Chapter 5.3, Problem 34E

Solution

To match: The given graphs with a pair of the given equations.

The equations (a) and (g) are the correct match for the graph.

Given information:

The given graph is as follows,

  

Figure (1)

Formula used:

Use the formula for cosine of difference.

  $\cos \left(u-v\right)=\cos u\cos v+\sin u\sin v$

Calculation:

The equation of sine and cosine function with $\left|a\right|=1$ and no vertical translation is given by,

  $\begin{array}{c}y=\sin \left[b\left(x-c\right)\right]\\ y=\cos \left[b\left(x-c\right)\right]\end{array}$

In the interval $\left[0,2\pi \right)$ , there are two cycles of the graph so the period is $\frac{2\pi }{\pi }=\pi$ . Therefore, $b=2$ . So, the equations will be as follows.

  $\begin{array}{c}y=\sin \left[2\left(x-c\right)\right]\\ y=\cos \left[2\left(x-c\right)\right]\end{array}$

This result matches with (d) if only a horizontal translation is involved or matches (a) if there are both reflection across the $x$ -axis and a horizontal translation.

For (d), it can be written as $y=\sin \left[2\left(x-2.5\right)\right]$ , which does not match the graph since the horizontal translation is $2.5$ units to the right and the graph does not cross near $2.5$ .

For (a), it can be written as $y=\cos \left[-2\left(x-1.5\right)\right]$ , which match the graph since cosine curve will be reflected and the minimum is very near to the horizontal translation is $1.5$ units to the right. Therefore, it can be concluded that equation (a) is representing the graph.

So, the equation that represents the graph will be,

  $y=\cos \left(3-2x\right)$

Use the formula for cosine of difference to write the another equation as,

  $\begin{array}{c}y=\cos 3\cos 2x+\sin 3\sin 2x\\ =\cos \left(3-2x\right)\end{array}$

Therefore, the equations (a) and (g) are the correct match for the graph.