{"id":408,"date":"2021-03-25T16:48:17","date_gmt":"2021-03-25T16:48:17","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&p=408"},"modified":"2021-12-09T18:34:22","modified_gmt":"2021-12-09T18:34:22","slug":"polar-coordinates-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/polar-coordinates-2\/","title":{"raw":"Problem Set: Polar Coordinates","rendered":"Problem Set: Polar Coordinates"},"content":{"raw":"

In the following exercises, plot the point whose polar coordinates are given by first constructing the angle [latex]\\theta [\/latex] and then marking off the distance r<\/em> along the ray.<\/p>\r\n\r\n

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1.\u00a0<\/strong>[latex]\\left(3,\\frac{\\pi }{6}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"645457\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"645457\"]<\/span><\/p>\r\n\"On\r\n\r\n[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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2.\u00a0<\/strong>[latex]\\left(-2,\\frac{5\\pi }{3}\\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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3.\u00a0<\/strong>[latex]\\left(0,\\frac{7\\pi }{6}\\right)[\/latex]<\/p>\r\n[reveal-answer q=\"280496\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"280496\"]\r\n\r\n\"On\r\n\r\n[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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4.\u00a0<\/strong>[latex]\\left(-4,\\frac{3\\pi }{4}\\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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5.\u00a0<\/strong>[latex]\\left(1,\\frac{\\pi }{4}\\right)[\/latex]<\/p>\r\n[reveal-answer q=\"900381\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"900381\"]\r\n\r\n\"On\r\n\r\n[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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6.\u00a0<\/strong>[latex]\\left(2,\\frac{5\\pi }{6}\\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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7.\u00a0<\/strong>[latex]\\left(1,\\frac{\\pi }{2}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n[reveal-answer q=\"569634\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"569634\"]<\/span>\r\n\r\n\"On\r\n\r\n[\/hidden-answer]<\/span><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, consider the polar graph below. Give two sets of polar coordinates for each point.<\/p>\r\n\r\n\"The<\/span>\r\n

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8.\u00a0<\/strong>Coordinates of point A<\/em>.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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9.\u00a0<\/strong>Coordinates of point B<\/em>.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"35355\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"35355\"][latex]B\\begin{array}{cc}\\left(3,\\frac{\\text{-}\\pi }{3}\\right)\\hfill & B\\left(-3,\\frac{2\\pi }{3}\\right)\\hfill \\end{array}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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10.\u00a0<\/strong>Coordinates of point C<\/em>.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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11.\u00a0<\/strong>Coordinates of point D<\/em>.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"836415\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"836415\"][latex]D\\left(5,\\frac{7\\pi }{6}\\right)D\\left(-5,\\frac{\\pi }{6}\\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in [latex]\\left(0,2\\pi \\right][\/latex]. Round to three decimal places.<\/p>\r\n\r\n

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12.\u00a0<\/strong>[latex]\\left(2,2\\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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13.\u00a0<\/strong>[latex]\\left(3,-4\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"30104\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"30104\"][latex]\\begin{array}{cc}\\left(5,-0.927\\right)\\hfill & \\left(-5,-0.927+\\pi \\right)\\hfill \\end{array}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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14.\u00a0<\/strong>[latex]\\left(8,15\\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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15.\u00a0<\/strong>[latex]\\left(-6,8\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"961062\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"961062\"][latex]\\left(10,-0.927\\right)\\left(-10,-0.927+\\pi \\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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16.\u00a0<\/strong>[latex]\\left(4,3\\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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17.\u00a0<\/strong>[latex]\\left(3,\\text{-}\\sqrt{3}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"624461\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"624461\"][latex]\\left(2\\sqrt{3},-0.524\\right)\\left(-2\\sqrt{3},-0.524+\\pi \\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find rectangular coordinates for the given point in polar coordinates.<\/p>\r\n\r\n

