{"id":346,"date":"2026-02-16T20:55:49","date_gmt":"2026-02-16T20:55:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/?post_type=chapter&p=346"},"modified":"2026-02-18T16:54:03","modified_gmt":"2026-02-18T16:54:03","slug":"polar-functions-get-stronger-answer-key","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/polar-functions-get-stronger-answer-key\/","title":{"raw":"Polar Functions: Get Stronger Answer Key","rendered":"Polar Functions: Get Stronger Answer Key"},"content":{"raw":"

Polar Coordinates<\/h1>\r\n1.\u00a0For polar coordinates, the point in the plane depends on the angle from the positive x-<\/em>axis and distance from the origin, while in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there are infinite representations.\r\n\r\n3.\u00a0Determine [latex]\\theta [\/latex] for the point, then move [latex]r[\/latex] units from the pole to plot the point. If [latex]r[\/latex] is negative, move [latex]r[\/latex] units from the pole in the opposite direction but along the same angle. The point is a distance of [latex]r[\/latex] away from the origin at an angle of [latex]\\theta [\/latex] from the polar axis.\r\n\r\n5.\u00a0The point [latex]\\left(-3,\\frac{\\pi }{2}\\right)[\/latex] has a positive angle but a negative radius and is plotted by moving to an angle of [latex]\\frac{\\pi }{2}[\/latex] and then moving 3 units in the negative direction. This places the point 3 units down the negative y<\/em>-axis. The point [latex]\\left(3,-\\frac{\\pi }{2}\\right)[\/latex] has a negative angle and a positive radius and is plotted by first moving to an angle of [latex]-\\frac{\\pi }{2}[\/latex] and then moving 3 units down, which is the positive direction for a negative angle. The point is also 3 units down the negative y-axis.\r\n\r\n7.\u00a0[latex]\\left(-5,0\\right)[\/latex]\r\n\r\n9.\u00a0[latex]\\left(-\\frac{3\\sqrt{3}}{2},-\\frac{3}{2}\\right)[\/latex]\r\n\r\n11.\u00a0[latex]\\left(2\\sqrt{5}, 0.464\\right)[\/latex]\r\n\r\n13.\u00a0[latex]\\left(\\sqrt{34},5.253\\right)[\/latex]\r\n\r\n17.\u00a0[latex]r=4\\csc \\theta [\/latex]\r\n\r\n19.\u00a0[latex]r=\\sqrt[3]{\\frac{sin\\theta }{2co{s}^{4}\\theta }}[\/latex]\r\n\r\n21.\u00a0[latex]r=3\\cos \\theta [\/latex]\r\n\r\n23.\u00a0[latex]r=\\frac{3\\sin \\theta }{\\cos \\left(2\\theta \\right)}[\/latex]\r\n\r\n25.\u00a0[latex]r=\\frac{9\\sin \\theta }{{\\cos }^{2}\\theta }[\/latex]\r\n\r\n27.\u00a0[latex]r=\\sqrt{\\frac{1}{9\\cos \\theta \\sin \\theta }}[\/latex]\r\n\r\n29.\u00a0[latex]{x}^{2}+{y}^{2}=4x[\/latex] or [latex]\\frac{{\\left(x - 2\\right)}^{2}}{4}+\\frac{{y}^{2}}{4}=1[\/latex]; circle\r\n\r\n31.\u00a0[latex]3y+x=6[\/latex]; line\r\n\r\n33.\u00a0[latex]y=3[\/latex];\u00a0line\r\n\r\n35.\u00a0[latex]xy=4[\/latex]; hyperbola\r\n\r\n37.\u00a0[latex]{x}^{2}+{y}^{2}=4[\/latex]; circle\r\n\r\n39.\u00a0[latex]x - 5y=3[\/latex]; line\r\n\r\n41.\u00a0[latex]\\left(3,\\frac{3\\pi }{4}\\right)[\/latex]\r\n\r\n43.\u00a0[latex]\\left(5,\\pi \\right)[\/latex]\r\n\r\n45.\r\n\"Polar\r\n\r\n49.\r\n\"Polar\r\n\r\n51.\r\n\"Polar\r\n\r\n55.\u00a0[latex]r=\\frac{6}{5\\cos \\theta -\\sin \\theta }[\/latex]\r\n\"Plot\r\n\r\n57.\u00a0[latex]r=2\\sin \\theta [\/latex]\r\n\"Plot\r\n\r\n59.\u00a0[latex]r=\\frac{2}{\\cos \\theta }[\/latex]\r\n\"Plot\r\n\r\n61.\u00a0[latex]r=3\\cos \\theta [\/latex]\r\n\"Plot\r\n\r\n63.\u00a0[latex]{x}^{2}+{y}^{2}=16[\/latex]\r\n\"Plot\r\n\r\n65.\u00a0[latex]y=x[\/latex]\r\n\"Plot\r\n\r\n67.\u00a0[latex]{x}^{2}+{\\left(y+5\\right)}^{2}=25[\/latex]\r\n\"Plot\r\n\r\n69.\u00a0[latex]\\left(1.618,-1.176\\right)[\/latex]\r\n\r\n71.\u00a0[latex]\\left(10.630,131.186^\\circ \\right)[\/latex]\r\n\r\n73.\u00a0[latex]\\left(2,3.14\\right)or\\left(2,\\pi \\right)[\/latex]\r\n

