{"id":1816,"date":"2025-07-28T20:27:31","date_gmt":"2025-07-28T20:27:31","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1816"},"modified":"2025-08-13T03:06:09","modified_gmt":"2025-08-13T03:06:09","slug":"the-other-trigonometric-functions-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/the-other-trigonometric-functions-learn-it-3\/","title":{"raw":"The Other Trigonometric Functions: Learn It 3","rendered":"The Other Trigonometric Functions: Learn It 3"},"content":{"raw":"

Using Even and Odd Trigonometric Functions<\/h2>\r\nTo be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important difference among the functions in this regard.\r\n\r\n
[latex]f\\left(x\\right)={x}^{2}[\/latex] is an even function<\/strong>, a function such that two inputs that are opposites have the same output. That means [latex]f\\left(-x\\right)=f\\left(x\\right)[\/latex]. \"Graph<\/span>\r\n

The function [latex]f\\left(x\\right)={x}^{2}[\/latex] is an even function.<\/p>\r\n[latex]f\\left(x\\right)={x}^{3}[\/latex] is an odd function<\/strong>, one such that two inputs that are opposites have outputs that are also opposites. That means [latex]f\\left(-x\\right)=-f\\left(x\\right)[\/latex].\r\n\r\n \"Graph<\/span>\r\n

The function [latex]f\\left(x\\right)={x}^{3}[\/latex] is an odd function.<\/p>\r\n\r\n<\/section>Consider the function [latex]f\\left(x\\right)={x}^{2}[\/latex]. The graph of the function is symmetrical about the y<\/em>-axis. All along the curve, any two points with opposite x<\/em>-values have the same function value. This matches the result of calculation: [latex]{\\left(4\\right)}^{2}={\\left(-4\\right)}^{2}[\/latex], [latex]{\\left(-5\\right)}^{2}={\\left(5\\right)}^{2}[\/latex],\u00a0and so on. So We can test whether a trigonometric function is even or odd by drawing a unit circle<\/strong> with a positive and a negative angle. The sine of the positive angle is [latex]y[\/latex]. The sine of the negative angle is \u2212y<\/em>. The sine function<\/strong>, then, is an odd function. We can test each of the six trigonometric functions in this fashion. The results are shown in in the table below.\r\n\r\n\"Graph\r\n\r\n\r\n\r\n\r\n
[latex]\\begin{array}{l}\\sin t=y\\hfill \\\\ \\sin \\left(-t\\right)=-y\\hfill \\\\ \\sin t\\ne \\sin \\left(-t\\right)\\hfill \\end{array}[\/latex]<\/td>\r\n[latex]\\begin{array}{l}\\text{cos}t=x\\hfill \\\\ \\cos \\left(-t\\right)=x\\hfill \\\\ \\cos t=\\cos \\left(-t\\right)\\hfill \\end{array}[\/latex]<\/td>\r\n[latex]\\begin{array}{l}\\text{tan}\\left(t\\right)=\\frac{y}{x}\\hfill \\\\ \\tan \\left(-t\\right)=-\\frac{y}{x}\\hfill \\\\ \\tan t\\ne \\tan \\left(-t\\right)\\hfill \\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]\\begin{array}{l}\\sec t=\\frac{1}{x}\\hfill \\\\ \\sec \\left(-t\\right)=\\frac{1}{x}\\hfill \\\\ \\sec t=\\sec \\left(-t\\right)\\hfill \\end{array}[\/latex]<\/td>\r\n[latex]\\begin{array}{l}\\csc t=\\frac{1}{y}\\hfill \\\\ \\csc \\left(-t\\right)=\\frac{1}{-y}\\hfill \\\\ \\csc t\\ne \\csc \\left(-t\\right)\\hfill \\end{array}[\/latex]<\/td>\r\n[latex]\\begin{array}{l}\\cot t=\\frac{x}{y}\\hfill \\\\ \\cot \\left(-t\\right)=\\frac{x}{-y}\\hfill \\\\ \\cot t\\ne cot\\left(-t\\right)\\hfill \\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
<\/div>\r\n
\r\n

even and odd trigonometric functions<\/h3>\r\n
    \r\n \t
  • An even function is one in which [latex]f\\left(-x\\right)=f\\left(x\\right)[\/latex].<\/li>\r\n \t
  • An odd function is one in which [latex]f\\left(-x\\right)=-f\\left(x\\right)[\/latex].<\/li>\r\n<\/ul>\r\nCosine and secant are even:\r\n\r\n[latex]\\begin{gathered}\\cos \\left(-t\\right)=\\cos t \\\\ \\sec \\left(-t\\right)=\\sec t \\end{gathered}[\/latex]\r\n\r\nSine, tangent, cosecant, and cotangent are odd:\r\n\r\n[latex]\\begin{gathered}\\sin \\left(-t\\right)=-\\sin t \\\\ \\tan \\left(-t\\right)=-\\tan t \\\\ \\csc \\left(-t\\right)=-\\csc t \\\\ \\cot \\left(-t\\right)=-\\cot t \\end{gathered}[\/latex]*\r\n\r\n<\/section><\/div>\r\n
    If the [latex]\\sec t=2[\/latex], what is the [latex]\\sec (-t)[\/latex]?[reveal-answer q=\"5363\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"5363\"]\r\n\r\nSecant is an even function. The secant of an angle is the same as the secant of its opposite. So if the secant of angle t<\/em> is 2, the secant of [latex]-t[\/latex] is also 2.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
    If the [latex]\\cot t=\\sqrt{3}[\/latex], what is [latex]\\cot (-t)[\/latex]?[reveal-answer q=\"840134\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"840134\"][latex]-\\sqrt{3}[\/latex][\/hidden-answer]<\/section>","rendered":"

