{"id":17332,"date":"2020-04-20T03:54:10","date_gmt":"2020-04-20T03:54:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/?post_type=chapter&p=17332"},"modified":"2020-05-21T05:28:44","modified_gmt":"2020-05-21T05:28:44","slug":"chapter-9-solutions-to-odd-numbered-problems-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/chapter\/chapter-9-solutions-to-odd-numbered-problems-2\/","title":{"raw":"Chapter 6 Solutions to Odd-Numbered Problems","rendered":"Chapter 6 Solutions to Odd-Numbered Problems"},"content":{"raw":"

Section 6.1 Solutions<\/h2>\r\n1.\u00a0The altitude extends from any vertex to the opposite side or to the line containing the opposite side at a 90\u00b0 angle.\r\n\r\n3.\u00a0When the known values are the side opposite the missing angle and another side and its opposite angle.\r\n\r\n5.\u00a0A triangle with two given sides and a non-included angle.\r\n\r\n7.\u00a0[latex] \\beta =72^\\circ ,a\\approx 12.0,b\\approx 19.9[\/latex]\r\n\r\n9.\u00a0[latex] \\gamma =20^\\circ ,b\\approx 4.5,c\\approx 1.6[\/latex]\r\n\r\n11.\u00a0[latex]b\\approx 3.78[\/latex]\r\n\r\n13.\u00a0[latex]c\\approx 13.70[\/latex]\r\n\r\n15.\u00a0one triangle, [latex]\\alpha \\approx 50.3^\\circ ,\\beta \\approx 16.7^\\circ ,a\\approx 26.7[\/latex]\r\n\r\n17.\u00a0two triangles, [latex] \\gamma \\approx 54.3^\\circ ,\\beta \\approx 90.7^\\circ ,b\\approx 20.9[\/latex] or [latex] {\\gamma }^{\\prime }\\approx 125.7^\\circ ,{\\beta }^{\\prime }\\approx 19.3^\\circ ,{b}^{\\prime }\\approx 6.9[\/latex]\r\n\r\n19.\u00a0two triangles, [latex] \\beta \\approx 75.7^\\circ , \\gamma \\approx 61.3^\\circ ,b\\approx 9.9[\/latex] or [latex] {\\beta }^{\\prime }\\approx 18.3^\\circ ,{\\gamma }^{\\prime }\\approx 118.7^\\circ ,{b}^{\\prime }\\approx 3.2[\/latex]\r\n\r\n21.\u00a0two triangles, [latex]\\alpha \\approx 143.2^\\circ ,\\beta \\approx 26.8^\\circ ,a\\approx 17.3[\/latex] or [latex]{\\alpha }^{\\prime }\\approx 16.8^\\circ ,{\\beta }^{\\prime }\\approx 153.2^\\circ ,{a}^{\\prime }\\approx 8.3[\/latex]\r\n\r\n23.\u00a0no triangle possible\r\n\r\n25.\u00a0[latex]A\\approx 47.8^\\circ [\/latex] or [latex]{A}^{\\prime }\\approx 132.2^\\circ [\/latex]\r\n\r\n27.\u00a0[latex]8.6[\/latex]\r\n\r\n29.\u00a0[latex]370.9[\/latex]\r\n\r\n31.\u00a0[latex]12.3[\/latex]\r\n\r\n33.\u00a0[latex]12.2 [\/latex]\r\n\r\n35.\u00a0[latex]16.0 [\/latex]\r\n\r\n37.\u00a0[latex]29.7^\\circ [\/latex]\r\n\r\n39.\u00a0[latex]x=76.9^\\circ \\text{or }x=103.1^\\circ [\/latex]\r\n\r\n41.\u00a0[latex]110.6^\\circ [\/latex]\r\n\r\n43.\u00a0[latex]A\\approx 39.4,\\text{ }C\\approx 47.6,\\text{ }BC\\approx 20.7 [\/latex]\r\n\r\n45.\u00a0[latex]57.1[\/latex]\r\n\r\n47.\u00a0[latex]42.0 [\/latex]\r\n\r\n49.\u00a0[latex]430.2 [\/latex]\r\n\r\n51.\u00a0[latex]10.1[\/latex]\r\n\r\n53.\u00a0[latex]AD\\approx \\text{ }13.8[\/latex]\r\n\r\n55.\u00a0[latex]AB\\approx 2.8 [\/latex]\r\n\r\n57.\u00a0[latex]L\\approx 49.7,\\text{ }N\\approx 56.3,\\text{ }LN\\approx 5.8[\/latex]\r\n\r\n59.\u00a051.4 feet\r\n\r\n61.\u00a0The distance from the satellite to station [latex]A[\/latex] is approximately 1716 miles. The satellite is approximately 1706 miles above the ground.\r\n\r\n63.\u00a02.6\u00a0ft\r\n\r\n65.\u00a05.6\u00a0km\r\n\r\n67.\u00a0371\u00a0ft\r\n\r\n69.\u00a05936\u00a0ft\r\n\r\n71.\u00a024.1\u00a0ft\r\n\r\n73.\u00a019,056\u00a0ft2\r\n\r\n75.\u00a0445,624\u00a0square\u00a0miles\r\n\r\n77.\u00a08.65\u00a0ft2\r\n

