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I am studying trigonometry on Khan Academy and came across this problem:

The daily low temperature in Guangzhou, China, varies over time in a periodic way that can be modeled by a trigonometric function. The period of change is exactly $1$ year. The temperature peaks around July 26 at $78°F$, and has its minimum half a year later at $49°F$. Assuming a year is exactly 365 days, July 26 is $\frac { 206 }{ 365 } $ ​​ of a year after January 1. Find the formula of the trigonometric function that models the daily low temperature $T$ in Guangzhou t years after January 1, 2015. Define the function using radians.

$T(t)=$


So the steps I took are:

1) Finding the amplitude: $$\frac { 78-49 }{ 2 } =14.5$$

2) Finding the midline: $$\frac { 78+49 }{ 2 } =63.5$$

3) Figuring out whether to use cosine or sine:

I figured that I can treat July 26th as the beginning of the year and then shift the function to make it the 206th day 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 $$\frac { 2\pi }{ 365 } $$

5) The function without the shift is now: $$14.5cos(\frac { 2\pi }{ 365 } u)+63.5$$

6) Now I must find the value of $u$ in order to properly shift the function. I imagine that this must be $t-206$ since it is $206$ 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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It looks like you mostly have the right idea. The thing you forgot is that $t$ is the number of years, not days. The answer you should get is $$ 14.5 \cos \left(2 \pi \left(t - \frac{206}{365} \right)\right) + 63.5 $$

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  • $\begingroup$ can you please explain this? $\endgroup$ – Cherry_Developer Dec 1 '14 at 3:19
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    $\begingroup$ All of your problems come down to the fact that you're not keeping track of the units of $t$. Let's start here: what is the period of our function, and why is it not $365$? $\endgroup$ – Ben Grossmann Dec 1 '14 at 3:23
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    $\begingroup$ You're starting to get it, but your phrasing is strange so I'll spell it out: $T(t)$ is a function that takes the number of years since January 1, 2015 ($t$) and gives the temperature at that time. The statement "the temperature has a period of one year" is the same as saying "at every number of years $t$, $T(t+1)$ (the temperature one year after $t$) will be equal to $T(t)$ (the temperature at time $t$)". $\endgroup$ – Ben Grossmann Dec 1 '14 at 3:30
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    $\begingroup$ If $t$ measured days instead of years, then it would indeed make sense to say that the period is 365, since we would have to increase $t$ by 365 in order to get the same value for $T$. $\endgroup$ – Ben Grossmann Dec 1 '14 at 3:31
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    $\begingroup$ Exactly! you're all set. $\endgroup$ – Ben Grossmann Dec 1 '14 at 3:32

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