Chapter 6.5, Problem 11CFU

To determine

To write: an equation of cosine function.

Answer to Problem 11CFU

  $y=\pm 0.6\cos \left(0.16\pi \theta +0.34\pi \right)+7$

Explanation of Solution

Given information:

  $\begin{array}{c}\left|A\right|=0.6\\ \text{Period}=\frac{2\pi }{k}=12.4\\ \text{Phase shift}=-\frac{c}{k}=-2.13\\ h=7\end{array}$

Calculation:

The equation of cosine function is as follows:

  $y=A\cos \left(k\theta +c\right)+h$...........(1)

Here $A$ is a coefficient whose absolute value is termed as amplitude, c is a constant, $k=\frac{2\pi }{\text{Period}}$ and h is a vertical shift.

Evaluate the values of $A\text{ and }k$ as shown below.

  $\begin{array}{c}\left|A\right|=0.6\\ A=\pm 0.6\\ \\ k=\frac{2\pi }{\text{Period}}\\ =\frac{2\pi }{12.4}\\ =0.16\pi \end{array}$

Put the value of k in the formula given below and solve c as follows:

  $\begin{array}{c}-\frac{c}{k}=-2.13\\ -\frac{c}{0.16\pi }=-2.13\\ c=\frac{-2.13\left(0.16\pi \right)}{-1}\\ =0.34\pi \end{array}$

Now put the values in equation (1) to get the equation for cosine function.

  $y=\pm 0.6\cos \left(0.16\pi \theta +0.34\pi \right)+7$

Thus, the equation is $y=\pm 0.6\cos \left(0.16\pi \theta +0.34\pi \right)+7$ .