Chapter 11.4, Problem 76E

To determine

To write the sinusoidal function that models the temperature.

Answer to Problem 76E

  $y=13\sin \left(\frac{\pi }{2}t-\pi \right)+77$

Explanation of Solution

Given:

The given table is

  

Calculation:

Sinusoidal function:

The sine and cosine ratios as functions

These functions are the basis for study of all periodic functions

  

The general form of the sinusoidal function is

  $y=A\sin (kt+C)+h$

Amplitude is

  $\begin{array}{l}A=\frac{{90}^{\circ }-{64}^{\circ }}{2}={13}^{\circ }\\ h=\frac{{90}^{\circ }+{64}^{\circ }}{2}={77}^{\circ }\\ \frac{2\pi }{k}=4\\ k=\frac{\pi }{2}\end{array}$

The general form is

  $\begin{array}{l}y=13\sin \left(\frac{\pi }{2}t+C\right)+77\\ 64=13\sin \left(\frac{\pi }{2}t+C\right)+77\\ C={\sin }^{-1}\left(\frac{13}{13}\right)-\frac{\pi }{2}\\ C=-\pi \end{array}$

Therefore, the general form of the function is

. $y=13\sin \left(\frac{\pi }{2}t-\pi \right)+77$