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Harmonic motion graphs may be enclosed by bounding curves. When a function has a varying amplitude , such that the amplitude rises and falls multiple times within a period, we can determine the bounding curves from part of the function.
Graph the function $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{cos}(2\pi x)\mathrm{cos}(16\pi x).$
The graph produced by this function will be shown in two parts. The first graph will be the exact function $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ (see [link] ), and the second graph is the exact function $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ plus a bounding function (see [link] . The graphs look quite different.
Access these online resources for additional instruction and practice with trigonometric applications.
Visit this website for additional practice questions from Learningpod.
Standard form of sinusoidal equation | $y=A\text{\hspace{0.17em}}\mathrm{sin}\left(Bt-C\right)+D\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}y=A\text{\hspace{0.17em}}\mathrm{cos}\left(Bt-C\right)+D$ |
Simple harmonic motion | $d=a\text{\hspace{0.17em}}\mathrm{cos}\left(\omega t\right)\text{or}d=a\text{\hspace{0.17em}}\mathrm{sin}\left(\omega t\right)$ |
Damped harmonic motion | $$f\left(t\right)=a{e}^{-c}{}^{t}\mathrm{sin}(\omega t)\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}f\left(t\right)=a{e}^{-ct}\mathrm{cos}\left(\omega t\right)$$ |
Explain what types of physical phenomena are best modeled by sinusoidal functions. What are the characteristics necessary?
Physical behavior should be periodic, or cyclical.
What information is necessary to construct a trigonometric model of daily temperature? Give examples of two different sets of information that would enable modeling with an equation.
If we want to model cumulative rainfall over the course of a year, would a sinusoidal function be a good model? Why or why not?
Since cumulative rainfall is always increasing, a sinusoidal function would not be ideal here.
Explain the effect of a damping factor on the graphs of harmonic motion functions.
For the following exercises, find a possible formula for the trigonometric function represented by the given table of values.
$x$ | $y$ |
$0$ | $-4$ |
$3$ | $-1$ |
$6$ | $2$ |
$9$ | $-1$ |
$12$ | $-4$ |
$15$ | $-1$ |
$18$ | $2$ |
$y=-3\mathrm{cos}\left(\frac{\pi}{6}x\right)-1$
$x$ | $y$ |
$0$ | $5$ |
$2$ | $1$ |
$4$ | $-3$ |
$6$ | $1$ |
$8$ | $5$ |
$10$ | $1$ |
$12$ | $-3$ |
$x$ | $y$ |
$0$ | $2$ |
$\frac{\pi}{4}$ | $7$ |
$\frac{\pi}{2}$ | $2$ |
$\frac{3\pi}{4}$ | $-3$ |
$\pi $ | $2$ |
$\frac{5\pi}{4}$ | $7$ |
$\frac{3\pi}{2}$ | $2$ |
$5\mathrm{sin}(2x)+2$
$x$ | $y$ |
$0$ | $2$ |
$\frac{\pi}{4}$ | $7$ |
$\frac{\pi}{2}$ | $2$ |
$\frac{3\pi}{4}$ | $-3$ |
$\pi $ | $2$ |
$\frac{5\pi}{4}$ | $7$ |
$\frac{3\pi}{2}$ | $2$ |
$x$ | $y$ |
$0$ | $1$ |
$1$ | $-3$ |
$2$ | $-7$ |
$3$ | $-3$ |
$4$ | $1$ |
$5$ | $-3$ |
$6$ | $-7$ |
$4\mathrm{cos}\left(\frac{x\pi}{2}\right)-3$
$x$ | $y$ |
$0$ | $-2$ |
$1$ | $4$ |
$2$ | $10$ |
$3$ | $4$ |
$4$ | $-2$ |
$5$ | $4$ |
$6$ | $10$ |
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