Section 6.2 Solutions<\/h2>\r\n1.\u00a0two sides and the angle opposite the missing side\r\n\r\n3.\u00a0[latex]s[\/latex] is the semi-perimeter, which is half the perimeter of the triangle.\r\n\r\n5.\u00a0The Law of Cosines must be used for any oblique (non-right) triangle.\r\n\r\n7.\u00a011.3\r\n\r\n9.\u00a034.7\r\n\r\n11.\u00a026.7\r\n\r\n13.\u00a0257.4\r\n\r\n15.\u00a0not possible\r\n\r\n17.\u00a095.5\u00b0\r\n\r\n19.\u00a026.9\u00b0\r\n\r\n21.\u00a0[latex]B\\approx 45.9^\\circ ,C\\approx 99.1^\\circ ,a\\approx 6.4[\/latex]\r\n\r\n23.\u00a0[latex]A\\approx 20.6^\\circ ,B\\approx 38.4^\\circ ,c\\approx 51.1[\/latex]\r\n\r\n25.\u00a0[latex]A\\approx 37.8^\\circ ,B\\approx 43.8,C\\approx 98.4^\\circ [\/latex]\r\n\r\n27.\u00a0177.56 in2\r\n\r\n29.\u00a00.04 m2\r\n\r\n31.\u00a00.91 yd2\r\n\r\n33.\u00a03.0\r\n\r\n35.\u00a029.1\r\n\r\n37.\u00a00.5\r\n\r\n39.\u00a070.7\u00b0\r\n\r\n41.\u00a077.4\u00b0\r\n\r\n43.\u00a025.0\r\n\r\n45.\u00a09.3\r\n\r\n47.\u00a043.52\r\n\r\n49.\u00a01.41\r\n\r\n51.\u00a00.14\r\n\r\n53.\u00a018.3\r\n\r\n55.\u00a048.98\r\n\r\n57.\r\n\"A\r\n\r\n59.\u00a07.62\r\n\r\n61.\u00a085.1\r\n\r\n63.\u00a024.0 km\r\n\r\n65.\u00a099.9 ft\r\n\r\n67.\u00a037.3 miles\r\n\r\n69.\u00a02371 miles\r\n\r\n71.\r\n\"Angle\r\n\r\n73.\u00a0599.8 miles\r\n\r\n75.\u00a065.4 cm2\r\n\r\n77.\u00a0468 ft2\r\n

