{"id":2361,"date":"2025-08-13T00:47:35","date_gmt":"2025-08-13T00:47:35","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2361"},"modified":"2026-02-18T16:51:11","modified_gmt":"2026-02-18T16:51:11","slug":"polar-functions-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/polar-functions-get-stronger\/","title":{"raw":"Polar Functions: Get Stronger","rendered":"Polar Functions: Get Stronger"},"content":{"raw":"

Polar Coordinates<\/h1>\r\n1. How are polar coordinates different from rectangular coordinates?\r\n\r\n3. Explain how polar coordinates are graphed.\r\n\r\n5. Explain why the points [latex]\\left(-3,\\frac{\\pi }{2}\\right)[\/latex] and [latex]\\left(3,-\\frac{\\pi }{2}\\right)[\/latex] are the same.\r\n\r\nFor the following exercises, convert the given polar coordinates to Cartesian coordinates with [latex]r>0[\/latex] and [latex]0\\le \\theta \\le 2\\pi [\/latex]. Remember to consider the quadrant in which the given point is located when determining [latex]\\theta [\/latex] for the point.\r\n\r\n7. [latex]\\left(5,\\pi \\right)[\/latex]\r\n\r\n9. [latex]\\left(-3,\\frac{\\pi }{6}\\right)[\/latex]\r\n\r\nFor the following exercises, convert the given Cartesian coordinates to polar coordinates with [latex]r>0,0\\le \\theta <2\\pi [\/latex]. Remember to consider the quadrant in which the given point is located.\r\n\r\n11. [latex]\\left(4,2\\right)[\/latex]\r\n\r\n13. [latex]\\left(3,-5\\right)[\/latex]\r\n\r\nFor the following exercises, convert the given Cartesian equation to a polar equation.\r\n\r\n17. [latex]y=4[\/latex]\r\n\r\n19. [latex]y=2{x}^{4}[\/latex]\r\n\r\n21. [latex]{x}^{2}+{y}^{2}=3x[\/latex]\r\n\r\n23. [latex]{x}^{2}-{y}^{2}=3y[\/latex]\r\n\r\n25. [latex]{x}^{2}=9y[\/latex]\r\n\r\n27. [latex]9xy=1[\/latex]\r\n\r\nFor the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.\r\n\r\n29. [latex]r=4\\cos \\theta [\/latex]\r\n\r\n31. [latex]r=\\frac{6}{\\cos \\theta +3\\sin \\theta }[\/latex]\r\n\r\n33. [latex]r=3\\csc \\theta [\/latex]\r\n\r\n35. [latex]{r}^{2}=4\\sec \\theta \\csc \\theta [\/latex]\r\n\r\n37. [latex]{r}^{2}=4[\/latex]\r\n\r\n39. [latex]r=\\frac{3}{\\cos \\theta -5\\sin \\theta }[\/latex]\r\n\r\nFor the following exercises, find the polar coordinates of the point.\r\n\r\n41.\r\n\"Polar\r\n\r\n43.\r\n\"Polar\r\n\r\nFor the following exercises, plot the points.\r\n\r\n45. [latex]\\left(-2,\\frac{\\pi }{3}\\right)[\/latex]\r\n\r\n49. [latex]\\left(5,\\frac{\\pi }{2}\\right)[\/latex]\r\n\r\n51. [latex]\\left(3,\\frac{5\\pi }{6}\\right)[\/latex]\r\n\r\nFor the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.\r\n\r\n55. [latex]5x-y=6[\/latex]\r\n\r\n57. [latex]{x}^{2}+{\\left(y - 1\\right)}^{2}=1[\/latex]\r\n\r\n59. [latex]x=2[\/latex]\r\n\r\n61. [latex]{x}^{2}+{y}^{2}=3x[\/latex]\r\n\r\nFor the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.\r\n\r\n63. [latex]r=-4[\/latex]\r\n\r\n65. [latex]\\theta =\\frac{\\pi }{4}[\/latex]\r\n\r\n67. [latex]r=-10\\sin \\theta [\/latex]\r\n\r\n69. Use a graphing calculator to find the rectangular coordinates of [latex]\\left(2,-\\frac{\\pi }{5}\\right)[\/latex]. Round to the nearest thousandth.\r\n\r\n71. Use a graphing calculator to find the polar coordinates of [latex]\\left(-7,8\\right)[\/latex] in degrees. Round to the nearest thousandth.\r\n\r\n73. Use a graphing calculator to find the polar coordinates of [latex]\\left(-2,0\\right)[\/latex] in radians. Round to the nearest hundredth.\r\n

