Chapter 5.3, Problem 84E

To determine

For the curve whose graph is shown, find all the possible ways of expressing it as a sine curve $y=a\sin \left(x-b\right)$ or as a cosine curve $y=a\cos \left(x-b\right)$ . Explain why you think you have found all possible choices for $a$ and $b$ in each case.

Answer to Problem 84E

  $\begin{array}{l}y=5\sin \left(x-b\right) \text{ where }b=\left(2n+1\right)\pi \\ y=-5\sin \left(x-b\right) \text{ where }b=2n\pi \\ y=5\cos \left(x-b\right) \text{ where }b=\frac{\pi }{2}+\left(2n+1\right)\pi \\ y=-5\cos \left(x-b\right) \text{ where }b=\frac{\pi }{2}+2n\pi \end{array}$

  $a$ remains unchanged and $b$ represents all the possible phase shifts that express the curve.

Explanation of Solution

Calculation:

The following equations describe generalized sine and cosine curves.

  $y=a\sin k\left(x-b\right)$ and $y=a\cos k\left(x-b\right)$

Where $a$ is the amplitudeof the curve, $2\pi /k$ the period, and $b$ is the phase shift factor.

The Sine and cosine curves are collectively referred to as $\text{sinusoidal}$ which are functions whose graphs are periodic and shaped like waves.

Such curves have amplitude $a$

Which is the highest value a sinusoidal function attains along the $y-axis$ .

These functions are also periodic and repeat themselves every $2\pi /k$ , where $k>0$ . Finally, sinusoidal functions can be shifted horizontally by a factor $b$ .

Any sine curve can be represented as a cosine curve with the proper values of $b$ ,and any cosine curve can be represented as a sine function with the appropriate phase shifts.

To illustrate, consider the curve,

  

A glance at the standard form of sine and cosine functions tells us that the curve has amplitude $5$ and period $2\pi$ .

Graphing the equation of the sine function in the form $y=5\sin \left(x-b\right)$ ,where $b=\pi$ ,gives this curve.

Graphing the cosine function $y=5\cos \left(x-b\right)$ ,where $b=-\pi /2$ produces the same curve.

When plotted, all the following equations will produce the same curve.

  $\begin{array}{l}y=5\sin \left(x-b\right) \text{ where }b=\left(2n+1\right)\pi \\ y=-5\sin \left(x-b\right) \text{ where }b=2n\pi \\ y=5\cos \left(x-b\right) \text{ where }b=\frac{\pi }{2}+\left(2n+1\right)\pi \\ y=-5\cos \left(x-b\right) \text{ where }b=\frac{\pi }{2}+2n\pi \end{array}$

Hence, in these equations, $a$ remains unchanged and $b$ represents all the possible phase shifts that express the curve.