Section 6.3 Solutions<\/h2>\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.\r\na) [latex]\\left(5,-\\frac{4\\pi}{3}\\right)[\/latex]\r\nb) [latex]\\left(-5,\\frac{5\\pi}{3}\\right)[\/latex]\r\nc) [latex]\\left(5,\\frac{8\\pi}{3}\\right)[\/latex]\r\n\r\n9.\r\na) [latex]\\left(3,-\\frac{5\\pi}{4}\\right)[\/latex]\r\nb) [latex]\\left(-3,\\frac{7\\pi}{4}\\right)[\/latex]\r\nc) [latex]\\left(3,\\frac{11\\pi}{4}\\right)[\/latex]\r\n\r\n11.\r\na) [latex]\\left(4,-135^\\circ\\right)[\/latex]\r\nb) [latex]\\left(-4,45^\\circ\\right)[\/latex]\r\nc) [latex]\\left(4,585^\\circ\\right)[\/latex]\r\n\r\n13.\r\na) [latex]\\left(5,-60^\\circ\\right)[\/latex]\r\nb) [latex]\\left(-5,120^\\circ\\right)[\/latex]\r\nc) [latex]\\left(5,480^\\circ\\right)[\/latex]\r\n\r\n15.\u00a0[latex]\\left(-5,0\\right)[\/latex]\r\n\r\n17.\u00a0[latex]\\left(-\\frac{3\\sqrt{3}}{2},-\\frac{3}{2}\\right)[\/latex]\r\n\r\n19.\u00a0[latex]\\left(2\\sqrt{5}, 0.464\\right)[\/latex]\r\n\r\n21.\u00a0[latex]\\left(\\sqrt{34},5.253\\right)[\/latex]\r\n\r\n23.\u00a0[latex]\\left(8\\sqrt{2},\\frac{\\pi }{4}\\right)[\/latex]\r\n\r\n25.\u00a0[latex]r=4\\csc \\theta [\/latex]\r\n\r\n27.\u00a0[latex]r=\\sqrt[3]{\\frac{sin\\theta }{2co{s}^{4}\\theta }}[\/latex]\r\n\r\n29.\u00a0[latex]r=3\\cos \\theta [\/latex]\r\n\r\n31.\u00a0[latex]r=\\frac{3\\sin \\theta }{\\cos \\left(2\\theta \\right)}[\/latex]\r\n\r\n33.\u00a0[latex]r=\\frac{9\\sin \\theta }{{\\cos }^{2}\\theta }[\/latex]\r\n\r\n35.\u00a0[latex]r=\\sqrt{\\frac{1}{9\\cos \\theta \\sin \\theta }}[\/latex]\r\n\r\n37.\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\n39.\u00a0[latex]3y+x=6[\/latex]; line\r\n\r\n41.\u00a0[latex]y=3[\/latex];\u00a0line\r\n\r\n43.\u00a0[latex]xy=4[\/latex]; hyperbola\r\n\r\n45.\u00a0[latex]{x}^{2}+{y}^{2}=4[\/latex]; circle\r\n\r\n47.\u00a0[latex]x - 5y=3[\/latex]; line\r\n\r\n49.\u00a0[latex]\\left(3,\\frac{3\\pi }{4}\\right)[\/latex]\r\n\r\n51.\u00a0[latex]\\left(5,\\pi \\right)[\/latex]\r\n\r\n53.\r\n\"Polar\r\n\r\n55.\r\n\"Polar\r\n\r\n57.\r\n\"Polar\r\n\r\n59.\r\n\"Polar\r\n\r\n61.\r\n\"Polar\r\n\r\n63.\u00a0[latex]r=\\frac{6}{5\\cos \\theta -\\sin \\theta }[\/latex]\r\n\"Plot\r\n\r\n65.\u00a0[latex]r=2\\sin \\theta [\/latex]\r\n\"Plot\r\n\r\n67.\u00a0[latex]r=\\frac{2}{\\cos \\theta }[\/latex]\r\n\"Plot\r\n\r\n69.\u00a0[latex]r=3\\cos \\theta [\/latex]\r\n\"Plot\r\n\r\n71.\u00a0[latex]{x}^{2}+{y}^{2}=16[\/latex]\r\n\"Plot\r\n\r\n73.\u00a0[latex]y=x[\/latex]\r\n\"Plot\r\n\r\n75.\u00a0[latex]{x}^{2}+{\\left(y+5\\right)}^{2}=25[\/latex]\r\n\"Plot\r\n\r\n77.\u00a0A vertical line with [latex]a[\/latex] units left of the y-axis.\r\n\r\n79.\u00a0A horizontal line with [latex]a[\/latex] units below the x-axis.\r\n