Polar Coordinates: Graphs<\/h1>\r\n1. Describe the three types of symmetry in polar graphs, and compare them to the symmetry of the Cartesian plane.\r\n\r\n3. What are the steps to follow when graphing polar equations?\r\n\r\n5. What part of the equation determines the shape of the graph of a polar equation?\r\n\r\nFor the following exercises, test the equation for symmetry.\r\n\r\n7. [latex]r=3 - 3\\cos \\theta [\/latex]\r\n\r\n9. [latex]r=3\\sin 2\\theta [\/latex]\r\n\r\n11. [latex]r=2\\theta [\/latex]\r\n\r\n13. [latex]r=\\frac{2}{\\theta }[\/latex]\r\n\r\n15. [latex]r=\\sqrt{5\\sin 2\\theta }[\/latex]\r\n\r\nFor the following exercises, graph the polar equation. Identify the name of the shape.\r\n\r\n17. [latex]r=4\\sin \\theta [\/latex]\r\n\r\n19. [latex]r=2 - 2\\cos \\theta [\/latex]\r\n\r\n21. [latex]r=3+3\\sin \\theta [\/latex]\r\n\r\n23. [latex]r=7+4\\sin \\theta [\/latex]\r\n\r\n25. [latex]r=5+4\\cos \\theta [\/latex]\r\n\r\n27. [latex]r=1+3\\sin \\theta [\/latex]\r\n\r\n29. [latex]r=5+7\\sin \\theta [\/latex]\r\n\r\n31. [latex]r=5+6\\cos \\theta [\/latex]\r\n\r\n33. [latex]{r}^{2}=10\\cos \\left(2\\theta \\right)[\/latex]\r\n\r\n35. [latex]{r}^{2}=10\\sin \\left(2\\theta \\right)[\/latex]\r\n\r\n37. [latex]r=3\\text{cos}\\left(2\\theta \\right)[\/latex]\r\n\r\n39. [latex]r=4\\text{sin}\\left(4\\theta \\right)[\/latex]\r\n\r\n41. [latex]r=-\\theta [\/latex]\r\n\r\n43. [latex]r=-3\\theta [\/latex]\r\n\r\nFor the following exercises, use a graphing calculator to sketch the graph of the polar equation.\r\n\r\n45.\u00a0[latex]r=\\frac{1}{\\sqrt{\\theta }}[\/latex]\r\n\r\n47. [latex]r=2\\sqrt{1-{\\sin }^{2}\\theta }[\/latex] , a hippopede\r\n\r\n49. [latex]r=2-\\sin \\left(2\\theta \\right)[\/latex]\r\n\r\n51. [latex]r=\\theta +1[\/latex]\r\n\r\n53. [latex]r=\\theta \\cos \\theta [\/latex]\r\n\r\n61. On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs.\r\n[latex]\\begin{array}{l}\\begin{array}{l}\\\\ {r}_{1}=3\\sin \\left(3\\theta \\right)\\end{array}\\hfill \\\\ {r}_{2}=2\\sin \\left(3\\theta \\right)\\hfill \\\\ {r}_{3}=\\sin \\left(3\\theta \\right)\\hfill \\end{array}[\/latex]\r\n\r\n63. On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs.\r\n[latex]\\begin{array}{l}\\begin{array}{l}\\\\ {r}_{1}=3\\theta \\end{array}\\hfill \\\\ {r}_{2}=2\\theta \\hfill \\\\ {r}_{3}=\\theta \\hfill \\end{array}[\/latex]\r\n\r\nFor the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.\r\n\r\n65. [latex]{r}_{1}=6 - 4\\cos \\theta ,{r}_{2}=4[\/latex]\r\n\r\n67. [latex]{r}_{1}=1+\\cos \\theta ,{r}_{2}=3\\cos \\theta [\/latex]\r\n\r\n69. [latex]{r}_{1}={\\sin }^{2}\\left(2\\theta \\right),{r}_{2}=1-\\cos \\left(4\\theta \\right)[\/latex]\r\n\r\n71. [latex]{r}_{1}{}^{2}=\\sin \\theta ,{r}_{2}{}^{2}=\\cos \\theta [\/latex]\r\n

