Chapter 10.7, Problem 1MRPS

Solution

(a)

To find: Graph the function $P=3\sin 880\pi t+4\cos 880\pi t$ in the viewing window $0\le t\le 0.005,-6\le P\le 6$ by graphing calculator.

It is observed from the graph that function attains maximum value at $t=0.002505$ which is local maxima of the function.

Given information:

The given function is:

  $P=3\sin 880\pi t+4\cos 880\pi t$

The graph by the graphing calculator in the viewing window is given below:

  

It is observed from the graph that function attains maximum value at $t=0.002505$ which is local maxima of the function.

(b)

To find: write the given model in the form of $y=a\cos b(x-h)+k$

The required function is $y=5\cos \left(x-\left(-36.86\right)\right)$ .

Given information:

The given function is:

  $P=3\sin 880\pi t+4\cos 880\pi t$

Linear combination of the sine and cosine with equal argument can be written as:

  $A\cos x+B\sin x=C\cos (x+D)\mathrm{.....}(1)$

Here $C=\sqrt{A^2+B^2}$ and $D={\tan }^{-1}\left(\frac{-B}{A}\right)$ .

Compare the given equation with equation $(1)$ gives:

  $A=4,B=3$

Substitute these values in $C=\sqrt{A^2+B^2}$ gives:

  $\begin{array}{l}C=\sqrt{3^2+4^2}\\ C=\sqrt{9+16}\\ C=\sqrt{25}\\ C=5\end{array}$

Now find $D$ by substituting the values in $D={\tan }^{-1}\left(-\frac{B}{A}\right)$ :

  $\begin{array}{l}D={\tan }^{-1}\left(-\frac{3}{4}\right)\\ D=-36.86\end{array}$

Substitute the values of $C$ and $D$ in $A\cos x+B\sin x=C\cos (x+D)$ gives:

  $4\cos x+3\sin x=5\cos (x+36.86)$

The function will be: $y=5\cos (x+36.86)$ .

(c)

To find: Graph the model found in part (b).

The graph is given.

Given information:

The given function is:

  $y=5\cos (x-36.86)$

The graph is given below:

  

It is clear from the graph in part (a) and (c) that both functions are same.

The function will be: $y=5\cos (x+36.86)$ .