The day length in Juneau, Alaska, varies over time in a periodic way that can be modeled by a trigonometric function. Assume the length of the year (which is the period of change) is exactly 365365365 days. The longest day of the year is June 212121, and it's 1096.51096.51096, point, 5 minutes long. The shortest day of the year is half a year later, and it's 382.5382.5382, point, 5 minutes long. Note that June 212121 is 171171171 days after January 111. Find the formula of the trigonometric function that models the length LLL of the day ttt days after January 111. Define the function using radians.

L = 

376.5sin[(2πx/365)-(168 ×2π/365)] +720

Step-by-step explanation: Given that

Number of days  = 365

Then each day must be  2π /365 rads

The midline is in minutes = 12 × 60 = 720 minutes  

Amplitude = 1096.5 - 720 = 376.5

Let use the sine function

Let January 1st   =  "Day 0"

that is, where t = 0

Using the basic function   

L = 376.5 sin [  ( 2πt/ 365 ) ]     

The shortest day of the year is half a year later, and it's 382.5 minutes long. Which is on December 21

Then, we need to find the phase shift

The phase shift equation is

ps = 360 × td / p,

where ps is the phase shift in degrees, td is the time difference between waves and p is the wave period.

td = 720 - 382.5 = 337.5

Ps = (360 × 337.5)/720 = 168.75

So, the phase shift  can be expressed as  168 × 2π/ 365

Therefore, function L can be expressed as  L = 

376.5sin[(2πx/365)-(168 ×2π/365)] +720

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