Section 6.4 Solutions<\/h2>\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\n55.\u00a0They are both spirals, but not quite the same.\r\n\r\n57.\u00a0Both graphs are curves with 2 loops. The equation with a coefficient of [latex]\\theta [\/latex] has two loops on the left, the equation with a coefficient of 2 has two loops side by side. Graph these from 0 to [latex]4\\pi [\/latex] to get a better picture.\r\n\r\n59.\u00a0When the width of the domain is increased, more petals of the flower are visible.\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

Section 6.5 Solutions<\/h2>\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\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.\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]\r\n

Section 6.6 Solutions<\/h2>\r\n1.\u00a0lowercase, bold letter, usually u<\/em><\/strong>, v<\/em><\/strong>, w<\/strong><\/em>\r\n\r\n3.\u00a0They are unit vectors. They are used to represent the horizontal and vertical components of a vector. They each have a magnitude of 1.\r\n\r\n5.\u00a0The first number always represents the coefficient of the i<\/em><\/strong>, and the second represents the j<\/strong><\/em>.\r\n\r\n7.\u00a0[latex]\\langle 7,\u22125\\rangle[\/latex]\r\n\r\n9. not equal\r\n\r\n11. equal\r\n\r\n13. equal\r\n\r\n15. [latex]7\\boldsymbol{i}\u22123\\boldsymbol{j}[\/latex]\r\n\r\n17. [latex]\u22126\\boldsymbol{i}\u22122\\boldsymbol{j}[\/latex]\r\n\r\n19. [latex]\\boldsymbol{u}+\\boldsymbol{v}=\\langle\u22125,5\\rangle,\\boldsymbol{u}\u2212\\boldsymbol{v}=\\langle\u22121,3\\rangle,2\\boldsymbol{u}\u22123\\boldsymbol{v}=\\langle 0,5\\rangle[\/latex]\r\n\r\n21. [latex]\u221210\\boldsymbol{i}\u20134\\boldsymbol{j}[\/latex]\r\n\r\n23. [latex]\u2212\\frac{2\\sqrt{29}}{29}\\boldsymbol{i}+\\frac{5\\sqrt{29}}{29}\\boldsymbol{j}[\/latex]\r\n\r\n25. [latex]\u2013\\frac{2\\sqrt{229}}{229}\\boldsymbol{i}+\\frac{15\\sqrt{229}}{229}\\boldsymbol{j}[\/latex]\r\n\r\n27. [latex]\u2013\\frac{7\\sqrt{2}}{10}\\boldsymbol{i}+\\frac{\\sqrt{2}}{10}\\boldsymbol{j}[\/latex]\r\n\r\n29. [latex]|\\boldsymbol{v}|=7.810,\\theta=39.806^{\\circ}[\/latex]\r\n\r\n31. [latex]|\\boldsymbol{v}|=7.211,\\theta=236.310^{\\circ}[\/latex]\r\n\r\n33.\u00a0\u22126\r\n\r\n35.\u00a0\u221212\r\n\r\n37.\r\n\"\"\r\n\r\n39.\r\n\"Plot\r\n\r\n41.\r\n\"Plot\r\n\r\n43.\r\n\"Plot\r\n\r\n45.\r\n\"Vector\r\n\r\n47. [latex]\\langle 4,1\\rangle[\/latex]\r\n\r\n49. [latex]\\boldsymbol{v}=\u22127\\boldsymbol{i}+3\\boldsymbol{j}[\/latex]\r\n\"Vector\r\n\r\n51. [latex]3\\sqrt{2}\\boldsymbol{i}+3\\sqrt{2}\\boldsymbol{j}[\/latex]\r\n\r\n53. [latex]\\boldsymbol{i}\u2212\\sqrt{3}\\boldsymbol{j}[\/latex]\r\n\r\n55. Magnitude: 29.05 pounds, Direction: 130.44 degrees\r\n\r\n57. Magnitude: 8.29 pounds, Direction: -44.56 degrees\r\n\r\n59. a. 58.7; b. 12.5\r\n\r\n61. [latex]x=7.13[\/latex] pounds, [latex]y=3.63[\/latex] pounds\r\n\r\n63.\u00a0[latex]x=2.87[\/latex] pounds, [latex]y=4.10[\/latex] pounds\r\n\r\n65. 4.635 miles, [latex]17.764^{\\circ}[\/latex] N of E\r\n\r\n67.\u00a017 miles. 10.318 miles\r\n\r\n69.\u00a0Distance: 2.868. Direction: [latex]86.474^{\\circ}[\/latex] North of West, or [latex]3.526^{\\circ}[\/latex] West of North\r\n\r\n71. [latex]4.924^{\\circ}[\/latex]. 659 km\/hr\r\n\r\n73. [latex]4.424^{\\circ}[\/latex]\r\n\r\n75. (0.081, 8.602)\r\n\r\n77. [latex]21.801^{\\circ}[\/latex], relative to the car\u2019s forward direction\r\n\r\n79.\u00a0parallel: 16.28, perpendicular: 47.28 pounds\r\n\r\n81.\u00a019.35 pounds, [latex]231.54^{\\circ}[\/latex] from the horizontal\r\n\r\n83.\u00a05.1583 pounds, [latex]75.8^{\\circ}[\/latex] from the horizontal","rendered":"