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18.\u00a0<\/strong>[latex]\\left(2,\\frac{5\\pi }{4}\\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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19.\u00a0<\/strong>[latex]\\left(-2,\\frac{\\pi }{6}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"330808\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"330808\"][latex]\\left(\\begin{array}{cc}\\text{-}\\sqrt{3},\\hfill & -1\\hfill \\end{array}\\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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20.\u00a0<\/strong>[latex]\\left(5,\\frac{\\pi }{3}\\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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21.\u00a0<\/strong>[latex]\\left(1,\\frac{7\\pi }{6}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"486917\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"486917\"][latex]\\left(\\begin{array}{cc}-\\frac{\\sqrt{3}}{2},\\hfill & \\frac{-1}{2}\\hfill \\end{array}\\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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22.\u00a0<\/strong>[latex]\\left(-3,\\frac{3\\pi }{4}\\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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23.\u00a0<\/strong>[latex]\\left(0,\\frac{\\pi }{2}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"477501\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"477501\"][latex]\\left(\\begin{array}{cc}0,\\hfill & 0\\hfill \\end{array}\\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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24.\u00a0<\/strong>[latex]\\left(-4.5,6.5\\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, determine whether the graphs of the polar equation are symmetric with respect to the [latex]x[\/latex] -axis, the [latex]y[\/latex] -axis, or the origin.<\/p>\r\n\r\n

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25.\u00a0<\/strong>[latex]r=3\\sin\\left(2\\theta \\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"807910\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"807910\"]Symmetry with respect to the x-axis, y-axis, and origin.[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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26.\u00a0<\/strong>[latex]{r}^{2}=9\\cos\\theta [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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27.\u00a0<\/strong>[latex]r=\\cos\\left(\\frac{\\theta }{5}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"434444\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"434444\"]Symmetric with respect to x-axis only.[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

28.\u00a0<\/strong>[latex]r=2\\sec\\theta [\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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29.\u00a0<\/strong>[latex]r=1+\\cos\\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"991819\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"991819\"]Symmetry with respect to x-axis only.[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation.<\/p>\r\n\r\n

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30.\u00a0<\/strong>[latex]r=3[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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31.\u00a0<\/strong>[latex]\\theta =\\frac{\\pi }{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"484471\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"484471\"]Line [latex]y=x[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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32.\u00a0<\/strong>[latex]r=\\sec\\theta [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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33.\u00a0<\/strong>[latex]r=\\csc\\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"399579\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"399579\"][latex]y=1[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, convert the rectangular equation to polar form and sketch its graph.<\/p>\r\n\r\n

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34.\u00a0<\/strong>[latex]{x}^{2}+{y}^{2}=16[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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35.\u00a0<\/strong>[latex]{x}^{2}-{y}^{2}=16[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"774452\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"774452\"]Hyperbola; polar form [latex]{r}^{2}\\cos\\left(2\\theta \\right)=16[\/latex] or [latex]{r}^{2}=16\\sec\\theta [\/latex].<\/span><\/p>\r\n\"A\r\n\r\n[\/hidden-answer]<\/span><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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36.\u00a0<\/strong>[latex]x=8[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, convert the rectangular equation to polar form and sketch its graph.<\/p>\r\n\r\n

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37.\u00a0<\/strong>[latex]3x-y=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"615408\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"615408\"][latex]r=\\frac{2}{3\\cos\\theta -\\sin\\theta }[\/latex]<\/span><\/p>\r\n\"A\r\n\r\n[\/hidden-answer]<\/span><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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38.\u00a0<\/strong>[latex]{y}^{2}=4x[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, convert the polar equation to rectangular form and sketch its graph.<\/p>\r\n\r\n

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\r\n\r\n39.\u00a0<\/strong>[latex]r=4\\sin\\theta [\/latex]\r\n\r\n[reveal-answer q=\"6915\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"6915\"]<\/span>\r\n\r\n[latex]{x}^{2}+{y}^{2}=4y[\/latex]<\/span>\r\n\r\n\"A\r\n\r\n[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n
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40.<\/strong>\u00a0[latex]r=6\\cos\\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n41.\u00a0<\/strong>[latex]r=\\theta [\/latex]<\/span>\r\n

[reveal-answer q=\"975307\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"975307\"][latex]x\\tan\\sqrt{{x}^{2}+{y}^{2}}=y[\/latex]<\/span><\/p>\r\n\"A\r\n\r\n[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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42.\u00a0<\/strong>[latex]r=\\cot\\theta \\csc\\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, sketch a graph of the polar equation and identify any symmetry.<\/p>\r\n\r\n

