{"id":1920,"date":"2025-07-30T21:25:18","date_gmt":"2025-07-30T21:25:18","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1920"},"modified":"2025-10-13T19:59:01","modified_gmt":"2025-10-13T19:59:01","slug":"graphs-of-the-other-trigonometric-functions-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-the-other-trigonometric-functions-learn-it-5\/","title":{"raw":"Graphs of the Other Trigonometric Functions: Learn It 5","rendered":"Graphs of the Other Trigonometric Functions: Learn It 5"},"content":{"raw":"

Analyzing the Graph of y = cot x and Its Variations<\/h2>\r\nThe last trigonometric function we need to explore is cotangent<\/strong>. The cotangent is defined by the reciprocal identity<\/strong> [latex]\\cot x=\\frac{1}{\\tan x}[\/latex]. Notice that the function is undefined when the tangent function is 0, leading to a vertical asymptote in the graph at 0, \u03c0, etc. Since the output of the tangent function is all real numbers, the output of the cotangent function<\/strong> is also all real numbers.\r\n\r\nWe can graph [latex]y=\\cot x[\/latex] by observing the graph of the tangent function because these two functions are reciprocals of one another. Where the graph of the tangent function decreases, the graph of the cotangent function increases. Where the graph of the tangent function increases, the graph of the cotangent function decreases.\r\n\r\nThe cotangent graph has vertical asymptotes at each value of x<\/em> where [latex]\\tan x=0[\/latex]; we show these in the graph below with dashed lines. Since the cotangent is the reciprocal of the tangent, [latex]\\cot x[\/latex] has vertical asymptotes at all values of x<\/em> where [latex]\\tan x=0[\/latex] , and [latex]\\cot x=0[\/latex] at all values of x where tan x has its vertical asymptotes.\r\n
\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"A The cotangent function[\/caption]\r\n\r\n
\r\n

features of the graph of y<\/em> = A<\/em>cot(Bx<\/em>)<\/h3>\r\n
    \r\n \t
  • The amplitude is |A<\/em>|.<\/li>\r\n \t
  • The period is [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\r\n \t
  • The domain is [latex]x\\ne\\frac{\\pi}{|B|}k[\/latex], where k<\/em> is an integer.<\/li>\r\n \t
  • The range is [latex](-\\infty,\\infty)[\/latex].<\/li>\r\n \t
  • The asymptotes occur at [latex]x=\\frac{\\pi}{|B|}k[\/latex], where k<\/em> is an integer.<\/li>\r\n \t
  • [latex]y=A\\cot(Bx)[\/latex] is an odd function.<\/li>\r\n<\/ul>\r\n<\/section><\/figure>\r\n

    Graphing Variations of y<\/em> = cot x<\/em><\/h2>\r\nWe can transform the graph of the cotangent in much the same way as we did for the tangent. The equation becomes the following.\r\n
    \r\n
    [latex]y=A\\cot(Bx\u2212C)+D[\/latex]<\/div>\r\n<\/div>\r\n
    How To: Given a modified cotangent function of the form [latex]f(x)=A\\cot(Bx)[\/latex], graph one period.<\/strong>\r\n
      \r\n \t
    1. Express the function in the form [latex]f(x)=A\\cot(Bx)[\/latex].<\/li>\r\n \t
    2. Identify the stretching factor, |A<\/em>|.<\/li>\r\n \t
    3. Identify the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\r\n \t
    4. Draw the graph of [latex]y=A\\tan(Bx)[\/latex].<\/li>\r\n \t
    5. Plot any two reference points.<\/li>\r\n \t
    6. Use the reciprocal relationship between tangent and cotangent to draw the graph of [latex]y=A\\cot(Bx)[\/latex].<\/li>\r\n \t
    7. Sketch the asymptotes.<\/li>\r\n<\/ol>\r\n<\/section><\/div>\r\n
      Determine the stretching factor, period, and phase shift of [latex]y=3\\cot(4x)[\/latex], and then sketch a graph.\r\n\r\n[reveal-answer q=\"32362\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"32362\"]\r\n\r\nStep 1.<\/strong> Expressing the function in the form [latex]f(x)=A\\cot(Bx)[\/latex] gives [latex]f(x)=3\\cot(4x)[\/latex].\r\n\r\nStep 2.<\/strong> The stretching factor is [latex]|A|=3[\/latex].\r\n\r\nStep 3.<\/strong> The period is [latex]P=\\frac{\\pi}{4}[\/latex].\r\n\r\nStep 4.<\/strong> Sketch the graph of [latex]y=3\\tan(4x)[\/latex].\r\n\r\nStep 5.<\/strong> Plot two reference points. Two such points are [latex]\\left(\\frac{\\pi}{16}\\text{, }3\\right)[\/latex] and [latex]\\left(\\frac{3\\pi}{16}\\text{, }\u22123\\right)[\/latex].\r\n\r\nStep 6.<\/strong> Use the reciprocal relationship to draw [latex]y=3\\cot(4x)[\/latex].\r\n\r\nStep 7.<\/strong> Sketch the asymptotes, [latex]x=0[\/latex], [latex]x=\\frac{\\pi}{4}[\/latex].\r\n\r\nThe orange graph shows [latex]y=3\\tan(4x)[\/latex] and the blue graph shows [latex]y=3\\cot(4x)[\/latex].\r\n\r\n \r\n\r\n\"A\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
      How To: Given a modified cotangent function of the form [latex]f(x)=A\\cot(Bx\u2212C)+D[\/latex], graph one period.<\/strong>\r\n
        \r\n \t
      1. Express the function in the form [latex]f(x)=A\\cot(Bx\u2212C)+D[\/latex].<\/li>\r\n \t
      2. Identify the stretching factor, |A<\/em>|.<\/li>\r\n \t
      3. Identify the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\r\n \t
      4. Identify the phase shift, [latex]\\frac{C}{B}[\/latex].<\/li>\r\n \t
      5. Draw the graph of [latex]y=A\\tan(Bx)[\/latex] shifted to the right by [latex]\\frac{C}{B}[\/latex] and up by D<\/em>.<\/li>\r\n \t
      6. Sketch the asymptotes [latex]x =\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where k<\/em> is an integer.<\/li>\r\n \t
      7. Plot any three reference points and draw the graph through these points.<\/li>\r\n<\/ol>\r\n<\/section>
        Sketch a graph of one period of the function [latex]f(x)=4\\cot\\left(\\frac{\\pi}{8}x\u2212\\frac{\\pi}{2}\\right)\u22122[\/latex].\r\n\r\n[reveal-answer q=\"706245\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"706245\"]\r\n\r\nStep 1.<\/strong> The function is already written in the general form [latex]f(x)=A\\cot(Bx\u2212C)+D[\/latex].\r\n\r\nStep 2.<\/strong>\u00a0[latex]A=4[\/latex], so the stretching factor is 4.\r\n\r\nStep 3.<\/strong>\u00a0[latex]B=\\frac{\\pi}{8}[\/latex], so the period is [latex]P=\\frac{\\pi}{|B|}=\\frac{\\pi}{\\frac{\\pi}{8}}=8[\/latex].\r\n\r\nStep 4.<\/strong>\u00a0[latex]C=\\frac{\\pi}{2}[\/latex], so the phase shift is [latex]\\frac{C}{B}=\\frac{\\frac{\\pi}{2}}{\\frac{\\pi}{8}}=4[\/latex].\r\n\r\nStep 5.<\/strong> We draw [latex]f(x)=4\\tan\\left(\\frac{\\pi}{8}x\u2212\\frac{\\pi}{2}\\right)\u22122[\/latex].\r\n\r\nStep 6-7.<\/strong> Three points we can use to guide the graph are (6,2), (8,\u22122), and (10,\u22126). We use the reciprocal relationship of tangent and cotangent to draw [latex]f(x)=4\\cot\\left(\\frac{\\pi}{8}x\u2212\\frac{\\pi}{2}\\right)\u22122[\/latex].\r\n\r\nStep 8.<\/strong> The vertical asymptotes are [latex]x=4[\/latex] and [latex]x=12[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"A One period of a modified cotangent function.[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"