Section 6.1 Solutions<\/h2>\n

1.\u00a0The altitude extends from any vertex to the opposite side or to the line containing the opposite side at a 90\u00b0 angle.<\/p>\n

3.\u00a0When the known values are the side opposite the missing angle and another side and its opposite angle.<\/p>\n

5.\u00a0A triangle with two given sides and a non-included angle.<\/p>\n

7.\u00a0[latex]\\beta =72^\\circ ,a\\approx 12.0,b\\approx 19.9[\/latex]<\/p>\n

9.\u00a0[latex]\\gamma =20^\\circ ,b\\approx 4.5,c\\approx 1.6[\/latex]<\/p>\n

11.\u00a0[latex]b\\approx 3.78[\/latex]<\/p>\n

13.\u00a0[latex]c\\approx 13.70[\/latex]<\/p>\n

15.\u00a0one triangle, [latex]\\alpha \\approx 50.3^\\circ ,\\beta \\approx 16.7^\\circ ,a\\approx 26.7[\/latex]<\/p>\n

17.\u00a0two triangles, [latex]\\gamma \\approx 54.3^\\circ ,\\beta \\approx 90.7^\\circ ,b\\approx 20.9[\/latex] or [latex]{\\gamma }^{\\prime }\\approx 125.7^\\circ ,{\\beta }^{\\prime }\\approx 19.3^\\circ ,{b}^{\\prime }\\approx 6.9[\/latex]<\/p>\n

19.\u00a0two triangles, [latex]\\beta \\approx 75.7^\\circ , \\gamma \\approx 61.3^\\circ ,b\\approx 9.9[\/latex] or [latex]{\\beta }^{\\prime }\\approx 18.3^\\circ ,{\\gamma }^{\\prime }\\approx 118.7^\\circ ,{b}^{\\prime }\\approx 3.2[\/latex]<\/p>\n

21.\u00a0two triangles, [latex]\\alpha \\approx 143.2^\\circ ,\\beta \\approx 26.8^\\circ ,a\\approx 17.3[\/latex] or [latex]{\\alpha }^{\\prime }\\approx 16.8^\\circ ,{\\beta }^{\\prime }\\approx 153.2^\\circ ,{a}^{\\prime }\\approx 8.3[\/latex]<\/p>\n

23.\u00a0no triangle possible<\/p>\n

25.\u00a0[latex]A\\approx 47.8^\\circ[\/latex] or [latex]{A}^{\\prime }\\approx 132.2^\\circ[\/latex]<\/p>\n