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43.\u00a0<\/strong>[latex]r=1+\\sin\\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n[reveal-answer q=\"579647\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"579647\"]\r\n\r\n\"A\r\n\r\ny-axis symmetry[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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44.\u00a0<\/strong>[latex]r=3 - 2\\cos\\theta [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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45.\u00a0<\/strong>[latex]r=2 - 2\\sin\\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"466651\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"466651\"]<\/p>\r\n\"A\r\n\r\ny-axis symmetry[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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46.\u00a0<\/strong>[latex]r=5 - 4\\sin\\theta [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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47.\u00a0<\/strong>[latex]r=3\\cos\\left(2\\theta \\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"494010\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"494010\"]<\/p>\r\n\"A\r\n\r\nx- and y-axis symmetry and symmetry about the pole[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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48.\u00a0<\/strong>[latex]r=3\\sin\\left(2\\theta \\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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49.\u00a0<\/strong>[latex]r=2\\cos\\left(3\\theta \\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"186344\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"186344\"]<\/p>\r\n\"A\r\n\r\nx-axis symmetry[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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50.\u00a0<\/strong>[latex]r=3\\cos\\left(\\frac{\\theta }{2}\\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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51.\u00a0<\/strong>[latex]{r}^{2}=4\\cos\\left(2\\theta \\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"705899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"705899\"]<\/p>\r\n\"The\r\n\r\nx- and y-axis symmetry and symmetry about the pole[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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52.\u00a0<\/strong>[latex]{r}^{2}=4\\sin\\theta [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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53.\u00a0<\/strong>[latex]r=2\\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"655999\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"655999\"]<\/p>\r\n\"A\r\n\r\nno symmetry[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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54. [T]<\/strong> The graph of [latex]r=2\\cos\\left(2\\theta \\right)\\sec\\left(\\theta \\right)[\/latex]. is called a strophoid.<\/em> Use a graphing utility to sketch the graph, and, from the graph, determine the asymptote.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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55. [T]<\/strong> Use a graphing utility and sketch the graph of [latex]r=\\frac{6}{2\\sin\\theta -3\\cos\\theta }[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"124678\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124678\"]<\/p>\r\n\"A\r\n\r\na line[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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56. [T]<\/strong> Use a graphing utility to graph [latex]r=\\frac{1}{1-\\cos\\theta }[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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57. [T]<\/strong> Use technology to graph [latex]r={e}^{\\sin\\left(\\theta \\right)}-2\\cos\\left(4\\theta \\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"583557\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"583557\"]<\/span><\/p>\r\n\"A\r\n\r\n<\/span>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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58. [T]<\/strong> Use technology to plot [latex]r=\\sin\\left(\\frac{3\\theta }{7}\\right)[\/latex] (use the interval [latex]0\\le \\theta \\le 14\\pi [\/latex]).<\/div>\r\n<\/div>\r\n<\/div>\r\n
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59.\u00a0<\/strong>Without using technology, sketch the polar curve [latex]\\theta =\\frac{2\\pi }{3}[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"231360\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"231360\"]<\/span><\/p>\r\n\"A\r\n\r\n<\/span>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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60. [T]<\/strong> Use a graphing utility to plot [latex]r=\\theta \\sin\\theta [\/latex] for [latex]\\text{-}\\pi \\le \\theta \\le \\pi [\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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61. [T]<\/strong> Use technology to plot [latex]r={e}^{-0.1\\theta }[\/latex] for [latex]-10\\le \\theta \\le 10[\/latex].<\/p>\r\n[reveal-answer q=\"532887\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"532887\"]\r\n\r\n\"A\r\n\r\n[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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62. [T]<\/strong> There is a curve known as the \"Black Hole<\/em>.\" Use technology to plot [latex]r={e}^{-0.01\\theta }[\/latex] for [latex]-100\\le \\theta \\le 100[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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63. [T]<\/strong> Use the results of the preceding two problems to explore the graphs of [latex]r={e}^{-0.001\\theta }[\/latex] and [latex]r={e}^{-0.0001\\theta }[\/latex] for [latex]|\\theta |>100[\/latex].<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"940072\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"940072\"]Answers vary. One possibility is the spiral lines become closer together and the total number of spirals increases.[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

