{"id":1422,"date":"2023-06-05T14:51:34","date_gmt":"2023-06-05T14:51:34","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-polar-form-of-complex-numbers\/"},"modified":"2023-06-05T14:51:34","modified_gmt":"2023-06-05T14:51:34","slug":"solutions-for-polar-form-of-complex-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-polar-form-of-complex-numbers\/","title":{"raw":"Solutions 61: Polar Form of Complex Numbers","rendered":"Solutions 61: Polar Form of Complex Numbers"},"content":{"raw":"\n

Solutions to Odd-Numbered Answers<\/h2>\n1. a<\/em> is the real part, b<\/em> is the imaginary part, and [latex]i=\\sqrt{\u22121}[\/latex]\n\n3. Polar form converts the real and imaginary part of the complex number in polar form using [latex]x=r\\cos\\theta[\/latex] and [latex]y=r\\sin\\theta[\/latex]\n\n5. [latex]z^{n}=r^{n}\\left(\\cos\\left(n\\theta\\right)+i\\sin\\left(n\\theta\\right)\\right)[\/latex]. It is used to simplify polar form when a number has been raised to a power.\n\n7. [latex]5\\sqrt{2}[\/latex]\n\n9. [latex]\\sqrt{38}[\/latex]\n\n11. [latex]\\sqrt{14.45}[\/latex]\n\n13. [latex]4\\sqrt{5}\\text{cis}\\left(333.4^{\\circ}\\right)[\/latex]\n\n15. [latex]2\\text{cis}\\left(\\frac{\\pi}{6}\\right)[\/latex]\n\n17. [latex]\\frac{7\\sqrt{3}}{2}+i\\frac{7}{2}[\/latex]\n\n19. [latex]\u22122\\sqrt{3}\u22122i[\/latex]\n\n21. [latex]\u22121.5\u2212i\\frac{3\\sqrt{3}}{2}[\/latex]\n\n23. [latex]4\\sqrt{3}\\text{cis}\\left(198^{\\circ}\\right)[\/latex]\n\n25. [latex]\\frac{3}{4}\\text{cis}\\left(180^{\\circ}\\right)[\/latex]\n\n27. [latex]5\\sqrt{3}\\text{cis}\\left(\\frac{17\\pi}{24}\\right)[\/latex]\n\n29. [latex]7\\text{cis}\\left(70^{\\circ}\\right)[\/latex]\n\n31. [latex]5\\text{cis}\\left(80^{\\circ}\\right)[\/latex]\n\n33. [latex]5\\text{cis}\\left(\\frac{\\pi}{3}\\right)[\/latex]\n\n35. [latex]125\\text{cis}\\left(135^{\\circ}\\right)[\/latex]\n\n37. [latex]9\\text{cis}\\left(240^{\\circ}\\right)[\/latex]\n\n39. [latex]\\text{cis}\\left(\\frac{3\\pi}{4}\\right)[\/latex]\n\n41. [latex]3\\text{cis}\\left(80^{\\circ}\\right)\\text{, }3\\text{cis}\\left(200^{\\circ}\\right)\\text{, }3\\text{cis}\\left(320^{\\circ}\\right)[\/latex]\n\n43. [latex]2\\sqrt[3]{4}\\text{cis}\\left(\\frac{2\\pi}{9}\\right)\\text{, }2\\sqrt[3]{4}\\text{cis}\\left(\\frac{8\\pi}{9}\\right)\\text{, }2\\sqrt[3]{4}\\text{cis}\\left(\\frac{14\\pi}{9}\\right)[\/latex]\n\n45. [latex]2\\sqrt{2}\\text{cis}\\left(\\frac{7\\pi}{8}\\right)\\text{, }2\\sqrt{2}\\text{cis}\\left(\\frac{15\\pi}{8}\\right)[\/latex]\n\n47.\n\"Plot\n\n49.\n\"Plot\n\n51.\n\"Plot\n\n53.\n\"Plot\n\n55.\n\"Plot\n\n57. [latex]3.61e^{\u22120.59i}[\/latex]\n\n59. [latex]\u22122+3.46i[\/latex]\n\n61. [latex]\u22124.33\u22122.50i[\/latex]\n","rendered":"

