{"id":2027,"date":"2025-07-31T23:54:23","date_gmt":"2025-07-31T23:54:23","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2027"},"modified":"2025-08-13T03:22:45","modified_gmt":"2025-08-13T03:22:45","slug":"modeling-with-trigonometric-equations-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/modeling-with-trigonometric-equations-learn-it-5\/","title":{"raw":"Modeling with Trigonometric Equations: Learn It 5","rendered":"Modeling with Trigonometric Equations: Learn It 5"},"content":{"raw":"

Modeling the same periodic function with sine and cosine<\/h2>\r\n
\r\n

sine vs. cosine<\/h3>\r\nAny periodic phenomenon can be modeled using either a sine or cosine function. The key is understanding that cosine and sine functions are phase shifts<\/strong> of each other:\r\n\r\n[latex]\\cos(x) = \\sin(x + \\frac{\\pi}{2})[\/latex] and [latex]\\sin(x) = \\cos(x - \\frac{\\pi}{2})[\/latex].\r\n\r\n \r\n\r\nWhen modeling real-world periodic behavior, you can choose whichever function makes the mathematics simpler or aligns better with your reference point.\r\n\r\n<\/section>\r\n

Both sine and cosine functions have the same basic shape - they're both sinusoidal. The only difference is their starting point:<\/p>\r\n\r\n

    \r\n \t
  • Cosine<\/strong> starts at its maximum value when the input is 0<\/li>\r\n \t
  • Sine<\/strong> starts at its middle value (crossing the axis) when the input is 0<\/li>\r\n<\/ul>\r\n
    \r\n

    Suppose the temperature in Phoenix, Arizona follows this pattern:<\/p>\r\n\r\n

      \r\n \t
    • Maximum temperature: 85\u00b0F at 3:00 PM ([latex]t = 15[\/latex] hours)<\/li>\r\n \t
    • Minimum temperature: 65\u00b0F at 3:00 AM ([latex]t = 3[\/latex] hours)<\/li>\r\n \t
    • The pattern repeats every 24 hours<\/li>\r\n<\/ul>\r\n

      Let's find both sine and cosine functions to model this behavior.<\/p>\r\n

      Step 1: Identify the parameters<\/p>\r\n\r\n

        \r\n \t
      • Amplitude: [latex]A = \\frac{85 - 65}{2} = 10[\/latex]<\/li>\r\n \t
      • Midline: [latex]D = \\frac{85 + 65}{2} = 75[\/latex]<\/li>\r\n \t
      • Period = 24, so [latex]B = \\frac{2\\pi}{24} = \\frac{\\pi}{12}[\/latex]<\/li>\r\n<\/ul>\r\n

        Step 2: Write the cosine function<\/p>\r\n

        Since cosine starts at its maximum, and our maximum occurs at [latex]t = 15[\/latex]: [latex]T(t) = 10 \\cos\\left(\\frac{\\pi}{12}(t - 15)\\right) + 75[\/latex]<\/p>\r\n

        Step 3: Write the equivalent sine function<\/p>\r\n

        For sine to reach maximum at [latex]t = 15[\/latex], we need: [latex]\\sin\\left(\\frac{\\pi}{12}(t - C)\\right) = 1[\/latex] This happens when [latex]\\frac{\\pi}{12}(t - C) = \\frac{\\pi}{2}[\/latex] So: [latex]t - C = 6[\/latex], which means [latex]C = 15 - 6 = 9[\/latex]<\/p>\r\n

        [latex]T(t) = 10 \\sin\\left(\\frac{\\pi}{12}(t - 9)\\right) + 75[\/latex]<\/p>\r\n

        Verification: Both functions give [latex]T(15) = 85\u00b0F[\/latex] and [latex]T(3) = 65\u00b0F[\/latex]<\/p>\r\n\r\n<\/section>

        \r\n

        Question Help: Converting Between Sine and Cosine<\/strong><\/p>\r\n

        When you have one trigonometric function and need to write the equivalent using the other:<\/p>\r\n\r\n