Polar Form of Complex Numbers<\/h1>\r\n1. A complex number is [latex]a+bi[\/latex]. Explain each part.\r\n\r\n3. How is a complex number converted to polar form?\r\n\r\nFor the following exercises, find the absolute value of the given complex number.\r\n\r\n7. [latex]\u22127+i[\/latex]\r\n\r\n9. [latex]\\sqrt{2}\u22126i[\/latex]\r\n\r\n11. [latex]2.2\u22123.1i[\/latex]\r\n\r\nFor the following exercises, write the complex number in polar form.\r\n\r\n13. [latex]8\u22124i[\/latex]\r\n\r\n15. [latex]\\sqrt{3}+i[\/latex]\r\n\r\nFor the following exercises, convert the complex number from polar to rectangular form.\r\n\r\n17. [latex]z=7\\text{cis}\\left(\\frac{\\pi}{6}\\right)[\/latex]\r\n\r\n19. [latex]z=4\\text{cis}\\left(\\frac{7\\pi}{6}\\right)[\/latex]\r\n\r\n21. [latex]z=3\\text{cis}\\left(240^{\\circ}\\right)[\/latex]\r\n\r\nFor the following exercises, find z1<\/sub>z2<\/sub> in polar form.\r\n\r\n23. [latex]z_{1}=2\\sqrt{3}\\text{cis}\\left(116^{\\circ}\\right)\\text{; }\\left(118^{\\circ}\\right)[\/latex]\r\n\r\n25. [latex]z_{1}=3\\text{cis}\\left(120^{\\circ}\\right)\\text{; }z_{2}=\\frac{1}{4}\\text{cis}\\left(60^{\\circ}\\right)[\/latex]\r\n\r\n27. [latex]z_{1}=\\sqrt{5}\\text{cis}\\left(\\frac{5\\pi}{8}\\right)\\text{; }z_{2}=\\sqrt{15}\\text{cis}\\left(\\frac{\\pi}{12}\\right)[\/latex]\r\n\r\nFor the following exercises, find [latex]\\frac{z_{1}}{z_{2}}[\/latex] in polar form.\r\n\r\n29. [latex]z_{1}=21\\text{cis}\\left(135^{\\circ}\\right)\\text{; }z_{2}=3\\text{cis}\\left(65^{\\circ}\\right)[\/latex]\r\n\r\n31. [latex]z_{1}=15\\text{cis}\\left(120^{\\circ}\\right)\\text{; }z_{2}=3\\text{cis}\\left(40^{\\circ}\\right)[\/latex]\r\n\r\n33. [latex]z_{1}=5\\sqrt{2}\\text{cis}\\left(\\pi\\right)\\text{; }z_{2}=\\sqrt{2}\\text{cis}\\left(\\frac{2\\pi}{3}\\right)[\/latex]\r\n\r\nFor the following exercises, find the powers of each complex number in polar form.\r\n\r\n35. Find [latex]z^{3}[\/latex] when [latex]z=5\\text{cis}\\left(45^{\\circ}\\right)[\/latex].\r\n\r\n37. Find [latex]z^{2}[\/latex] when [latex]z=3\\text{cis}\\left(120^{\\circ}\\right)[\/latex].\r\n\r\n39. Find [latex]z^{4}[\/latex] when [latex]z=\\text{cis}\\left(\\frac{3\\pi}{16}\\right)[\/latex].\r\n\r\nFor the following exercises, evaluate each root.\r\n\r\n41. Evaluate the cube root of\u00a0z<\/em> when [latex]z=27\\text{cis}\\left(240^{\\circ}\\right)[\/latex].\r\n\r\n43. Evaluate the cube root of\u00a0z<\/em> when [latex]z=32\\text{cis}\\left(\\frac{2\\pi}{3}\\right)[\/latex].\r\n\r\n45. Evaluate the cube root of\u00a0z<\/em> when [latex]z=8\\text{cis}\\left(\\frac{7\\pi}{4}\\right)[\/latex].\r\n\r\nFor the following exercises, plot the complex number in the complex plane.\r\n\r\n47. [latex]\u22123\u22123i[\/latex]\r\n\r\n49. [latex]\u22121\u22125i[\/latex]\r\n\r\n51. [latex]2i[\/latex]\r\n\r\n53. [latex]6\u22122i[\/latex]\r\n\r\n55. [latex]1\u22124i[\/latex]\r\n\r\nFor the following exercises, find all answers rounded to the nearest hundredth.\r\n\r\n57. Use the rectangular to polar feature on the graphing calculator to change [latex]3\u22122i[\/latex]\r\n\r\n59. Use the polar to rectangular feature on the graphing calculator to change [latex]4\\text{cis}\\left(120^{\\circ}\\right)[\/latex] to rectangular form.\r\n\r\n61. Use the polar to rectangular feature on the graphing calculator to change [latex]5\\text{cis}\\left(210^{\\circ}\\right)[\/latex] to rectangular form.","rendered":"