27.\u00a0[latex]8.6[\/latex]<\/p>\n

29.\u00a0[latex]370.9[\/latex]<\/p>\n

31.\u00a0[latex]12.3[\/latex]<\/p>\n

33.\u00a0[latex]12.2[\/latex]<\/p>\n

35.\u00a0[latex]16.0[\/latex]<\/p>\n

37.\u00a0[latex]29.7^\\circ[\/latex]<\/p>\n

39.\u00a0[latex]x=76.9^\\circ \\text{or }x=103.1^\\circ[\/latex]<\/p>\n

41.\u00a0[latex]110.6^\\circ[\/latex]<\/p>\n

43.\u00a0[latex]A\\approx 39.4,\\text{ }C\\approx 47.6,\\text{ }BC\\approx 20.7[\/latex]<\/p>\n

45.\u00a0[latex]57.1[\/latex]<\/p>\n

47.\u00a0[latex]42.0[\/latex]<\/p>\n

49.\u00a0[latex]430.2[\/latex]<\/p>\n

51.\u00a0[latex]10.1[\/latex]<\/p>\n

53.\u00a0[latex]AD\\approx \\text{ }13.8[\/latex]<\/p>\n

55.\u00a0[latex]AB\\approx 2.8[\/latex]<\/p>\n

57.\u00a0[latex]L\\approx 49.7,\\text{ }N\\approx 56.3,\\text{ }LN\\approx 5.8[\/latex]<\/p>\n

59.\u00a051.4 feet<\/p>\n

61.\u00a0The distance from the satellite to station [latex]A[\/latex] is approximately 1716 miles. The satellite is approximately 1706 miles above the ground.<\/p>\n

63.\u00a02.6\u00a0ft<\/p>\n

65.\u00a05.6\u00a0km<\/p>\n

67.\u00a0371\u00a0ft<\/p>\n

69.\u00a05936\u00a0ft<\/p>\n

71.\u00a024.1\u00a0ft<\/p>\n

73.\u00a019,056\u00a0ft2<\/p>\n

75.\u00a0445,624\u00a0square\u00a0miles<\/p>\n

77.\u00a08.65\u00a0ft2<\/p>\n

Section 6.2 Solutions<\/h2>\n

1.\u00a0two sides and the angle opposite the missing side<\/p>\n

3.\u00a0[latex]s[\/latex] is the semi-perimeter, which is half the perimeter of the triangle.<\/p>\n

5.\u00a0The Law of Cosines must be used for any oblique (non-right) triangle.<\/p>\n

7.\u00a011.3<\/p>\n

9.\u00a034.7<\/p>\n

11.\u00a026.7<\/p>\n

13.\u00a0257.4<\/p>\n

15.\u00a0not possible<\/p>\n

17.\u00a095.5\u00b0<\/p>\n

19.\u00a026.9\u00b0<\/p>\n

21.\u00a0[latex]B\\approx 45.9^\\circ ,C\\approx 99.1^\\circ ,a\\approx 6.4[\/latex]<\/p>\n

23.\u00a0[latex]A\\approx 20.6^\\circ ,B\\approx 38.4^\\circ ,c\\approx 51.1[\/latex]<\/p>\n

25.\u00a0[latex]A\\approx 37.8^\\circ ,B\\approx 43.8,C\\approx 98.4^\\circ[\/latex]<\/p>\n