In the following exercises, plot the point whose polar coordinates are given by first constructing the angle [latex]\\theta[\/latex] and then marking off the distance r<\/em> along the ray.<\/p>\n

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1.\u00a0<\/strong>[latex]\\left(3,\\frac{\\pi }{6}\\right)[\/latex]<\/p>\n<\/div>\n

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<\/p>\n

Show Solution<\/span><\/p>\n
<\/span><\/p>\n

\"On<\/p>\n

<\/div>\n<\/div>\n

<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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2.\u00a0<\/strong>[latex]\\left(-2,\\frac{5\\pi }{3}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n
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3.\u00a0<\/strong>[latex]\\left(0,\\frac{7\\pi }{6}\\right)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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\"On<\/p>\n

<\/div>\n<\/div>\n

<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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4.\u00a0<\/strong>[latex]\\left(-4,\\frac{3\\pi }{4}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n
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5.\u00a0<\/strong>[latex]\\left(1,\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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\"On<\/p>\n

<\/div>\n<\/div>\n

<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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6.\u00a0<\/strong>[latex]\\left(2,\\frac{5\\pi }{6}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n
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7.\u00a0<\/strong>[latex]\\left(1,\\frac{\\pi }{2}\\right)[\/latex]<\/p>\n<\/div>\n

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<\/p>\n

Show Solution<\/span><\/p>\n
<\/span><\/p>\n

\"On<\/p>\n

<\/div>\n<\/div>\n

<\/span><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, consider the polar graph below. Give two sets of polar coordinates for each point.<\/p>\n


\n\"The<\/span><\/p>\n

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8.\u00a0<\/strong>Coordinates of point A<\/em>.<\/div>\n<\/div>\n<\/div>\n
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9.\u00a0<\/strong>Coordinates of point B<\/em>.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]B\\begin{array}{cc}\\left(3,\\frac{\\text{-}\\pi }{3}\\right)\\hfill & B\\left(-3,\\frac{2\\pi }{3}\\right)\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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10.\u00a0<\/strong>Coordinates of point C<\/em>.<\/div>\n<\/div>\n<\/div>\n
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11.\u00a0<\/strong>Coordinates of point D<\/em>.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]D\\left(5,\\frac{7\\pi }{6}\\right)D\\left(-5,\\frac{\\pi }{6}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in [latex]\\left(0,2\\pi \\right][\/latex]. Round to three decimal places.<\/p>\n

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12.\u00a0<\/strong>[latex]\\left(2,2\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n
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13.\u00a0<\/strong>[latex]\\left(3,-4\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\begin{array}{cc}\\left(5,-0.927\\right)\\hfill & \\left(-5,-0.927+\\pi \\right)\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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14.\u00a0<\/strong>[latex]\\left(8,15\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n
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15.\u00a0<\/strong>[latex]\\left(-6,8\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\left(10,-0.927\\right)\\left(-10,-0.927+\\pi \\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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16.\u00a0<\/strong>[latex]\\left(4,3\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n
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17.\u00a0<\/strong>[latex]\\left(3,\\text{-}\\sqrt{3}\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\left(2\\sqrt{3},-0.524\\right)\\left(-2\\sqrt{3},-0.524+\\pi \\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find rectangular coordinates for the given point in polar coordinates.<\/p>\n

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18.\u00a0<\/strong>[latex]\\left(2,\\frac{5\\pi }{4}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n
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19.\u00a0<\/strong>[latex]\\left(-2,\\frac{\\pi }{6}\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\left(\\begin{array}{cc}\\text{-}\\sqrt{3},\\hfill & -1\\hfill \\end{array}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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20.\u00a0<\/strong>[latex]\\left(5,\\frac{\\pi }{3}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n
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21.\u00a0<\/strong>[latex]\\left(1,\\frac{7\\pi }{6}\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\left(\\begin{array}{cc}-\\frac{\\sqrt{3}}{2},\\hfill & \\frac{-1}{2}\\hfill \\end{array}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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22.\u00a0<\/strong>[latex]\\left(-3,\\frac{3\\pi }{4}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n
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23.\u00a0<\/strong>[latex]\\left(0,\\frac{\\pi }{2}\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\left(\\begin{array}{cc}0,\\hfill & 0\\hfill \\end{array}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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24.\u00a0<\/strong>[latex]\\left(-4.5,6.5\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n