        Analyzing the Graph of y = cot x and Its Variations<\/h2>\n

        The last trigonometric function we need to explore is cotangent<\/strong>. The cotangent is defined by the reciprocal identity<\/strong> [latex]\\cot x=\\frac{1}{\\tan x}[\/latex]. Notice that the function is undefined when the tangent function is 0, leading to a vertical asymptote in the graph at 0, \u03c0, etc. Since the output of the tangent function is all real numbers, the output of the cotangent function<\/strong> is also all real numbers.<\/p>\n

        We can graph [latex]y=\\cot x[\/latex] by observing the graph of the tangent function because these two functions are reciprocals of one another. Where the graph of the tangent function decreases, the graph of the cotangent function increases. Where the graph of the tangent function increases, the graph of the cotangent function decreases.<\/p>\n

        The cotangent graph has vertical asymptotes at each value of x<\/em> where [latex]\\tan x=0[\/latex]; we show these in the graph below with dashed lines. Since the cotangent is the reciprocal of the tangent, [latex]\\cot x[\/latex] has vertical asymptotes at all values of x<\/em> where [latex]\\tan x=0[\/latex] , and [latex]\\cot x=0[\/latex] at all values of x where tan x has its vertical asymptotes.<\/p>\n

        \n
        \"A
        The cotangent function<\/figcaption><\/figure>\n
        \n

        features of the graph of y<\/em> = A<\/em>cot(Bx<\/em>)<\/h3>\n
          \n
        • The amplitude is |A<\/em>|.<\/li>\n
        • The period is [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\n
        • The domain is [latex]x\\ne\\frac{\\pi}{|B|}k[\/latex], where k<\/em> is an integer.<\/li>\n
        • The range is [latex](-\\infty,\\infty)[\/latex].<\/li>\n
        • The asymptotes occur at [latex]x=\\frac{\\pi}{|B|}k[\/latex], where k<\/em> is an integer.<\/li>\n
        • [latex]y=A\\cot(Bx)[\/latex] is an odd function.<\/li>\n<\/ul>\n<\/section>\n<\/figure>\n

          Graphing Variations of y<\/em> = cot x<\/em><\/h2>\n

          We can transform the graph of the cotangent in much the same way as we did for the tangent. The equation becomes the following.<\/p>\n

          \n
          [latex]y=A\\cot(Bx\u2212C)+D[\/latex]<\/div>\n<\/div>\n
          \n
          How To: Given a modified cotangent function of the form [latex]f(x)=A\\cot(Bx)[\/latex], graph one period.<\/strong><\/p>\n
            \n
          1. Express the function in the form [latex]f(x)=A\\cot(Bx)[\/latex].<\/li>\n
          2. Identify the stretching factor, |A<\/em>|.<\/li>\n
          3. Identify the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\n
          4. Draw the graph of [latex]y=A\\tan(Bx)[\/latex].<\/li>\n
          5. Plot any two reference points.<\/li>\n
          6. Use the reciprocal relationship between tangent and cotangent to draw the graph of [latex]y=A\\cot(Bx)[\/latex].<\/li>\n
          7. Sketch the asymptotes.<\/li>\n<\/ol>\n<\/section>\n<\/div>\n
            Determine the stretching factor, period, and phase shift of [latex]y=3\\cot(4x)[\/latex], and then sketch a graph.<\/p>\n