7.6 Modeling with trigonometric equations (Page 6/14)

Precalculus

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Bounding curves in harmonic motion

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.

Graphing an oscillating cosine curve

Graph the function $f (x) = \cos (2 π x) \cos (16 π x) .$

The graph produced by this function will be shown in two parts. The first graph will be the exact function $f (x)$ (see [link] ), and the second graph is the exact function $f (x)$ plus a bounding function (see [link] . The graphs look quite different.

Graph of f(x) = cos(2pi*x)cos(16pi*x), a sinusoidal function that increases and decreases its amplitude periodically.

Graph of f(x) = cos(2pi*x)cos(16pi*x), a sinusoidal function that increases and decreases its amplitude periodically. There is also a bonding function drawn over it in red, which makes the whole image look like a DNA (double helix) piece stretched along the x-axis.

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Media

Access these online resources for additional instruction and practice with trigonometric applications.

Visit this website for additional practice questions from Learningpod.

Key equations


Standard form of sinusoidal equation	$y = A \sin (B t - C) + D or y = A \cos (B t - C) + D$
Simple harmonic motion	$d = a \cos (ω t) or d = a \sin (ω t)$
Damped harmonic motion	$f (t) = a e^{- c}^{t} \sin (ω t) or f (t) = a e^{- c t} \cos (ω t)$

Key concepts

Sinusoidal functions are represented by the sine and cosine graphs. In standard form, we can find the amplitude, period, and horizontal and vertical shifts. See [link] and [link] .
Use key points to graph a sinusoidal function. The five key points include the minimum and maximum values and the midline values. See [link] .
Periodic functions can model events that reoccur in set cycles, like the phases of the moon, the hands on a clock, and the seasons in a year. See [link] , [link] , [link] and [link] .
Harmonic motion functions are modeled from given data. Similar to periodic motion applications, harmonic motion requires a restoring force. Examples include gravitational force and spring motion activated by weight. See [link] .
Damped harmonic motion is a form of periodic behavior affected by a damping factor. Energy dissipating factors, like friction, cause the displacement of the object to shrink. See [link] , [link] , [link] , [link] , and [link] .
Bounding curves delineate the graph of harmonic motion with variable maximum and minimum values. See [link] .

Section exercises

Verbal

Explain what types of physical phenomena are best modeled by sinusoidal functions. What are the characteristics necessary?

Physical behavior should be periodic, or cyclical.

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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.

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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.

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Explain the effect of a damping factor on the graphs of harmonic motion functions.

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Algebraic

For the following exercises, find a possible formula for the trigonometric function represented by the given table of values.


$x$	$y$
$0$	$- 4$
$3$	$- 1$
$6$	$2$
$9$	$- 1$
$12$	$- 4$
$15$	$- 1$
$18$	$2$

$y = - 3 \cos (\frac{π}{6} x) - 1$

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$x$	$y$
$0$	$5$
$2$	$1$
$4$	$- 3$
$6$	$1$
$8$	$5$
$10$	$1$
$12$	$- 3$

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$x$	$y$
$0$	$2$
$\frac{π}{4}$	$7$
$\frac{π}{2}$	$2$
$\frac{3 π}{4}$	$- 3$
$π$	$2$
$\frac{5 π}{4}$	$7$
$\frac{3 π}{2}$	$2$

$5 \sin (2 x) + 2$

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$x$	$y$
$0$	$2$
$\frac{π}{4}$	$7$
$\frac{π}{2}$	$2$
$\frac{3 π}{4}$	$- 3$
$π$	$2$
$\frac{5 π}{4}$	$7$
$\frac{3 π}{2}$	$2$

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$x$	$y$
$0$	$1$
$1$	$- 3$
$2$	$- 7$
$3$	$- 3$
$4$	$1$
$5$	$- 3$
$6$	$- 7$

$4 \cos (\frac{x π}{2}) - 3$

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$x$	$y$
$0$	$- 2$
$1$	$4$
$2$	$10$
$3$	$4$
$4$	$- 2$
$5$	$4$
$6$	$10$

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Source: OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6

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