    Using Even and Odd Trigonometric Functions<\/h2>\n

    To be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important difference among the functions in this regard.<\/p>\n

    [latex]f\\left(x\\right)={x}^{2}[\/latex] is an even function<\/strong>, a function such that two inputs that are opposites have the same output. That means [latex]f\\left(-x\\right)=f\\left(x\\right)[\/latex]. \"Graph<\/span><\/p>\n

    The function [latex]f\\left(x\\right)={x}^{2}[\/latex] is an even function.<\/p>\n

    [latex]f\\left(x\\right)={x}^{3}[\/latex] is an odd function<\/strong>, one such that two inputs that are opposites have outputs that are also opposites. That means [latex]f\\left(-x\\right)=-f\\left(x\\right)[\/latex].<\/p>\n

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

    The function [latex]f\\left(x\\right)={x}^{3}[\/latex] is an odd function.<\/p>\n<\/section>\n

    Consider the function [latex]f\\left(x\\right)={x}^{2}[\/latex]. The graph of the function is symmetrical about the y<\/em>-axis. All along the curve, any two points with opposite x<\/em>-values have the same function value. This matches the result of calculation: [latex]{\\left(4\\right)}^{2}={\\left(-4\\right)}^{2}[\/latex], [latex]{\\left(-5\\right)}^{2}={\\left(5\\right)}^{2}[\/latex],\u00a0and so on. So We can test whether a trigonometric function is even or odd by drawing a unit circle<\/strong> with a positive and a negative angle. The sine of the positive angle is [latex]y[\/latex]. The sine of the negative angle is \u2212y<\/em>. The sine function<\/strong>, then, is an odd function. We can test each of the six trigonometric functions in this fashion. The results are shown in in the table below.<\/p>\n

    \"Graph<\/p>\n\n\n\n\n
    [latex]\\begin{array}{l}\\sin t=y\\hfill \\\\ \\sin \\left(-t\\right)=-y\\hfill \\\\ \\sin t\\ne \\sin \\left(-t\\right)\\hfill \\end{array}[\/latex]<\/td>\n[latex]\\begin{array}{l}\\text{cos}t=x\\hfill \\\\ \\cos \\left(-t\\right)=x\\hfill \\\\ \\cos t=\\cos \\left(-t\\right)\\hfill \\end{array}[\/latex]<\/td>\n[latex]\\begin{array}{l}\\text{tan}\\left(t\\right)=\\frac{y}{x}\\hfill \\\\ \\tan \\left(-t\\right)=-\\frac{y}{x}\\hfill \\\\ \\tan t\\ne \\tan \\left(-t\\right)\\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n
    [latex]\\begin{array}{l}\\sec t=\\frac{1}{x}\\hfill \\\\ \\sec \\left(-t\\right)=\\frac{1}{x}\\hfill \\\\ \\sec t=\\sec \\left(-t\\right)\\hfill \\end{array}[\/latex]<\/td>\n[latex]\\begin{array}{l}\\csc t=\\frac{1}{y}\\hfill \\\\ \\csc \\left(-t\\right)=\\frac{1}{-y}\\hfill \\\\ \\csc t\\ne \\csc \\left(-t\\right)\\hfill \\end{array}[\/latex]<\/td>\n[latex]\\begin{array}{l}\\cot t=\\frac{x}{y}\\hfill \\\\ \\cot \\left(-t\\right)=\\frac{x}{-y}\\hfill \\\\ \\cot t\\ne cot\\left(-t\\right)\\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
    <\/div>\n
    \n
    \n

    even and odd trigonometric functions<\/h3>\n
      \n
    • An even function is one in which [latex]f\\left(-x\\right)=f\\left(x\\right)[\/latex].<\/li>\n
    • An odd function is one in which [latex]f\\left(-x\\right)=-f\\left(x\\right)[\/latex].<\/li>\n<\/ul>\n

      Cosine and secant are even:<\/p>\n

      [latex]\\begin{gathered}\\cos \\left(-t\\right)=\\cos t \\\\ \\sec \\left(-t\\right)=\\sec t \\end{gathered}[\/latex]<\/p>\n

      Sine, tangent, cosecant, and cotangent are odd:<\/p>\n

      [latex]\\begin{gathered}\\sin \\left(-t\\right)=-\\sin t \\\\ \\tan \\left(-t\\right)=-\\tan t \\\\ \\csc \\left(-t\\right)=-\\csc t \\\\ \\cot \\left(-t\\right)=-\\cot t \\end{gathered}[\/latex]*<\/p>\n<\/section>\n<\/div>\n

      If the [latex]\\sec t=2[\/latex], what is the [latex]\\sec (-t)[\/latex]?<\/p>\n