Solutions to Odd-Numbered Answers<\/h2>\n

1. a<\/em> is the real part, b<\/em> is the imaginary part, and [latex]i=\\sqrt{\u22121}[\/latex]<\/p>\n

3. Polar form converts the real and imaginary part of the complex number in polar form using [latex]x=r\\cos\\theta[\/latex] and [latex]y=r\\sin\\theta[\/latex]<\/p>\n

5. [latex]z^{n}=r^{n}\\left(\\cos\\left(n\\theta\\right)+i\\sin\\left(n\\theta\\right)\\right)[\/latex]. It is used to simplify polar form when a number has been raised to a power.<\/p>\n

7. [latex]5\\sqrt{2}[\/latex]<\/p>\n

9. [latex]\\sqrt{38}[\/latex]<\/p>\n

11. [latex]\\sqrt{14.45}[\/latex]<\/p>\n

13. [latex]4\\sqrt{5}\\text{cis}\\left(333.4^{\\circ}\\right)[\/latex]<\/p>\n

15. [latex]2\\text{cis}\\left(\\frac{\\pi}{6}\\right)[\/latex]<\/p>\n

17. [latex]\\frac{7\\sqrt{3}}{2}+i\\frac{7}{2}[\/latex]<\/p>\n

19. [latex]\u22122\\sqrt{3}\u22122i[\/latex]<\/p>\n

21. [latex]\u22121.5\u2212i\\frac{3\\sqrt{3}}{2}[\/latex]<\/p>\n

23. [latex]4\\sqrt{3}\\text{cis}\\left(198^{\\circ}\\right)[\/latex]<\/p>\n

25. [latex]\\frac{3}{4}\\text{cis}\\left(180^{\\circ}\\right)[\/latex]<\/p>\n

27. [latex]5\\sqrt{3}\\text{cis}\\left(\\frac{17\\pi}{24}\\right)[\/latex]<\/p>\n

29. [latex]7\\text{cis}\\left(70^{\\circ}\\right)[\/latex]<\/p>\n

31. [latex]5\\text{cis}\\left(80^{\\circ}\\right)[\/latex]<\/p>\n

33. [latex]5\\text{cis}\\left(\\frac{\\pi}{3}\\right)[\/latex]<\/p>\n

35. [latex]125\\text{cis}\\left(135^{\\circ}\\right)[\/latex]<\/p>\n

37. [latex]9\\text{cis}\\left(240^{\\circ}\\right)[\/latex]<\/p>\n

39. [latex]\\text{cis}\\left(\\frac{3\\pi}{4}\\right)[\/latex]<\/p>\n

41. [latex]3\\text{cis}\\left(80^{\\circ}\\right)\\text{, }3\\text{cis}\\left(200^{\\circ}\\right)\\text{, }3\\text{cis}\\left(320^{\\circ}\\right)[\/latex]<\/p>\n

43. [latex]2\\sqrt[3]{4}\\text{cis}\\left(\\frac{2\\pi}{9}\\right)\\text{, }2\\sqrt[3]{4}\\text{cis}\\left(\\frac{8\\pi}{9}\\right)\\text{, }2\\sqrt[3]{4}\\text{cis}\\left(\\frac{14\\pi}{9}\\right)[\/latex]<\/p>\n

45. [latex]2\\sqrt{2}\\text{cis}\\left(\\frac{7\\pi}{8}\\right)\\text{, }2\\sqrt{2}\\text{cis}\\left(\\frac{15\\pi}{8}\\right)[\/latex]<\/p>\n

47.
\n\"Plot<\/p>\n

49.
\n\"Plot<\/p>\n

51.
\n\"Plot<\/p>\n

53.
\n\"Plot<\/p>\n

55.
\n\"Plot<\/p>\n

57. [latex]3.61e^{\u22120.59i}[\/latex]<\/p>\n

59. [latex]\u22122+3.46i[\/latex]<\/p>\n

61. [latex]\u22124.33\u22122.50i[\/latex]<\/p>\n\n\t\t\t

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Candela Citations<\/h3>\n\t\t\t\t\t
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CC licensed content, Specific attribution<\/div>