27.\u00a0177.56 in2<\/p>\n

29.\u00a00.04 m2<\/p>\n

31.\u00a00.91 yd2<\/p>\n

33.\u00a03.0<\/p>\n

35.\u00a029.1<\/p>\n

37.\u00a00.5<\/p>\n

39.\u00a070.7\u00b0<\/p>\n

41.\u00a077.4\u00b0<\/p>\n

43.\u00a025.0<\/p>\n

45.\u00a09.3<\/p>\n

47.\u00a043.52<\/p>\n

49.\u00a01.41<\/p>\n

51.\u00a00.14<\/p>\n

53.\u00a018.3<\/p>\n

55.\u00a048.98<\/p>\n

57.
\n\"A<\/p>\n

59.\u00a07.62<\/p>\n

61.\u00a085.1<\/p>\n

63.\u00a024.0 km<\/p>\n

65.\u00a099.9 ft<\/p>\n

67.\u00a037.3 miles<\/p>\n

69.\u00a02371 miles<\/p>\n

71.
\n\"Angle<\/p>\n

73.\u00a0599.8 miles<\/p>\n

75.\u00a065.4 cm2<\/p>\n

77.\u00a0468 ft2<\/p>\n

Section 6.3 Solutions<\/h2>\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.
\na) [latex]\\left(5,-\\frac{4\\pi}{3}\\right)[\/latex]
\nb) [latex]\\left(-5,\\frac{5\\pi}{3}\\right)[\/latex]
\nc) [latex]\\left(5,\\frac{8\\pi}{3}\\right)[\/latex]<\/p>\n

9.
\na) [latex]\\left(3,-\\frac{5\\pi}{4}\\right)[\/latex]
\nb) [latex]\\left(-3,\\frac{7\\pi}{4}\\right)[\/latex]
\nc) [latex]\\left(3,\\frac{11\\pi}{4}\\right)[\/latex]<\/p>\n

11.
\na) [latex]\\left(4,-135^\\circ\\right)[\/latex]
\nb) [latex]\\left(-4,45^\\circ\\right)[\/latex]
\nc) [latex]\\left(4,585^\\circ\\right)[\/latex]<\/p>\n

13.
\na) [latex]\\left(5,-60^\\circ\\right)[\/latex]
\nb) [latex]\\left(-5,120^\\circ\\right)[\/latex]
\nc) [latex]\\left(5,480^\\circ\\right)[\/latex]<\/p>\n

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

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

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

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

23.\u00a0[latex]\\left(8\\sqrt{2},\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n

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

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

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

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

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

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

37.\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

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

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

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

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

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

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

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

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

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

57.
\n\"Polar<\/p>\n

59.
\n\"Polar<\/p>\n

61.
\n\"Polar<\/p>\n

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

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

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

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

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

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

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

77.\u00a0A vertical line with [latex]a[\/latex] units left of the y-axis.<\/p>\n

79.\u00a0A horizontal line with [latex]a[\/latex] units below the x-axis.<\/p>\n

Section 6.4 Solutions<\/h2>\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

55.\u00a0They are both spirals, but not quite the same.<\/p>\n

57.\u00a0Both graphs are curves with 2 loops. The equation with a coefficient of [latex]\\theta[\/latex] has two loops on the left, the equation with a coefficient of 2 has two loops side by side. Graph these from 0 to [latex]4\\pi[\/latex] to get a better picture.<\/p>\n

59.\u00a0When the width of the domain is increased, more petals of the flower are visible.<\/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

Section 6.5 Solutions<\/h2>\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

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.\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

Section 6.6 Solutions<\/h2>\n

1.\u00a0lowercase, bold letter, usually u<\/em><\/strong>, v<\/em><\/strong>, w<\/strong><\/em><\/p>\n

3.\u00a0They are unit vectors. They are used to represent the horizontal and vertical components of a vector. They each have a magnitude of 1.<\/p>\n

5.\u00a0The first number always represents the coefficient of the i<\/em><\/strong>, and the second represents the j<\/strong><\/em>.<\/p>\n

7.\u00a0[latex]\\langle 7,\u22125\\rangle[\/latex]<\/p>\n

9. not equal<\/p>\n

11. equal<\/p>\n

13. equal<\/p>\n

15. [latex]7\\boldsymbol{i}\u22123\\boldsymbol{j}[\/latex]<\/p>\n

17. [latex]\u22126\\boldsymbol{i}\u22122\\boldsymbol{j}[\/latex]<\/p>\n

19. [latex]\\boldsymbol{u}+\\boldsymbol{v}=\\langle\u22125,5\\rangle,\\boldsymbol{u}\u2212\\boldsymbol{v}=\\langle\u22121,3\\rangle,2\\boldsymbol{u}\u22123\\boldsymbol{v}=\\langle 0,5\\rangle[\/latex]<\/p>\n