          \r\n \t
        1. From cosine to sine: Replace cos with sin and adjust the phase shift by subtracting [latex]\\frac{1}{4}[\/latex] the period from the argument<\/li>\r\n \t
        2. From sine to cosine: Replace sin with cos and adjust the phase shift by adding latex]\\frac{1}{4}[\/latex] to the argument<\/li>\r\n \t
        3. Check your work: Both functions should give the same values at key points (maximum, minimum, zeros)<\/li>\r\n<\/ol>\r\n<\/section>
          \r\n

          A Ferris wheel has a diameter of 50 feet with its center 30 feet above the ground. The wheel completes one revolution every 8 minutes. If a rider starts at the bottom of the wheel when t = 0:<\/p>\r\n\r\n

            \r\n \t
          1. Write a cosine function h(t) to model the rider's height above ground.<\/li>\r\n \t
          2. Write an equivalent sine function for the same motion.<\/li>\r\n<\/ol>\r\n

            Answer: [latex]h(t) = -25 \\cos\\left(\\frac{\\pi}{4}t\\right) + 30[\/latex] and [latex]h(t) = 25 \\sin\\left(\\frac{\\pi}{4}(t - 2)\\right) + 30[\/latex]<\/p>\r\n\r\n<\/section>

            When modeling real-world periodic behavior, choose the function (sine or cosine) that makes your reference point most natural:\r\n
              \r\n \t
            • Use cosine if your phenomenon starts at a maximum or minimum value<\/li>\r\n \t
            • Use sine if your phenomenon starts at the middle value (crossing the midline)<\/li>\r\n<\/ul>\r\nThis choice can make your phase shift calculation much simpler!\r\n\r\n<\/section>
              \r\n

              A sound wave oscillates with amplitude 0.5 units, frequency 440 Hz glossary: hertz, a unit measuring cycles per second, and passes through the origin [latex](0,0)[\/latex]\u00a0while increasing.<\/p>\r\n

              [reveal-answer q=\"686192\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"686192\"]Cosine approach: Since the wave passes through [latex](0,0)[\/latex] going up, and cosine starts at maximum, we need a negative cosine with appropriate shifting: [latex]y = -0.5 \\cos(2\\pi(440)t) + 0 = -0.5 \\cos(880\\pi t)[\/latex] Sine approach: Since sine naturally passes through latex[\/latex] going up: [latex]y = 0.5 \\sin(880\\pi t)[\/latex] Notice: The sine form is much simpler for this scenario![\/hidden-answer]<\/p>\r\n\r\n<\/section>","rendered":"

              Modeling the same periodic function with sine and cosine<\/h2>\n
              \n

              sine vs. cosine<\/h3>\n

              Any periodic phenomenon can be modeled using either a sine or cosine function. The key is understanding that cosine and sine functions are phase shifts<\/strong> of each other:<\/p>\n

              [latex]\\cos(x) = \\sin(x + \\frac{\\pi}{2})[\/latex] and [latex]\\sin(x) = \\cos(x - \\frac{\\pi}{2})[\/latex].<\/p>\n

               <\/p>\n

              When modeling real-world periodic behavior, you can choose whichever function makes the mathematics simpler or aligns better with your reference point.<\/p>\n<\/section>\n

              Both sine and cosine functions have the same basic shape – they’re both sinusoidal. The only difference is their starting point:<\/p>\n

                \n
              • Cosine<\/strong> starts at its maximum value when the input is 0<\/li>\n
              • Sine<\/strong> starts at its middle value (crossing the axis) when the input is 0<\/li>\n<\/ul>\n
                \n

                Suppose the temperature in Phoenix, Arizona follows this pattern:<\/p>\n

                  \n
                • Maximum temperature: 85\u00b0F at 3:00 PM ([latex]t = 15[\/latex] hours)<\/li>\n
                • Minimum temperature: 65\u00b0F at 3:00 AM ([latex]t = 3[\/latex] hours)<\/li>\n
                • The pattern repeats every 24 hours<\/li>\n<\/ul>\n