For the following exercises, determine whether the graphs of the polar equation are symmetric with respect to the [latex]x[\/latex] -axis, the [latex]y[\/latex] -axis, or the origin.<\/p>\n

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25.\u00a0<\/strong>[latex]r=3\\sin\\left(2\\theta \\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
Symmetry with respect to the x-axis, y-axis, and origin.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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26.\u00a0<\/strong>[latex]{r}^{2}=9\\cos\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n
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27.\u00a0<\/strong>[latex]r=\\cos\\left(\\frac{\\theta }{5}\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
Symmetric with respect to x-axis only.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
28.\u00a0<\/strong>[latex]r=2\\sec\\theta[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n
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29.\u00a0<\/strong>[latex]r=1+\\cos\\theta[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
Symmetry with respect to x-axis only.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation.<\/p>\n

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30.\u00a0<\/strong>[latex]r=3[\/latex]<\/div>\n<\/div>\n<\/div>\n
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31.\u00a0<\/strong>[latex]\\theta =\\frac{\\pi }{4}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
Line [latex]y=x[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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32.\u00a0<\/strong>[latex]r=\\sec\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n
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33.\u00a0<\/strong>[latex]r=\\csc\\theta[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]y=1[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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34.\u00a0<\/strong>[latex]{x}^{2}+{y}^{2}=16[\/latex]<\/div>\n<\/div>\n<\/div>\n
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35.\u00a0<\/strong>[latex]{x}^{2}-{y}^{2}=16[\/latex]<\/p>\n<\/div>\n

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<\/p>\n

Show Solution<\/span><\/p>\n
Hyperbola; polar form [latex]{r}^{2}\\cos\\left(2\\theta \\right)=16[\/latex] or [latex]{r}^{2}=16\\sec\\theta[\/latex].<\/span><\/p>\n

\"A<\/p>\n

<\/div>\n<\/div>\n

<\/span><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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36.\u00a0<\/strong>[latex]x=8[\/latex]<\/div>\n<\/div>\n<\/div>\n

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

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37.\u00a0<\/strong>[latex]3x-y=2[\/latex]<\/p>\n<\/div>\n

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<\/p>\n

Show Solution<\/span><\/p>\n
[latex]r=\\frac{2}{3\\cos\\theta -\\sin\\theta }[\/latex]<\/span><\/p>\n

\"A<\/p>\n

<\/div>\n<\/div>\n

<\/span><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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38.\u00a0<\/strong>[latex]{y}^{2}=4x[\/latex]<\/div>\n<\/div>\n<\/div>\n

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

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39.\u00a0<\/strong>[latex]r=4\\sin\\theta[\/latex]<\/p>\n

<\/p>\n

Show Solution<\/span><\/p>\n
<\/span><\/p>\n

[latex]{x}^{2}+{y}^{2}=4y[\/latex]<\/span><\/p>\n

\"A<\/p>\n

<\/div>\n<\/div>\n

<\/span><\/p>\n<\/div>\n<\/div>\n

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40.<\/strong>\u00a0[latex]r=6\\cos\\theta[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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41.\u00a0<\/strong>[latex]r=\\theta[\/latex]<\/span><\/p>\n

<\/p>\n

Show Solution<\/span><\/p>\n
[latex]x\\tan\\sqrt{{x}^{2}+{y}^{2}}=y[\/latex]<\/span><\/p>\n

\"A<\/p>\n

<\/div>\n<\/div>\n

<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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42.\u00a0<\/strong>[latex]r=\\cot\\theta \\csc\\theta[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, sketch a graph of the polar equation and identify any symmetry.<\/p>\n