Polar Coordinates: Graphs<\/h1>\r\n1.\u00a0Symmetry with respect to the polar axis is similar to symmetry about the [latex]x[\/latex] -axis, symmetry with respect to the pole is similar to symmetry about the origin, and symmetric with respect to the line [latex]\\theta =\\frac{\\pi }{2}[\/latex] is similar to symmetry about the [latex]y[\/latex] -axis.\r\n\r\n3.\u00a0Test for symmetry; find zeros, intercepts, and maxima; make a table of values. Decide the general type of graph, cardioid, lima\u00e7on, lemniscate, etc., then plot points at [latex]\\theta =0,\\frac{\\pi }{2},\\pi \\text{and }\\frac{3\\pi }{2}[\/latex], and sketch the graph.\r\n\r\n5.\u00a0The shape of the polar graph is determined by whether or not it includes a sine, a cosine, and constants in the equation.\r\n\r\n7.\u00a0symmetric with respect to the polar axis\r\n\r\n9.\u00a0symmetric with respect to the polar axis, symmetric with respect to the line [latex]\\theta =\\frac{\\pi }{2}[\/latex], symmetric with respect to the pole\r\n\r\n11.\u00a0no symmetry\r\n\r\n13.\u00a0no symmetry\r\n\r\n15.\u00a0symmetric with respect to the pole\r\n\r\n17.\u00a0circle\r\n\"Graph\r\n\r\n19.\u00a0cardioid\r\n\"Graph\r\n\r\n21.\u00a0cardioid\r\n\"Graph\r\n\r\n23.\u00a0one-loop\/dimpled lima\u00e7on\r\n\"Graph\r\n\r\n25.\u00a0one-loop\/dimpled lima\u00e7on\r\n\"Graph\r\n\r\n27.\u00a0inner loop\/two-loop lima\u00e7on\r\n\"Graph\r\n\r\n29.\u00a0inner loop\/two-loop lima\u00e7on\r\n\"Graph\r\n\r\n31.\u00a0inner loop\/two-loop lima\u00e7on\r\n\"Graph\r\n\r\n33.\u00a0lemniscate\r\n\"Graph\r\n\r\n35.\u00a0lemniscate\r\n\"Graph\r\n\r\n37.\u00a0rose curve\r\n\"Graph\r\n\r\n39.\u00a0rose curve\r\n\"Graph\r\n\r\n41.\u00a0Archimedes\u2019 spiral\r\n\"Graph\r\n\r\n43.\u00a0Archimedes\u2019 spiral\r\n\"Graph\r\n\r\n45.\r\n\"Graph\r\n\r\n47.\r\n\"Graph\r\n\r\n49.\r\n\"Graph\r\n\r\n51.\r\n\"Graph\r\n\r\n53.\r\n\"Graph\r\n\r\n61.\u00a0The graphs are three-petal, rose curves. The larger the coefficient, the greater the curve\u2019s distance from the pole.\r\n\r\n63.\u00a0The graphs are spirals. The smaller the coefficient, the tighter the spiral.\r\n\r\n65.\u00a0[latex]\\left(4,\\frac{\\pi }{3}\\right),\\left(4,\\frac{5\\pi }{3}\\right)[\/latex]\r\n\r\n67.\u00a0[latex]\\left(\\frac{3}{2},\\frac{\\pi }{3}\\right),\\left(\\frac{3}{2},\\frac{5\\pi }{3}\\right)[\/latex]\r\n\r\n69.\u00a0[latex]\\left(0,\\frac{\\pi }{2}\\right),\\left(0,\\pi \\right),\\left(0,\\frac{3\\pi }{2}\\right),\\left(0,2\\pi \\right)[\/latex]\r\n\r\n71.\u00a0[latex]\\left(\\frac{\\sqrt[4]{8}}{2},\\frac{\\pi }{4}\\right),\\left(\\frac{\\sqrt[4]{8}}{2},\\frac{5\\pi }{4}\\right)[\/latex]\r\nand at [latex]\\theta =\\frac{3\\pi }{4},\\frac{7\\pi }{4}[\/latex]\u00a0since [latex]r[\/latex] is squared\r\n

