{"id":2258,"date":"2025-08-12T16:34:55","date_gmt":"2025-08-12T16:34:55","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2258"},"modified":"2025-08-13T17:03:51","modified_gmt":"2025-08-13T17:03:51","slug":"graphing-in-polar-coordinates-learn-it-6","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphing-in-polar-coordinates-learn-it-6\/","title":{"raw":"Graphing in Polar Coordinates: Learn It 6","rendered":"Graphing in Polar Coordinates: Learn It 6"},"content":{"raw":"

Investigating the Archimedes\u2019 Spiral<\/h2>\r\nThe final polar equation we will discuss is the Archimedes\u2019 spiral, named for its discoverer, the Greek mathematician Archimedes (c. 287 BCE - c. 212 BCE), who is credited with numerous discoveries in the fields of geometry and mechanics.\r\n\r\n
\r\n

Archimedes\u2019 spiral<\/h3>\r\nThe formula that generates the graph of the Archimedes\u2019 spiral<\/strong> is given by [latex]r=\\theta [\/latex]\u00a0for [latex]\\theta \\ge 0[\/latex]. As [latex]\\theta [\/latex] increases, [latex]r[\/latex]\u00a0increases at a constant rate in an ever-widening, never-ending, spiraling path.\r\n\r\n\"Two\r\n\r\n<\/section>
How To: Given an Archimedes\u2019 spiral over [latex]\\left[0,2\\pi \\right][\/latex], sketch the graph.<\/strong>\r\n
    \r\n \t
  1. Make a table of values for [latex]r[\/latex] and [latex]\\theta [\/latex] over the given domain.<\/li>\r\n \t
  2. Plot the points and sketch the graph.<\/li>\r\n<\/ol>\r\n<\/section>
    Sketch the graph of [latex]r=\\theta [\/latex] over [latex]\\left[0,2\\pi \\right][\/latex].[reveal-answer q=\"329167\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"329167\"]As [latex]r[\/latex] is equal to [latex]\\theta [\/latex], the plot of the Archimedes\u2019 spiral begins at the pole at the point (0, 0). While the graph hints of symmetry, there is no formal symmetry with regard to passing the symmetry tests. Further, there is no maximum value, unless the domain is restricted.Create a table such as the one below.\r\n\r\n\r\n\r\n\r\n
    [latex]\\theta [\/latex] <\/strong><\/td>\r\n[latex]\\frac{\\pi }{4}[\/latex]<\/td>\r\n[latex]\\frac{\\pi }{2}[\/latex]<\/td>\r\n[latex]\\pi [\/latex]<\/td>\r\n[latex]\\frac{3\\pi }{2}[\/latex]<\/td>\r\n[latex]\\frac{7\\pi }{4}[\/latex]<\/td>\r\n[latex]2\\pi [\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]r[\/latex] <\/strong><\/td>\r\n0.785<\/td>\r\n1.57<\/td>\r\n3.14<\/td>\r\n4.71<\/td>\r\n5.50<\/td>\r\n6.28<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that the r<\/em>-values are just the decimal form of the angle measured in radians.\r\n\r\n\"Graph\r\n

    Analysis of the Solution<\/h4>\r\nThe domain of this polar curve is [latex]\\left[0,2\\pi \\right][\/latex]. In general, however, the domain of this function is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Graphing the equation of the Archimedes\u2019 spiral is rather simple, although the image makes it seem like it would be complex.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
    Sketch the graph of [latex]r=-\\theta [\/latex] over the interval [latex]\\left[0,4\\pi \\right][\/latex].[reveal-answer q=\"937587\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"937587\"]\"\"[\/hidden-answer]<\/section>","rendered":"

    Investigating the Archimedes\u2019 Spiral<\/h2>\n

    The final polar equation we will discuss is the Archimedes\u2019 spiral, named for its discoverer, the Greek mathematician Archimedes (c. 287 BCE – c. 212 BCE), who is credited with numerous discoveries in the fields of geometry and mechanics.<\/p>\n

    \n

    Archimedes\u2019 spiral<\/h3>\n

    The formula that generates the graph of the Archimedes\u2019 spiral<\/strong> is given by [latex]r=\\theta[\/latex]\u00a0for [latex]\\theta \\ge 0[\/latex]. As [latex]\\theta[\/latex] increases, [latex]r[\/latex]\u00a0increases at a constant rate in an ever-widening, never-ending, spiraling path.<\/p>\n

    \"Two<\/p>\n<\/section>\n

    How To: Given an Archimedes\u2019 spiral over [latex]\\left[0,2\\pi \\right][\/latex], sketch the graph.<\/strong><\/p>\n
      \n
    1. Make a table of values for [latex]r[\/latex] and [latex]\\theta[\/latex] over the given domain.<\/li>\n
    2. Plot the points and sketch the graph.<\/li>\n<\/ol>\n<\/section>\n
      Sketch the graph of [latex]r=\\theta[\/latex] over [latex]\\left[0,2\\pi \\right][\/latex].<\/p>\n