Polar Coordinates<\/h1>\n

1. How are polar coordinates different from rectangular coordinates?<\/p>\n

3. Explain how polar coordinates are graphed.<\/p>\n

5. Explain why the points [latex]\\left(-3,\\frac{\\pi }{2}\\right)[\/latex] and [latex]\\left(3,-\\frac{\\pi }{2}\\right)[\/latex] are the same.<\/p>\n

For the following exercises, convert the given polar coordinates to Cartesian coordinates with [latex]r>0[\/latex] and [latex]0\\le \\theta \\le 2\\pi[\/latex]. Remember to consider the quadrant in which the given point is located when determining [latex]\\theta[\/latex] for the point.<\/p>\n

7. [latex]\\left(5,\\pi \\right)[\/latex]<\/p>\n

9. [latex]\\left(-3,\\frac{\\pi }{6}\\right)[\/latex]<\/p>\n

For the following exercises, convert the given Cartesian coordinates to polar coordinates with [latex]r>0,0\\le \\theta <2\\pi[\/latex]. Remember to consider the quadrant in which the given point is located.\n\n11. [latex]\\left(4,2\\right)[\/latex]\n\n13. [latex]\\left(3,-5\\right)[\/latex]\n\nFor the following exercises, convert the given Cartesian equation to a polar equation.\n\n17. [latex]y=4[\/latex]\n\n19. [latex]y=2{x}^{4}[\/latex]\n\n21. [latex]{x}^{2}+{y}^{2}=3x[\/latex]\n\n23. [latex]{x}^{2}-{y}^{2}=3y[\/latex]\n\n25. [latex]{x}^{2}=9y[\/latex]\n\n27. [latex]9xy=1[\/latex]\n\nFor the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.\n\n29. [latex]r=4\\cos \\theta[\/latex]\n\n31. [latex]r=\\frac{6}{\\cos \\theta +3\\sin \\theta }[\/latex]\n\n33. [latex]r=3\\csc \\theta[\/latex]\n\n35. [latex]{r}^{2}=4\\sec \\theta \\csc \\theta[\/latex]\n\n37. [latex]{r}^{2}=4[\/latex]\n\n39. [latex]r=\\frac{3}{\\cos \\theta -5\\sin \\theta }[\/latex]\n\nFor the following exercises, find the polar coordinates of the point.\n\n41.\n\"Polar<\/p>\n

43.
\n\"Polar<\/p>\n

For the following exercises, plot the points.<\/p>\n

45. [latex]\\left(-2,\\frac{\\pi }{3}\\right)[\/latex]<\/p>\n

49. [latex]\\left(5,\\frac{\\pi }{2}\\right)[\/latex]<\/p>\n

51. [latex]\\left(3,\\frac{5\\pi }{6}\\right)[\/latex]<\/p>\n

For the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.<\/p>\n

55. [latex]5x-y=6[\/latex]<\/p>\n

57. [latex]{x}^{2}+{\\left(y - 1\\right)}^{2}=1[\/latex]<\/p>\n

59. [latex]x=2[\/latex]<\/p>\n

61. [latex]{x}^{2}+{y}^{2}=3x[\/latex]<\/p>\n

For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.<\/p>\n

63. [latex]r=-4[\/latex]<\/p>\n

65. [latex]\\theta =\\frac{\\pi }{4}[\/latex]<\/p>\n

67. [latex]r=-10\\sin \\theta[\/latex]<\/p>\n

69. Use a graphing calculator to find the rectangular coordinates of [latex]\\left(2,-\\frac{\\pi }{5}\\right)[\/latex]. Round to the nearest thousandth.<\/p>\n

71. Use a graphing calculator to find the polar coordinates of [latex]\\left(-7,8\\right)[\/latex] in degrees. Round to the nearest thousandth.<\/p>\n

73. Use a graphing calculator to find the polar coordinates of [latex]\\left(-2,0\\right)[\/latex] in radians. Round to the nearest hundredth.<\/p>\n