Polar Form of Complex Numbers<\/h1>\r\n1. a<\/em> is the real part,\u00a0b<\/em> is the imaginary part, and [latex]i=\\sqrt{\u22121}[\/latex]\r\n\r\n3.\u00a0Polar 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]\r\n\r\n7. [latex]5\\sqrt{2}[\/latex]\r\n\r\n9. [latex]\\sqrt{38}[\/latex]\r\n\r\n11. [latex]\\sqrt{14.45}[\/latex]\r\n\r\n13. [latex]4\\sqrt{5}\\text{cis}\\left(333.4^{\\circ}\\right)[\/latex]\r\n\r\n15. [latex]2\\text{cis}\\left(\\frac{\\pi}{6}\\right)[\/latex]\r\n\r\n17. [latex]\\frac{7\\sqrt{3}}{2}+i\\frac{7}{2}[\/latex]\r\n\r\n19.\u00a0[latex]\u22122\\sqrt{3}\u22122i[\/latex]\r\n\r\n21. [latex]\u22121.5\u2212i\\frac{3\\sqrt{3}}{2}[\/latex]\r\n\r\n23. [latex]4\\sqrt{3}\\text{cis}\\left(198^{\\circ}\\right)[\/latex]\r\n\r\n25. [latex]\\frac{3}{4}\\text{cis}\\left(180^{\\circ}\\right)[\/latex]\r\n\r\n27. [latex]5\\sqrt{3}\\text{cis}\\left(\\frac{17\\pi}{24}\\right)[\/latex]\r\n\r\n29. [latex]7\\text{cis}\\left(70^{\\circ}\\right)[\/latex]\r\n\r\n31. [latex]5\\text{cis}\\left(80^{\\circ}\\right)[\/latex]\r\n\r\n33. [latex]5\\text{cis}\\left(\\frac{\\pi}{3}\\right)[\/latex]\r\n\r\n35. [latex]125\\text{cis}\\left(135^{\\circ}\\right)[\/latex]\r\n\r\n37. [latex]9\\text{cis}\\left(240^{\\circ}\\right)[\/latex]\r\n\r\n39. [latex]\\text{cis}\\left(\\frac{3\\pi}{4}\\right)[\/latex]\r\n\r\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]\r\n\r\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]\r\n\r\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]\r\n\r\n47.\r\n\"Plot\r\n\r\n49.\r\n\"Plot\r\n\r\n51.\r\n\"Plot\r\n\r\n53.\r\n\"Plot\r\n\r\n55.\r\n\"Plot\r\n\r\n57. [latex]3.61e^{\u22120.59i}[\/latex]\r\n\r\n59. [latex]\u22122+3.46i[\/latex]\r\n\r\n61. [latex]\u22124.33\u22122.50i[\/latex]","rendered":"

Polar Coordinates<\/h1>\n

1.\u00a0For polar coordinates, the point in the plane depends on the angle from the positive x-<\/em>axis and distance from the origin, while in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there are infinite representations.<\/p>\n

3.\u00a0Determine [latex]\\theta[\/latex] for the point, then move [latex]r[\/latex] units from the pole to plot the point. If [latex]r[\/latex] is negative, move [latex]r[\/latex] units from the pole in the opposite direction but along the same angle. The point is a distance of [latex]r[\/latex] away from the origin at an angle of [latex]\\theta[\/latex] from the polar axis.<\/p>\n