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43.\u00a0<\/strong>[latex]r=1+\\sin\\theta[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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\"A<\/p>\n

y-axis symmetry<\/div>\n<\/div>\n

<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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44.\u00a0<\/strong>[latex]r=3 - 2\\cos\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n
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45.\u00a0<\/strong>[latex]r=2 - 2\\sin\\theta[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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\"A<\/p>\n

y-axis symmetry<\/div>\n<\/div>\n

<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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46.\u00a0<\/strong>[latex]r=5 - 4\\sin\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n
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47.\u00a0<\/strong>[latex]r=3\\cos\\left(2\\theta \\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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\"A<\/p>\n

x- and y-axis symmetry and symmetry about the pole<\/div>\n<\/div>\n

<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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48.\u00a0<\/strong>[latex]r=3\\sin\\left(2\\theta \\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n
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49.\u00a0<\/strong>[latex]r=2\\cos\\left(3\\theta \\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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\"A<\/p>\n

x-axis symmetry<\/div>\n<\/div>\n

<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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50.\u00a0<\/strong>[latex]r=3\\cos\\left(\\frac{\\theta }{2}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n
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51.\u00a0<\/strong>[latex]{r}^{2}=4\\cos\\left(2\\theta \\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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\"The<\/p>\n

x- and y-axis symmetry and symmetry about the pole<\/div>\n<\/div>\n

<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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52.\u00a0<\/strong>[latex]{r}^{2}=4\\sin\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n
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53.\u00a0<\/strong>[latex]r=2\\theta[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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\"A<\/p>\n

no symmetry<\/div>\n<\/div>\n

<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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54. [T]<\/strong> The graph of [latex]r=2\\cos\\left(2\\theta \\right)\\sec\\left(\\theta \\right)[\/latex]. is called a strophoid.<\/em> Use a graphing utility to sketch the graph, and, from the graph, determine the asymptote.<\/div>\n<\/div>\n<\/div>\n
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55. [T]<\/strong> Use a graphing utility and sketch the graph of [latex]r=\\frac{6}{2\\sin\\theta -3\\cos\\theta }[\/latex].<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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\"A<\/p>\n

a line<\/div>\n<\/div>\n

<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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56. [T]<\/strong> Use a graphing utility to graph [latex]r=\\frac{1}{1-\\cos\\theta }[\/latex].<\/div>\n<\/div>\n<\/div>\n
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57. [T]<\/strong> Use technology to graph [latex]r={e}^{\\sin\\left(\\theta \\right)}-2\\cos\\left(4\\theta \\right)[\/latex].<\/p>\n<\/div>\n

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<\/p>\n

Show Solution<\/span><\/p>\n
<\/span><\/p>\n

\"A<\/p>\n

<\/span>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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58. [T]<\/strong> Use technology to plot [latex]r=\\sin\\left(\\frac{3\\theta }{7}\\right)[\/latex] (use the interval [latex]0\\le \\theta \\le 14\\pi[\/latex]).<\/div>\n<\/div>\n<\/div>\n
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59.\u00a0<\/strong>Without using technology, sketch the polar curve [latex]\\theta =\\frac{2\\pi }{3}[\/latex].<\/p>\n<\/div>\n

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<\/p>\n

Show Solution<\/span><\/p>\n
<\/span><\/p>\n

\"A<\/p>\n

<\/span>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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60. [T]<\/strong> Use a graphing utility to plot [latex]r=\\theta \\sin\\theta[\/latex] for [latex]\\text{-}\\pi \\le \\theta \\le \\pi[\/latex].<\/div>\n<\/div>\n<\/div>\n
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61. [T]<\/strong> Use technology to plot [latex]r={e}^{-0.1\\theta }[\/latex] for [latex]-10\\le \\theta \\le 10[\/latex].<\/p>\n

Show Solution<\/span><\/p>\n
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\"A<\/p>\n

<\/div>\n<\/div>\n

<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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62. [T]<\/strong> There is a curve known as the “Black Hole<\/em>.” Use technology to plot [latex]r={e}^{-0.01\\theta }[\/latex] for [latex]-100\\le \\theta \\le 100[\/latex].<\/div>\n<\/div>\n<\/div>\n
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63. [T]<\/strong> Use the results of the preceding two problems to explore the graphs of [latex]r={e}^{-0.001\\theta }[\/latex] and [latex]r={e}^{-0.0001\\theta }[\/latex] for [latex]|\\theta |>100[\/latex].<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
Answers vary. One possibility is the spiral lines become closer together and the total number of spirals increases.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\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, Shared previously<\/div>