21. [latex]\u221210\\boldsymbol{i}\u20134\\boldsymbol{j}[\/latex]<\/p>\n

23. [latex]\u2212\\frac{2\\sqrt{29}}{29}\\boldsymbol{i}+\\frac{5\\sqrt{29}}{29}\\boldsymbol{j}[\/latex]<\/p>\n

25. [latex]\u2013\\frac{2\\sqrt{229}}{229}\\boldsymbol{i}+\\frac{15\\sqrt{229}}{229}\\boldsymbol{j}[\/latex]<\/p>\n

27. [latex]\u2013\\frac{7\\sqrt{2}}{10}\\boldsymbol{i}+\\frac{\\sqrt{2}}{10}\\boldsymbol{j}[\/latex]<\/p>\n

29. [latex]|\\boldsymbol{v}|=7.810,\\theta=39.806^{\\circ}[\/latex]<\/p>\n

31. [latex]|\\boldsymbol{v}|=7.211,\\theta=236.310^{\\circ}[\/latex]<\/p>\n

33.\u00a0\u22126<\/p>\n

35.\u00a0\u221212<\/p>\n

37.
\n\"\"<\/p>\n

39.
\n\"Plot<\/p>\n

41.
\n\"Plot<\/p>\n

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

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

47. [latex]\\langle 4,1\\rangle[\/latex]<\/p>\n

49. [latex]\\boldsymbol{v}=\u22127\\boldsymbol{i}+3\\boldsymbol{j}[\/latex]
\n\"Vector<\/p>\n

51. [latex]3\\sqrt{2}\\boldsymbol{i}+3\\sqrt{2}\\boldsymbol{j}[\/latex]<\/p>\n

53. [latex]\\boldsymbol{i}\u2212\\sqrt{3}\\boldsymbol{j}[\/latex]<\/p>\n

55. Magnitude: 29.05 pounds, Direction: 130.44 degrees<\/p>\n

57. Magnitude: 8.29 pounds, Direction: -44.56 degrees<\/p>\n

59. a. 58.7; b. 12.5<\/p>\n

61. [latex]x=7.13[\/latex] pounds, [latex]y=3.63[\/latex] pounds<\/p>\n

63.\u00a0[latex]x=2.87[\/latex] pounds, [latex]y=4.10[\/latex] pounds<\/p>\n

65. 4.635 miles, [latex]17.764^{\\circ}[\/latex] N of E<\/p>\n

67.\u00a017 miles. 10.318 miles<\/p>\n

69.\u00a0Distance: 2.868. Direction: [latex]86.474^{\\circ}[\/latex] North of West, or [latex]3.526^{\\circ}[\/latex] West of North<\/p>\n

71. [latex]4.924^{\\circ}[\/latex]. 659 km\/hr<\/p>\n

73. [latex]4.424^{\\circ}[\/latex]<\/p>\n

75. (0.081, 8.602)<\/p>\n

77. [latex]21.801^{\\circ}[\/latex], relative to the car\u2019s forward direction<\/p>\n

79.\u00a0parallel: 16.28, perpendicular: 47.28 pounds<\/p>\n

81.\u00a019.35 pounds, [latex]231.54^{\\circ}[\/latex] from the horizontal<\/p>\n

83.\u00a05.1583 pounds, [latex]75.8^{\\circ}[\/latex] from the horizontal<\/p>\n","protected":false},"author":264444,"menu_order":6,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-17332","chapter","type-chapter","status-publish","hentry"],"part":16602,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters\/17332","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/users\/264444"}],"version-history":[{"count":15,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters\/17332\/revisions"}],"predecessor-version":[{"id":17641,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters\/17332\/revisions\/17641"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/parts\/16602"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters\/17332\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/media?parent=17332"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=17332"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/contributor?post=17332"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/license?post=17332"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}