5.\u00a0The point [latex]\\left(-3,\\frac{\\pi }{2}\\right)[\/latex] has a positive angle but a negative radius and is plotted by moving to an angle of [latex]\\frac{\\pi }{2}[\/latex] and then moving 3 units in the negative direction. This places the point 3 units down the negative y<\/em>-axis. The point [latex]\\left(3,-\\frac{\\pi }{2}\\right)[\/latex] has a negative angle and a positive radius and is plotted by first moving to an angle of [latex]-\\frac{\\pi }{2}[\/latex] and then moving 3 units down, which is the positive direction for a negative angle. The point is also 3 units down the negative y-axis.<\/p>\n

7.\u00a0[latex]\\left(-5,0\\right)[\/latex]<\/p>\n

9.\u00a0[latex]\\left(-\\frac{3\\sqrt{3}}{2},-\\frac{3}{2}\\right)[\/latex]<\/p>\n

11.\u00a0[latex]\\left(2\\sqrt{5}, 0.464\\right)[\/latex]<\/p>\n

13.\u00a0[latex]\\left(\\sqrt{34},5.253\\right)[\/latex]<\/p>\n

17.\u00a0[latex]r=4\\csc \\theta[\/latex]<\/p>\n

19.\u00a0[latex]r=\\sqrt[3]{\\frac{sin\\theta }{2co{s}^{4}\\theta }}[\/latex]<\/p>\n

21.\u00a0[latex]r=3\\cos \\theta[\/latex]<\/p>\n

23.\u00a0[latex]r=\\frac{3\\sin \\theta }{\\cos \\left(2\\theta \\right)}[\/latex]<\/p>\n

25.\u00a0[latex]r=\\frac{9\\sin \\theta }{{\\cos }^{2}\\theta }[\/latex]<\/p>\n

27.\u00a0[latex]r=\\sqrt{\\frac{1}{9\\cos \\theta \\sin \\theta }}[\/latex]<\/p>\n

29.\u00a0[latex]{x}^{2}+{y}^{2}=4x[\/latex] or [latex]\\frac{{\\left(x - 2\\right)}^{2}}{4}+\\frac{{y}^{2}}{4}=1[\/latex]; circle<\/p>\n

31.\u00a0[latex]3y+x=6[\/latex]; line<\/p>\n

33.\u00a0[latex]y=3[\/latex];\u00a0line<\/p>\n

35.\u00a0[latex]xy=4[\/latex]; hyperbola<\/p>\n

37.\u00a0[latex]{x}^{2}+{y}^{2}=4[\/latex]; circle<\/p>\n

39.\u00a0[latex]x - 5y=3[\/latex]; line<\/p>\n

41.\u00a0[latex]\\left(3,\\frac{3\\pi }{4}\\right)[\/latex]<\/p>\n

43.\u00a0[latex]\\left(5,\\pi \\right)[\/latex]<\/p>\n

45.
\n\"Polar<\/p>\n

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

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

55.\u00a0[latex]r=\\frac{6}{5\\cos \\theta -\\sin \\theta }[\/latex]
\n\"Plot<\/p>\n

57.\u00a0[latex]r=2\\sin \\theta[\/latex]
\n\"Plot<\/p>\n

59.\u00a0[latex]r=\\frac{2}{\\cos \\theta }[\/latex]
\n\"Plot<\/p>\n

61.\u00a0[latex]r=3\\cos \\theta[\/latex]
\n\"Plot<\/p>\n

63.\u00a0[latex]{x}^{2}+{y}^{2}=16[\/latex]
\n\"Plot<\/p>\n

65.\u00a0[latex]y=x[\/latex]
\n\"Plot<\/p>\n

67.\u00a0[latex]{x}^{2}+{\\left(y+5\\right)}^{2}=25[\/latex]
\n\"Plot<\/p>\n

69.\u00a0[latex]\\left(1.618,-1.176\\right)[\/latex]<\/p>\n

71.\u00a0[latex]\\left(10.630,131.186^\\circ \\right)[\/latex]<\/p>\n

73.\u00a0[latex]\\left(2,3.14\\right)or\\left(2,\\pi \\right)[\/latex]<\/p>\n

Polar Coordinates: Graphs<\/h1>\n

1.\u00a0Symmetry with respect to the polar axis is similar to symmetry about the [latex]x[\/latex] -axis, symmetry with respect to the pole is similar to symmetry about the origin, and symmetric with respect to the line [latex]\\theta =\\frac{\\pi }{2}[\/latex] is similar to symmetry about the [latex]y[\/latex] -axis.<\/p>\n

