Chapter 4.5, Problem 46E

To determine

To graph:The given functions.

Explanation of Solution

Given information:The functions are $f\left(x\right)=\cos x$ and $g\left(x\right)=-\sin \left(x-5\frac{\pi }{2}\right)$ .

Formula used:

The generalized equation of sine and cosine function is

  $y=A\sin \left(Bx+C\right)+D\&y=A\cos \left(Bx+C\right)+D$

Here

  $\begin{array}{l}\text{Amplitude}=Amp=\text{|}A\text{|}\\ \text{Period}\text{of}\text{function}=T=\frac{2\pi }{B}\\ \text{Horizontal/Phase}\text{shift}=Bx+C=0\to x=-\frac{C}{B}\\ \text{Vertical}\text{shift}=D\end{array}$

Graph:

The generalized equation of sine function is

  $y=A\sin \left(Bx+C\right)+D$ ......... (1)

The generalized equation of cosine function is

  $y=A\cos \left(Bx+C\right)+D$ ......... (2)

The given functions are

  $f\left(x\right)=\cos x$ ......... (3)

  $g\left(x\right)=-\sin \left(x-5\frac{\pi }{2}\right)$ .........(4)

By comparing equation (2) & (3) $\begin{array}{l}A=1,B=1\\ \\ Amp=\left|A\right|=\left|1\right|=1\\ \\ T=\frac{2\pi }{B}=2\pi \end{array}$

By comparing equation (1) & (4) $\begin{array}{l}A=-1,B=1\&C=-5\frac{\pi }{2}\\ \\ Amp=\left|A\right|=|-1|=1\\ \\ T=\frac{2\pi }{B}=\frac{2\pi }{1}=2\pi \\ \\ \text{Horizontal}\text{shift=}Bx+C=0\to x=-\frac{C}{B}\\ =x=-\frac{\left(-5\frac{\pi }{2}\right)}{1}=5\frac{\pi }{2}\end{array}$

By using the results of the table the graph of the functions can be easily made.

The graph of the function $f\left(x\right)=\cos x$ is given below.

  

The graph of the function $g\left(x\right)=-\sin \left(x-5\frac{\pi }{2}\right)$ is given below.