Confusion with modeling a trigonometric function with phase shiftI am studying trigonometry on Khan Academy and came across this problem:The moon's distance from Earth varies in a periodic way that can be modeled by a trigonometric function.When the moon is at its perigee (closest point to Earth), it's about 363,000 km away. When it's at its apogee (farthest point from Earth), it's about 406,000 k away. The moon's apogees occur 27.3 days apart. The moon will reach its apogee on January 22, 2016. Find the formula of the trigonometric function that models the distance D between Earth and the moon t days after January 1, 2016. Define the function using radians. D ( t ) =So the steps I took are:Finding the amplitude: ( 406000 − 36300 ) / 2 = 21500Finding the midline: 21500 + 36300 = 384500Figuring out whether to use cosine or sine:I figured that I can treat January 2nd as the beginning of the year. So I used cosine. Since at 0, a cosine function is at its max value.4. The period: 1 year is a period so it must be 2 π / 3655. The function without the shift is now: 21500 cos ⁡ ( 2 π / 365 t ) + 3845006. Now I must find the value of u in order to properly shift the function. I imagine that this must be t − 27.3 since it is 27.3 days after January 1.I feel like I must be missing something here or got one of the steps wrong. Please guide me in the right direction.

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Confusion with modeling a trigonometric function with phase shiftI am studying trigonometry on Khan Academy and came across this problem:The moon&#039;s distance from Earth varies in a periodic way that can be modeled by a trigonometric function.When the moon is at its perigee (closest point to Earth), it&#039;s about 363,000 km away. When it&#039;s at its apogee (farthest point from Earth), it&#039;s about 406,000 k away. The moon&#039;s apogees occur 27.3 days apart. The moon will reach its apogee on January 22, 2016. Find the formula of the trigonometric function that models the distance D between Earth and the moon t days after January 1, 2016. Define the function using radians.  D  (  t  )  =So the steps I took are:Finding the amplitude:   (  406000  −  36300  )      /    2  =  21500Finding the midline:   21500  +  36300  =  384500Figuring out whether to use cosine or sine:I figured that I can treat January 2nd as the beginning of the year. So I used cosine. Since at 0, a cosine function is at its max value.4. The period: 1 year is a period so it must be   2  π      /    3655. The function without the shift is now:   21500  cos  ⁡  (  2  π      /    365  t  )  +  3845006. Now I must find the value of u in order to properly shift the function. I imagine that this must be   t  −  27.3 since it is 27.3 days after January 1.I feel like I must be missing something here or got one of the steps wrong. Please guide me in the right direction.

kuthiwenihca · Accepted Answer

Step 1First the period is wrong. It&#039;s not 1 year. You are given that it&#039;s 27.3 days. And the phase you get from knowing that an apogee is on January 22nd, which is 21 days after January 1st. So  d  =  21500  cos  ⁡  (  2  π      /    27.3  (  t  −      t    0    )  )  +  384500Step 2So knowing when you have the maximum (at 21 days), that&#039;s the phase. Just to check, plug in   t  =  21 in the above equation.

Kyler Oconnor · Accepted Answer

Step 1  cos  ⁡  t   is the same as   sin  ⁡  (  t  +            π      2        )  .    sin  ⁡  t   is the same as   cos  ⁡  (  t  −            π      2        )  .The sine of one number is the cosine of another, so it can be done either way with different phase shifts. The sine and cosine functions are just phase-shifted versions of each other.The cosine function reaches its peak when its argument is 0, so you can use a cosine function of the difference between any point in time and January 22.The period is 27.3 days, so the argument to the cosine function must increase by   2  π every time 27.3 days pass. If t is time measured in days, then (  (  t  −  January 22  )) is how many days have passed since January 22, and (  (  t  −  January 22  ))/27.3 is how many 27.3-day periods have passed since then, so   2  π  ⋅  (  t  −  January 22  )      /    27.3 increases by   2  π every time a 27.3-day period passes. Thus you need the cosine evaluated at that argument. Thus  384500  +  21500  cos  ⁡      (                  2        π        (        t        −        Jannuary 22        )            27.3        )  which is the same as                          384500        +        21500        sin        ⁡                  (                                    2              π              (              t              −              Jannuary 22              )                        27.3                    +                      π            2                    )                                    =                                    384500        +        21500        sin        ⁡                  (                                    2              π              (              t              −              (              somewhat earlier date              )              )                        27.3                    )                    Step 2That &quot;somewhat earlier date&quot; must be 27.3/4 days before January 22, i.e. a quarter of a full period earlier, since             π      2       is a quarter of   2  π  .The fraction of a day expressed by “⋯.3&#039;&#039; in “27.3&#039;&#039; represents   0.3   days  =  7.2   hours  .So                          27.3                  /                4         days        =        6.825         days                            =                                    7         days minus         7.2         hours                            =                                    7         hours and 12 minutes        .            So for example, if 12:00 noon on January 22 is when apogee occurs, then 7:12 pm on January 15.

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