3.\u00a0Test for symmetry; find zeros, intercepts, and maxima; make a table of values. Decide the general type of graph, cardioid, lima\u00e7on, lemniscate, etc., then plot points at [latex]\\theta =0,\\frac{\\pi }{2},\\pi \\text{and }\\frac{3\\pi }{2}[\/latex], and sketch the graph.<\/p>\n

5.\u00a0The shape of the polar graph is determined by whether or not it includes a sine, a cosine, and constants in the equation.<\/p>\n

7.\u00a0symmetric with respect to the polar axis<\/p>\n

9.\u00a0symmetric with respect to the polar axis, symmetric with respect to the line [latex]\\theta =\\frac{\\pi }{2}[\/latex], symmetric with respect to the pole<\/p>\n

11.\u00a0no symmetry<\/p>\n

13.\u00a0no symmetry<\/p>\n

15.\u00a0symmetric with respect to the pole<\/p>\n

17.\u00a0circle
\n\"Graph<\/p>\n

19.\u00a0cardioid
\n\"Graph<\/p>\n

21.\u00a0cardioid
\n\"Graph<\/p>\n

23.\u00a0one-loop\/dimpled lima\u00e7on
\n\"Graph<\/p>\n

25.\u00a0one-loop\/dimpled lima\u00e7on
\n\"Graph<\/p>\n

27.\u00a0inner loop\/two-loop lima\u00e7on
\n\"Graph<\/p>\n

29.\u00a0inner loop\/two-loop lima\u00e7on
\n\"Graph<\/p>\n

31.\u00a0inner loop\/two-loop lima\u00e7on
\n\"Graph<\/p>\n

33.\u00a0lemniscate
\n\"Graph<\/p>\n

35.\u00a0lemniscate
\n\"Graph<\/p>\n

37.\u00a0rose curve
\n\"Graph<\/p>\n

39.\u00a0rose curve
\n\"Graph<\/p>\n

41.\u00a0Archimedes\u2019 spiral
\n\"Graph<\/p>\n

43.\u00a0Archimedes\u2019 spiral
\n\"Graph<\/p>\n

45.
\n\"Graph<\/p>\n

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

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

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

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

61.\u00a0The graphs are three-petal, rose curves. The larger the coefficient, the greater the curve\u2019s distance from the pole.<\/p>\n

63.\u00a0The graphs are spirals. The smaller the coefficient, the tighter the spiral.<\/p>\n

65.\u00a0[latex]\\left(4,\\frac{\\pi }{3}\\right),\\left(4,\\frac{5\\pi }{3}\\right)[\/latex]<\/p>\n

67.\u00a0[latex]\\left(\\frac{3}{2},\\frac{\\pi }{3}\\right),\\left(\\frac{3}{2},\\frac{5\\pi }{3}\\right)[\/latex]<\/p>\n

69.\u00a0[latex]\\left(0,\\frac{\\pi }{2}\\right),\\left(0,\\pi \\right),\\left(0,\\frac{3\\pi }{2}\\right),\\left(0,2\\pi \\right)[\/latex]<\/p>\n

71.\u00a0[latex]\\left(\\frac{\\sqrt[4]{8}}{2},\\frac{\\pi }{4}\\right),\\left(\\frac{\\sqrt[4]{8}}{2},\\frac{5\\pi }{4}\\right)[\/latex]
\nand at [latex]\\theta =\\frac{3\\pi }{4},\\frac{7\\pi }{4}[\/latex]\u00a0since [latex]r[\/latex] is squared<\/p>\n

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

1. a<\/em> is the real part,\u00a0b<\/em> is the imaginary part, and [latex]i=\\sqrt{\u22121}[\/latex]<\/p>\n

3.\u00a0Polar 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

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.\u00a0[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","protected":false},"author":13,"menu_order":17,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":224,"module-header":"- Select Header -","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\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/346"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":3,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/346\/revisions"}],"predecessor-version":[{"id":366,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/346\/revisions\/366"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/224"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/346\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=346"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=346"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=346"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=346"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}