Chapter 7.5, Problem 38E

(a)

To determine

To find: The graph of $f$ and $g$ and intersection points graphically.

Answer to Problem 38E

The graph and intersection points are shown in Figure-(1).

Explanation of Solution

Given information:

The function is $f\left(x\right)=\sin x-1$ , $g\left(x\right)=\cos x$ with domain $\left[-2\pi ,2\pi \right]$ and range $\left[-2.5,1.5\right]$ .

Calculation:

Consider the function.

  $f\left(x\right)=\sin x-1$ and $g\left(x\right)=\cos x$ .

The graph of $f$ and $g$ is shown below.

  

Figure-(1)

The intersection points are shown in Figure-(1).

Therefore, the graph of $f$ and $g$ is shown in Figure-(1) and the intersection points are $\left(\frac{-3\pi }{2},0\right)$

  $\left(-\pi ,-1\right)$ , $\left(\frac{\pi }{2},0\right)$ and $\left(\pi ,-1\right)$ .

(b)

To determine

To find: The intersection points of $f$ and $g$ algebraically.

Answer to Problem 38E

The intersection points of $f$ and $g$ algebraically are $\left(0,\pi ,-\pi ,2\pi ,-2\pi \right)$ and $\left(\frac{\pi }{2},\frac{-3\pi }{2}\right)$ .

Explanation of Solution

Given information:

The function is $f\left(x\right)=\sin x-1$ , $g\left(x\right)=\cos x$ with domain $\left[-2\pi ,2\pi \right]$ and range $\left[-2.5,1.5\right]$ .

Calculation:

Calculate intersection points by equating $f$ and $g$ .

  $\begin{array}{l}f\left(x\right)=g\left(x\right)\\ \sin x-1=\cos x\\ \sin x-1=\pm \sqrt{1-{\sin }^{2}x}\end{array}$

Square both the sides.

  $\begin{array}{c}{\left(\sin x-1\right)}^2={\left(\pm \sqrt{1-{\sin }^{2}x}\right)}^2\\ {\sin }^{2}x-2\sin x+1=1-{\sin }^{2}x\\ \sin x\left(\sin x-1\right)=0\end{array}$

Further solve.

  $\begin{array}{c}\sin x=0\\ x=\left(0,\pi ,-\pi ,2\pi ,-2\pi \right)\end{array}$

  $\begin{array}{c}\sin x-1=0\\ \sin x=1\\ x=\left(\frac{\pi }{2},\frac{-3\pi }{2}\right)\end{array}$

Therefore, the intersection points are $\left(0,\pi ,-\pi ,2\pi ,-2\pi \right)$ and $\left(\frac{\pi }{2},\frac{-3\pi }{2}\right)$ .