{"id":1797,"date":"2025-07-28T19:42:18","date_gmt":"2025-07-28T19:42:18","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1797"},"modified":"2025-08-13T03:04:01","modified_gmt":"2025-08-13T03:04:01","slug":"sine-and-cosine-functions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/sine-and-cosine-functions-learn-it-2\/","title":{"raw":"Sine and Cosine Functions: Learn It 2","rendered":"Sine and Cosine Functions: Learn It 2"},"content":{"raw":"

The Pythagorean Identity<\/h2>\r\nNow that we can define sine and cosine, we will learn how they relate to each other and the unit circle. Recall that the equation for the unit circle is [latex]{x}^{2}+{y}^{2}=1[\/latex]. Because [latex]x=\\cos t[\/latex] and [latex]y=\\sin t[\/latex], we can substitute for [latex]x[\/latex] and [latex]y[\/latex] to get [latex]{\\cos }^{2}t+{\\sin }^{2}t=1[\/latex]. This equation, [latex]{\\cos }^{2}t+{\\sin }^{2}t=1[\/latex], is known as the Pythagorean Identity<\/strong>.\r\n<\/span>\r\n\r\n\"Graph\r\n\r\nWe can use the Pythagorean Identity to find the cosine of an angle if we know the sine, or to find the sine of an angle if we know the cosine. However, because the equation yields two solutions, we need additional knowledge of the angle to choose the solution with the correct sign. If we know the quadrant where the angle is, we can easily choose the correct solution.\r\n\r\n
\r\n

Pythagorean identity<\/h3>\r\nThe Pythagorean Identity<\/strong> states that, for any real number [latex]t[\/latex],\r\n
[latex]{\\cos }^{2}t+{\\sin }^{2}t=1[\/latex]<\/div>\r\n<\/section>\r\n
How To: Given the sine of some angle [latex]t[\/latex] and its quadrant location, find the cosine of [latex]t[\/latex].<\/strong>\r\n
    \r\n \t
  1. Substitute the known value of [latex]\\sin \\left(t\\right)[\/latex] into the Pythagorean Identity.<\/li>\r\n \t
  2. Solve for [latex]\\cos \\left(t\\right)[\/latex].<\/li>\r\n \t
  3. Choose the solution with the appropriate sign for the x<\/em>-values in the quadrant where [latex]t[\/latex] is located.<\/li>\r\n<\/ol>\r\n<\/section><\/div>\r\n
    If [latex]\\sin \\left(t\\right)=\\frac{3}{7}[\/latex] and [latex]t[\/latex] is in the second quadrant, find [latex]\\cos \\left(t\\right)[\/latex].[reveal-answer q=\"81587\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"81587\"]If we drop a vertical line from the point on the unit circle corresponding to [latex]t[\/latex], we create a right triangle, from which we can see that the Pythagorean Identity is simply one case of the Pythagorean Theorem.\u00a0\r\n<\/span>\"Graph \r\n\r\nSubstituting the known value for sine into the Pythagorean Identity,\r\n

    [latex]\\begin{gathered}{\\cos }^{2}\\left(t\\right)+{\\sin }^{2}\\left(t\\right)=1 \\\\ {\\cos }^{2}\\left(t\\right)+\\frac{9}{49}=1 \\\\ {\\cos }^{2}\\left(t\\right)=\\frac{40}{49} \\\\ \\text{cos}\\left(t\\right)=\\pm \\sqrt{\\frac{40}{49}}=\\pm \\frac{\\sqrt{40}}{7}=\\pm \\frac{2\\sqrt{10}}{7} \\end{gathered}[\/latex]<\/p>\r\nBecause the angle is in the second quadrant, we know the x-<\/em>value is a negative real number, so the cosine is also negative. So\r\n[latex]\\text{cos}\\left(t\\right)=-\\frac{2\\sqrt{10}}{7}\\\\[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

    \r\n
    \r\n\r\nIf [latex]\\cos \\left(t\\right)=\\frac{24}{25}[\/latex] and [latex]t[\/latex] is in the fourth quadrant, find [latex]\\sin\\left(t\\right)[\/latex].\r\n\r\n[reveal-answer q=\"656903\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"656903\"]\r\n\r\n[latex]\\sin \\left(t\\right)=-\\frac{7}{25}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>
    [ohm_question hide_question_numbers=1]174837[\/ohm_question]<\/section>
    You'll need to use the quadrant to decide if your solution is positive or negative. Remember,\r\n
      \r\n \t
    • sine is positive in Quadrant I and II<\/li>\r\n \t
    • cosine is positive in Quadrant I and IV<\/li>\r\n<\/ul>\r\n<\/section> \r\n\r\n<\/section>","rendered":"

      The Pythagorean Identity<\/h2>\n

      Now that we can define sine and cosine, we will learn how they relate to each other and the unit circle. Recall that the equation for the unit circle is [latex]{x}^{2}+{y}^{2}=1[\/latex]. Because [latex]x=\\cos t[\/latex] and [latex]y=\\sin t[\/latex], we can substitute for [latex]x[\/latex] and [latex]y[\/latex] to get [latex]{\\cos }^{2}t+{\\sin }^{2}t=1[\/latex]. This equation, [latex]{\\cos }^{2}t+{\\sin }^{2}t=1[\/latex], is known as the Pythagorean Identity<\/strong>.
      \n<\/span><\/p>\n

      \"Graph<\/p>\n

      We can use the Pythagorean Identity to find the cosine of an angle if we know the sine, or to find the sine of an angle if we know the cosine. However, because the equation yields two solutions, we need additional knowledge of the angle to choose the solution with the correct sign. If we know the quadrant where the angle is, we can easily choose the correct solution.<\/p>\n

      \n

      Pythagorean identity<\/h3>\n

      The Pythagorean Identity<\/strong> states that, for any real number [latex]t[\/latex],<\/p>\n

      [latex]{\\cos }^{2}t+{\\sin }^{2}t=1[\/latex]<\/div>\n<\/section>\n
      \n
      How To: Given the sine of some angle [latex]t[\/latex] and its quadrant location, find the cosine of [latex]t[\/latex].<\/strong><\/p>\n
        \n
      1. Substitute the known value of [latex]\\sin \\left(t\\right)[\/latex] into the Pythagorean Identity.<\/li>\n
      2. Solve for [latex]\\cos \\left(t\\right)[\/latex].<\/li>\n
      3. Choose the solution with the appropriate sign for the x<\/em>-values in the quadrant where [latex]t[\/latex] is located.<\/li>\n<\/ol>\n<\/section>\n<\/div>\n
        If [latex]\\sin \\left(t\\right)=\\frac{3}{7}[\/latex] and [latex]t[\/latex] is in the second quadrant, find [latex]\\cos \\left(t\\right)[\/latex].<\/p>\n