                  Let’s find both sine and cosine functions to model this behavior.<\/p>\n

                  Step 1: Identify the parameters<\/p>\n

                    \n
                  • Amplitude: [latex]A = \\frac{85 - 65}{2} = 10[\/latex]<\/li>\n
                  • Midline: [latex]D = \\frac{85 + 65}{2} = 75[\/latex]<\/li>\n
                  • Period = 24, so [latex]B = \\frac{2\\pi}{24} = \\frac{\\pi}{12}[\/latex]<\/li>\n<\/ul>\n

                    Step 2: Write the cosine function<\/p>\n

                    Since cosine starts at its maximum, and our maximum occurs at [latex]t = 15[\/latex]: [latex]T(t) = 10 \\cos\\left(\\frac{\\pi}{12}(t - 15)\\right) + 75[\/latex]<\/p>\n

                    Step 3: Write the equivalent sine function<\/p>\n

                    For sine to reach maximum at [latex]t = 15[\/latex], we need: [latex]\\sin\\left(\\frac{\\pi}{12}(t - C)\\right) = 1[\/latex] This happens when [latex]\\frac{\\pi}{12}(t - C) = \\frac{\\pi}{2}[\/latex] So: [latex]t - C = 6[\/latex], which means [latex]C = 15 - 6 = 9[\/latex]<\/p>\n

                    [latex]T(t) = 10 \\sin\\left(\\frac{\\pi}{12}(t - 9)\\right) + 75[\/latex]<\/p>\n

                    Verification: Both functions give [latex]T(15) = 85\u00b0F[\/latex] and [latex]T(3) = 65\u00b0F[\/latex]<\/p>\n<\/section>\n

                    \n

                    Question Help: Converting Between Sine and Cosine<\/strong><\/p>\n

                    When you have one trigonometric function and need to write the equivalent using the other:<\/p>\n

                      \n
                    1. From cosine to sine: Replace cos with sin and adjust the phase shift by subtracting [latex]\\frac{1}{4}[\/latex] the period from the argument<\/li>\n
                    2. From sine to cosine: Replace sin with cos and adjust the phase shift by adding latex]\\frac{1}{4}[\/latex] to the argument<\/li>\n
                    3. Check your work: Both functions should give the same values at key points (maximum, minimum, zeros)<\/li>\n<\/ol>\n<\/section>\n
                      \n

                      A Ferris wheel has a diameter of 50 feet with its center 30 feet above the ground. The wheel completes one revolution every 8 minutes. If a rider starts at the bottom of the wheel when t = 0:<\/p>\n

                        \n
                      1. Write a cosine function h(t) to model the rider’s height above ground.<\/li>\n
                      2. Write an equivalent sine function for the same motion.<\/li>\n<\/ol>\n

                        Answer: [latex]h(t) = -25 \\cos\\left(\\frac{\\pi}{4}t\\right) + 30[\/latex] and [latex]h(t) = 25 \\sin\\left(\\frac{\\pi}{4}(t - 2)\\right) + 30[\/latex]<\/p>\n<\/section>\n

                        When modeling real-world periodic behavior, choose the function (sine or cosine) that makes your reference point most natural:<\/p>\n
                          \n
                        • Use cosine if your phenomenon starts at a maximum or minimum value<\/li>\n
                        • Use sine if your phenomenon starts at the middle value (crossing the midline)<\/li>\n<\/ul>\n

                          This choice can make your phase shift calculation much simpler!<\/p>\n<\/section>\n

                          \n

                          A sound wave oscillates with amplitude 0.5 units, frequency 440 Hz glossary: hertz, a unit measuring cycles per second, and passes through the origin [latex](0,0)[\/latex]\u00a0while increasing.<\/p>\n

                          \n