The daily low temperature in Guangzhou, China, varies over time in a periodic way that can be modeled by a trigonometric function. $\newline$ The period of change is exactly $1$ year. The temperature peaks around July $26$ at $78^{\circ} \mathrm{F}$ , and has its minimum half a year later at $49^{\circ} \mathrm{F}$ . Assuming a year is exactly $365$ days, July $26$ is $\frac{206}{365}$ of a year after January $1$ . $\newline$ 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. $\newline$ $T(t)=\square$

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Q. The daily low temperature in Guangzhou, China, varies over time in a periodic way that can be modeled by a trigonometric function.

\newline

The period of change is exactly

1

year. The temperature peaks around July

26

78^{\circ} \mathrm{F}

, and has its minimum half a year later at

49^{\circ} \mathrm{F}

. Assuming a year is exactly

365

days, July

26

\frac{206}{365}

of a year after January

1

\newline

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.

\newline

T(t)=\square

Calculate Average Temperature: The average temperature is the midpoint between the maximum and minimum temperatures. Calculate the average: $(78\,^\circ\mathrm{F} + 49\,^\circ\mathrm{F}) / 2$ .
Calculate Amplitude: The amplitude is half the distance between the maximum and minimum temperatures. Calculate the amplitude: $(78\degree\text{F} - 49\degree\text{F}) / 2$ .
Calculate Angular Frequency: The period of the function is $1$ year, which is $365$ days. To find the angular frequency, use the formula $\omega = \frac{2\pi}{\text{period}}$ . Calculate $\omega$ : $\frac{2\pi}{365}.$
Calculate Phase Shift: The function peaks at July $26$ , which is $(206)/(365)$ of a year after January $1$ . To find the phase shift, use the formula $\varphi = \omega \times (\text{days after January 1})$ . Calculate $\varphi$ : $2\pi \times (206)/(365)$ .
Model Temperature Function: The trigonometric function that models the temperature is of the form $T(t) = A \cdot \cos(\omega t + \varphi) + D$ , where $A$ is the amplitude, $\omega$ is the angular frequency, $\varphi$ is the phase shift, and $D$ is the average temperature. Substitute the values calculated in the previous steps.
Adjust Phase Shift: The function is $T(t) = 14.5 \times \cos\left(\frac{2\pi}{365}t + \frac{2\pi \times 206}{365}\right) + 63.5$ . However, we need to adjust the phase shift because the cosine function peaks at $0$ , and we want it to peak at $\frac{206}{365}$ of a year. To do this, we subtract the phase shift from $\pi$ , not add it.