Polar Coordinates: Graphs<\/h1>\n

1. Describe the three types of symmetry in polar graphs, and compare them to the symmetry of the Cartesian plane.<\/p>\n

3. What are the steps to follow when graphing polar equations?<\/p>\n

5. What part of the equation determines the shape of the graph of a polar equation?<\/p>\n

For the following exercises, test the equation for symmetry.<\/p>\n

7. [latex]r=3 - 3\\cos \\theta[\/latex]<\/p>\n

9. [latex]r=3\\sin 2\\theta[\/latex]<\/p>\n

11. [latex]r=2\\theta[\/latex]<\/p>\n

13. [latex]r=\\frac{2}{\\theta }[\/latex]<\/p>\n

15. [latex]r=\\sqrt{5\\sin 2\\theta }[\/latex]<\/p>\n

For the following exercises, graph the polar equation. Identify the name of the shape.<\/p>\n

17. [latex]r=4\\sin \\theta[\/latex]<\/p>\n

19. [latex]r=2 - 2\\cos \\theta[\/latex]<\/p>\n

21. [latex]r=3+3\\sin \\theta[\/latex]<\/p>\n

23. [latex]r=7+4\\sin \\theta[\/latex]<\/p>\n

25. [latex]r=5+4\\cos \\theta[\/latex]<\/p>\n

27. [latex]r=1+3\\sin \\theta[\/latex]<\/p>\n

29. [latex]r=5+7\\sin \\theta[\/latex]<\/p>\n

31. [latex]r=5+6\\cos \\theta[\/latex]<\/p>\n

33. [latex]{r}^{2}=10\\cos \\left(2\\theta \\right)[\/latex]<\/p>\n

35. [latex]{r}^{2}=10\\sin \\left(2\\theta \\right)[\/latex]<\/p>\n

37. [latex]r=3\\text{cos}\\left(2\\theta \\right)[\/latex]<\/p>\n

39. [latex]r=4\\text{sin}\\left(4\\theta \\right)[\/latex]<\/p>\n

41. [latex]r=-\\theta[\/latex]<\/p>\n

43. [latex]r=-3\\theta[\/latex]<\/p>\n

For the following exercises, use a graphing calculator to sketch the graph of the polar equation.<\/p>\n

45.\u00a0[latex]r=\\frac{1}{\\sqrt{\\theta }}[\/latex]<\/p>\n

47. [latex]r=2\\sqrt{1-{\\sin }^{2}\\theta }[\/latex] , a hippopede<\/p>\n

49. [latex]r=2-\\sin \\left(2\\theta \\right)[\/latex]<\/p>\n

51. [latex]r=\\theta +1[\/latex]<\/p>\n

53. [latex]r=\\theta \\cos \\theta[\/latex]<\/p>\n

61. On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs.
\n[latex]\\begin{array}{l}\\begin{array}{l}\\\\ {r}_{1}=3\\sin \\left(3\\theta \\right)\\end{array}\\hfill \\\\ {r}_{2}=2\\sin \\left(3\\theta \\right)\\hfill \\\\ {r}_{3}=\\sin \\left(3\\theta \\right)\\hfill \\end{array}[\/latex]<\/p>\n

63. On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs.
\n[latex]\\begin{array}{l}\\begin{array}{l}\\\\ {r}_{1}=3\\theta \\end{array}\\hfill \\\\ {r}_{2}=2\\theta \\hfill \\\\ {r}_{3}=\\theta \\hfill \\end{array}[\/latex]<\/p>\n

For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.<\/p>\n

65. [latex]{r}_{1}=6 - 4\\cos \\theta ,{r}_{2}=4[\/latex]<\/p>\n

67. [latex]{r}_{1}=1+\\cos \\theta ,{r}_{2}=3\\cos \\theta[\/latex]<\/p>\n

69. [latex]{r}_{1}={\\sin }^{2}\\left(2\\theta \\right),{r}_{2}=1-\\cos \\left(4\\theta \\right)[\/latex]<\/p>\n

71. [latex]{r}_{1}{}^{2}=\\sin \\theta ,{r}_{2}{}^{2}=\\cos \\theta[\/latex]<\/p>\n

Polar Form of Complex Numbers<\/h1>\n

1. A complex number is [latex]a+bi[\/latex]. Explain each part.<\/p>\n

3. How is a complex number converted to polar form?<\/p>\n

For the following exercises, find the absolute value of the given complex number.<\/p>\n

7. [latex]\u22127+i[\/latex]<\/p>\n

9. [latex]\\sqrt{2}\u22126i[\/latex]<\/p>\n

11. [latex]2.2\u22123.1i[\/latex]<\/p>\n

For the following exercises, write the complex number in polar form.<\/p>\n

13. [latex]8\u22124i[\/latex]<\/p>\n

15. [latex]\\sqrt{3}+i[\/latex]<\/p>\n

For the following exercises, convert the complex number from polar to rectangular form.<\/p>\n

