Chapter 7, Problem 25RE

(a)

To determine

To draw:The graph of the function 3 $f$ and

  $g$ if $f\left(x\right)=1-{\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)}^2$ and $g\left(x\right)=\sin x$ .

Answer to Problem 25RE

Thegraph of the functions $f\left(x\right)=1-{\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)}^2$ and $g\left(x\right)=\sin x$ are shown in the figure (1) and (2) respectively.

Explanation of Solution

Given information:

The functions are $f\left(x\right)=1-{\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)}^2$ and $g\left(x\right)=\sin x$ .

Graph:

Plot the graph of the function $f\left(x\right)=1-{\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)}^2$ as follows.

  

Figure (1)

Plot the graph of the function $g\left(x\right)=\sin x$ as follows.

  

Figure (2)

Interpretation:

In the figure (1), the graph of the function $f\left(x\right)=1-{\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)}^2$ is drawn. . The sine function and cosine function has a period of $2\pi$ .

In the figure (2), the graph of the function $g\left(x\right)=\sin x$ is drawn. The sine function has a period of $2\pi$ . The solution can be get by adding integer multiple of $2\pi$ in the solution.

Therefore, the graph of the functions $f\left(x\right)=1-{\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)}^2$ and $g\left(x\right)=\sin x$ are shown in the figure (1) and (2), respectively.

(b)

To determine

To check:The equation $f\left(x\right)=g\left(x\right)$ is an identity or not.

Answer to Problem 25RE

The graphs suggest that $f\left(x\right)=g\left(x\right)$ is an identity.

Explanation of Solution

Given information:

The functions are $f\left(x\right)=1-{\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)}^2$ and $g\left(x\right)=\sin x$ .

Graph:

Graph the functions $f\left(x\right)=1-{\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)}^2$ and $g\left(x\right)=\sin x$ on the same rectangle as shown below.

  

Figure (3)

As shown in the above graph, the graph of both the functions is completely overlapped. It means every point of a function is same as the corresponding point in another function. So, both the functions are effectively equal, even if their equations look different.

Therefore, the graphs suggest that $f\left(x\right)=g\left(x\right)$ is an identity.