\circ}\right)\right) \approx .0242-.0970 i\)
    6. \(\cos \left(\dfrac{2 \pi}{3}\right)+i \sin \left(\dfrac{2 \pi}{3}\right)=-\dfrac{1}{2}+\dfrac{\sqrt{3}}{2} i\)

    Exercise \(\PageIndex{7}\)

    Divide the complex numbers and write the answer in standard form \(a+bi\).

    1. \(\displaystyle\frac{18(\cos(\frac{\pi}{2})+i\sin(\frac{\pi}{2}))}{3(\cos(\frac{\pi}{6})+i\sin(\frac{\pi}{6}))}\)
    2. \(\displaystyle\frac{10(\cos(254^\circ)+i\sin(254^\circ))}{15(\cos(164^\circ)+i\sin(164^\circ))}\)
    3. \(\displaystyle\frac{\sqrt{24}(\cos(\frac{11\pi}{14})+i\sin(\frac{11\pi}{14}))}{\sqrt{6}(\cos(\frac{2\pi}{7})+i\sin(\frac{2\pi}{7}))}\)
    4. \(\displaystyle\frac{\cos(\frac{8\pi}{5})+i\sin(\frac{8\pi}{5})}{2(\cos(\frac{\pi}{10})+i\sin(\frac{\pi}{10}))}\)
    5. \(\displaystyle\frac{42(\cos(\frac{7\pi}{4})+i\sin(\frac{7\pi}{4}))}{7(\cos(\frac{5\pi}{12})+i\sin(\frac{5\pi}{12}))}\)
    6. \(\displaystyle\frac{30(\cos(-175^\circ)+i\sin(-175^\circ))}{18(\cos(144^\circ)+i\sin(144^\circ))}\)
    Answer
    1. \(6(\cos (\pi / 3)+i \sin (\pi / 3))=3+3 \sqrt{3} i\)
    2. \(\dfrac{2}{3}\left(\cos \left(90^{\circ}\right)+i \sin \left(90^{\circ}\right)\right)=\dfrac{2}{3} i\)
    3. \(2(\cos (\pi / 2)+i \sin (\pi / 2))=2 i\)
    4. \(\dfrac{1}{2}\left(\cos \left(\dfrac{3 \pi}{2}\right)+i \sin \left(\dfrac{3 \pi}{2}\right)\right)=-\dfrac{1}{2} i\)
    5. \(6\left(\cos \left(\dfrac{4 \pi}{3}\right)+i \sin \left(\dfrac{4 \pi}{3}\right)\right)=-3-3 \sqrt{3} i\)
    6. \(\dfrac{5}{3}\left(\cos \left(-319^{\circ}\right)+i \sin \left(-319^{\circ}\right)\right) \approx 1.258+1.093 i\)

    This page titled 21.3: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.