17. [latex]z=7\\text{cis}\\left(\\frac{\\pi}{6}\\right)[\/latex]<\/p>\n

19. [latex]z=4\\text{cis}\\left(\\frac{7\\pi}{6}\\right)[\/latex]<\/p>\n

21. [latex]z=3\\text{cis}\\left(240^{\\circ}\\right)[\/latex]<\/p>\n

For the following exercises, find z1<\/sub>z2<\/sub> in polar form.<\/p>\n

23. [latex]z_{1}=2\\sqrt{3}\\text{cis}\\left(116^{\\circ}\\right)\\text{; }\\left(118^{\\circ}\\right)[\/latex]<\/p>\n

25. [latex]z_{1}=3\\text{cis}\\left(120^{\\circ}\\right)\\text{; }z_{2}=\\frac{1}{4}\\text{cis}\\left(60^{\\circ}\\right)[\/latex]<\/p>\n

27. [latex]z_{1}=\\sqrt{5}\\text{cis}\\left(\\frac{5\\pi}{8}\\right)\\text{; }z_{2}=\\sqrt{15}\\text{cis}\\left(\\frac{\\pi}{12}\\right)[\/latex]<\/p>\n

For the following exercises, find [latex]\\frac{z_{1}}{z_{2}}[\/latex] in polar form.<\/p>\n

29. [latex]z_{1}=21\\text{cis}\\left(135^{\\circ}\\right)\\text{; }z_{2}=3\\text{cis}\\left(65^{\\circ}\\right)[\/latex]<\/p>\n

31. [latex]z_{1}=15\\text{cis}\\left(120^{\\circ}\\right)\\text{; }z_{2}=3\\text{cis}\\left(40^{\\circ}\\right)[\/latex]<\/p>\n

33. [latex]z_{1}=5\\sqrt{2}\\text{cis}\\left(\\pi\\right)\\text{; }z_{2}=\\sqrt{2}\\text{cis}\\left(\\frac{2\\pi}{3}\\right)[\/latex]<\/p>\n

For the following exercises, find the powers of each complex number in polar form.<\/p>\n

35. Find [latex]z^{3}[\/latex] when [latex]z=5\\text{cis}\\left(45^{\\circ}\\right)[\/latex].<\/p>\n

37. Find [latex]z^{2}[\/latex] when [latex]z=3\\text{cis}\\left(120^{\\circ}\\right)[\/latex].<\/p>\n

39. Find [latex]z^{4}[\/latex] when [latex]z=\\text{cis}\\left(\\frac{3\\pi}{16}\\right)[\/latex].<\/p>\n

For the following exercises, evaluate each root.<\/p>\n

41. Evaluate the cube root of\u00a0z<\/em> when [latex]z=27\\text{cis}\\left(240^{\\circ}\\right)[\/latex].<\/p>\n

43. Evaluate the cube root of\u00a0z<\/em> when [latex]z=32\\text{cis}\\left(\\frac{2\\pi}{3}\\right)[\/latex].<\/p>\n

45. Evaluate the cube root of\u00a0z<\/em> when [latex]z=8\\text{cis}\\left(\\frac{7\\pi}{4}\\right)[\/latex].<\/p>\n

For the following exercises, plot the complex number in the complex plane.<\/p>\n

47. [latex]\u22123\u22123i[\/latex]<\/p>\n

49. [latex]\u22121\u22125i[\/latex]<\/p>\n

51. [latex]2i[\/latex]<\/p>\n

53. [latex]6\u22122i[\/latex]<\/p>\n

55. [latex]1\u22124i[\/latex]<\/p>\n

For the following exercises, find all answers rounded to the nearest hundredth.<\/p>\n

57. Use the rectangular to polar feature on the graphing calculator to change [latex]3\u22122i[\/latex]<\/p>\n

59. Use the polar to rectangular feature on the graphing calculator to change [latex]4\\text{cis}\\left(120^{\\circ}\\right)[\/latex] to rectangular form.<\/p>\n

61. Use the polar to rectangular feature on the graphing calculator to change [latex]5\\text{cis}\\left(210^{\\circ}\\right)[\/latex] to rectangular form.<\/p>\n","protected":false},"author":67,"menu_order":26,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":247,"module-header":"practice","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2361"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2361\/revisions"}],"predecessor-version":[{"id":5716,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2361\/revisions\/5716"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/247"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2361\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2361"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2361"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2361"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2361"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}