{"id":2231,"date":"2025-08-11T17:25:09","date_gmt":"2025-08-11T17:25:09","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2231"},"modified":"2025-08-13T16:59:13","modified_gmt":"2025-08-13T16:59:13","slug":"polar-coordinates-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/polar-coordinates-learn-it-4\/","title":{"raw":"Polar Coordinates: Learn It 4","rendered":"Polar Coordinates: Learn It 4"},"content":{"raw":"

Transforming Equations between Polar and Rectangular Forms<\/h2>\r\nWe can now convert coordinates between polar and rectangular form. Converting equations can be more difficult, but it can be beneficial to be able to convert between the two forms. Since there are a number of polar equations that cannot be expressed clearly in Cartesian form, and vice versa, we can use the same procedures we used to convert points between the coordinate systems. We can then use a graphing calculator to graph either the rectangular form or the polar form of the equation.\r\n\r\n
How To: Given an equation in polar form, graph it using a graphing calculator.\r\n<\/strong>\r\n
    \r\n \t
  1. Change the MODE<\/strong> to POL<\/strong>, representing polar form.<\/li>\r\n \t
  2. Press the Y= <\/strong>button to bring up a screen allowing the input of six equations: [latex]{r}_{1},{r}_{2},...,{r}_{6}[\/latex].<\/li>\r\n \t
  3. Enter the polar equation, set equal to [latex]r[\/latex].<\/li>\r\n \t
  4. Press GRAPH.<\/strong><\/li>\r\n<\/ol>\r\n<\/section>
    Write the Cartesian equation [latex]{x}^{2}+{y}^{2}=9[\/latex] in polar form.[reveal-answer q=\"229215\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"229215\"]The goal is to eliminate [latex]x[\/latex] and [latex]y[\/latex] from the equation and introduce [latex]r[\/latex] and [latex]\\theta [\/latex]. Ideally, we would write the equation [latex]r[\/latex] as a function of [latex]\\theta [\/latex]. To obtain the polar form, we will use the relationships between [latex]\\left(x,y\\right)[\/latex] and [latex]\\left(r,\\theta \\right)[\/latex]. Since [latex]x=r\\cos \\theta [\/latex] and [latex]y=r\\sin \\theta [\/latex], we can substitute and solve for [latex]r[\/latex].\r\n

    [latex]\\begin{align}&{\\left(r\\cos \\theta \\right)}^{2}+{\\left(r\\sin \\theta \\right)}^{2}=9 \\\\ &{r}^{2}{\\cos }^{2}\\theta +{r}^{2}{\\sin }^{2}\\theta =9 \\\\ &{r}^{2}\\left({\\cos }^{2}\\theta +{\\sin }^{2}\\theta \\right)=9 \\\\ &{r}^{2}\\left(1\\right)=9&& {\\text{Substitute cos}}^{2}\\theta +{\\sin }^{2}\\theta =1. \\\\ &r=\\pm 3&& \\text{Use the square root property}. \\end{align}[\/latex]<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]\"Plotting (a) Cartesian form [latex]{x}^{2}+{y}^{2}=9[\/latex] (b) Polar form [latex]r=3[\/latex][\/caption]Thus, [latex]{x}^{2}+{y}^{2}=9,r=3[\/latex], and [latex]r=-3[\/latex] should generate the same graph.To graph a circle in rectangular form, we must first solve for [latex]y[\/latex].\r\n

    [latex]\\begin{gathered} {x}^{2}+{y}^{2}=9 \\\\ {y}^{2}=9-{x}^{2} \\\\ y=\\pm \\sqrt{9-{x}^{2}}\\end{gathered}[\/latex]<\/p>\r\nNote that this is two separate functions, since a circle fails the vertical line test. Therefore, we need to enter the positive and negative square roots into the calculator separately, as two equations in the form [latex]{Y}_{1}=\\sqrt{9-{x}^{2}}[\/latex] and [latex]{Y}_{2}=-\\sqrt{9-{x}^{2}}[\/latex]. Press GRAPH.<\/strong>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

    Rewrite the Cartesian equation<\/strong> [latex]{x}^{2}+{y}^{2}=6y[\/latex] as a polar equation.[reveal-answer q=\"483629\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"483629\"]This equation appears similar to the previous example, but it requires different steps to convert the equation.We can still follow the same procedures we have already learned and make the following substitutions:\r\n

    [latex]\\begin{align}&{r}^{2}=6y&& \\text{Use }{x}^{2}+{y}^{2}={r}^{2}. \\\\ &{r}^{2}=6r\\sin \\theta&& \\text{Substitute}y=r\\sin \\theta . \\\\ &{r}^{2}-6r\\sin \\theta =0&& \\text{Set equal to 0}. \\\\ &r\\left(r - 6\\sin \\theta \\right)=0&& \\text{Factor and solve}. \\\\ &r=0&& \\text{We reject }r=0,\\text{as it only represents one point, }\\left(0,0\\right). \\\\ &\\text{or }r=6\\sin \\theta \\end{align}[\/latex]<\/p>\r\nTherefore, the equations [latex]{x}^{2}+{y}^{2}=6y[\/latex] and [latex]r=6\\sin \\theta [\/latex] should give us the same graph.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]\"Plots (a) Cartesian form [latex]{x}^{2}+{y}^{2}=6y[\/latex] (b) polar form [latex]r=6\\sin \\theta [\/latex][\/caption]The Cartesian or rectangular equation<\/strong> is plotted on the rectangular grid, and the polar equation<\/strong> is plotted on the polar grid. Clearly, the graphs are identical.[\/hidden-answer]<\/section>

    Rewrite the Cartesian equation [latex]y=3x+2[\/latex] as a polar equation.[reveal-answer q=\"530698\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"530698\"]We will use the relationships [latex]x=r\\cos \\theta [\/latex] and [latex]y=r\\sin \\theta [\/latex].\r\n

    [latex]\\begin{align}&y=3x+2 \\\\ &r\\sin \\theta =3r\\cos \\theta +2 \\\\ &r\\sin \\theta -3r\\cos \\theta =2 \\\\ &r\\left(\\sin \\theta -3\\cos \\theta \\right)=2&& \\text{Isolate }r. \\\\ &r=\\frac{2}{\\sin \\theta -3\\cos \\theta }&& \\text{Solve for }r. \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

    Rewrite the Cartesian equation [latex]{y}^{2}=3-{x}^{2}[\/latex] in polar form.[reveal-answer q=\"187427\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"187427\"][latex]r=\\sqrt{3}[\/latex][\/hidden-answer]<\/section>
    [ohm_question hide_question_numbers=1]149349[\/ohm_question]<\/section>","rendered":"

    Transforming Equations between Polar and Rectangular Forms<\/h2>\n

    We can now convert coordinates between polar and rectangular form. Converting equations can be more difficult, but it can be beneficial to be able to convert between the two forms. Since there are a number of polar equations that cannot be expressed clearly in Cartesian form, and vice versa, we can use the same procedures we used to convert points between the coordinate systems. We can then use a graphing calculator to graph either the rectangular form or the polar form of the equation.<\/p>\n

    How To: Given an equation in polar form, graph it using a graphing calculator.
    \n<\/strong><\/p>\n
      \n
    1. Change the MODE<\/strong> to POL<\/strong>, representing polar form.<\/li>\n
    2. Press the Y= <\/strong>button to bring up a screen allowing the input of six equations: [latex]{r}_{1},{r}_{2},...,{r}_{6}[\/latex].<\/li>\n
    3. Enter the polar equation, set equal to [latex]r[\/latex].<\/li>\n
    4. Press GRAPH.<\/strong><\/li>\n<\/ol>\n<\/section>\n
      Write the Cartesian equation [latex]{x}^{2}+{y}^{2}=9[\/latex] in polar form.<\/p>\n