|Section 1 Exercises||2. The Six Trigonometric Functions||Trigonometric Functions Main Page||"RealWorld" Page||Everything for Calculus||Español|
Take a look at the following graph, which shows the approximate average daily high temperature in New York's Central Park. #
#Source: National Weather Service/The New York Times, January 7, 1996, p. 36.
Each year, the pattern repeats over and over, resulting in the following graph.
Cyclical behavior is common in the business world; just as there are seasonal fluctuations in the temperature in Central Park, there are seasonal fluctuations in the demand for surfing equipment, swimwear, snow shovels, and the list goes on. The following graph even suggests a cyclical behavior in employment at securities firms in the United States.*
* Source: Securities Industry Association/The New York Times, September 1, 1996, p. F9.
We model cyclical behavior using the sine and cosine functions. An easy way to describe these functions is as follows. Imagine a bicycle, wheel whose radius is one unit, with a marker attached to the rim of the rear wheel, as shown in the following figure.
|Press the button to bring up an animation showing $h(t)$ varying with time $t.$|
Bicycle Wheel DefinitionIf a wheel of radius $1$ unit rotates at a speed of $1$ unit of length per second, and is in the position shown in the figure at time $t = 0,$ then its height after $t$ seconds is given by
Geometric DefinitionThe sine of a real number $t$ is given by the $y-$coordinate (height) of the point $P$ in the following diagram, in which $t$ is the distance of the arc shown.
Using a calculator or a graphing calculator, plot the following pairs of graphs on the same set of axes:
(a) The usual graphing calculator format is:
The window might look like this:
(b) Here, the graphing calculator format is:
Using window coordinates
we get the following:
(c) Here, the graphing calculator format is:
Using the same window coordinates and color coding as in part (b), we obtain the following graph.
The graphs are shown in the window
Notice several things:
The following formula illustrates the result of combining some of the operations in the above example.
General Sine Curve
$C$ is the vertical offset (height of the baseline)
$P$ is the period or wavelength (the length of each cycle)
$ω$ is the angular frequency, given by $ω = 2π/P$
$α$ is the phase shift (the horizontal offset of the basepoint; where the curve crosses the baseline as it ascends)
The typical voltage V supplied by an electrical outlet in the U.S. is a sinusoidal function that oscillates between $− 165$ volts and $+165$ volts with a frequency of 60 cycles per second. Obtain an equation for the voltage as a function of time $t.$
What we are looking for is a function of the form
Referring to the above figure, let us look at the constants one-at-a-time.Amplitude $A$ and Vertical Offset $C:$ Since the voltage oscillates between $− 165$ volts and $+165$ volts, we see that $A = 165,$ and $C = 0.$ Period $P:$ Since the electric current oscillates $60$ times in one second, the length of time it takes to oscillate once is $1/60$ second. Thus, the period is $P = 1/60.$
Angular Frequency ω: This is given by the formula
See also p. 558 of Calculus Applied to the Real World or p. 1056 of Finite Mathematics and Calculus Applied to the Real World
An economist consulted by your temporary employment agency indicates that the demand for temporary employment (measured in thousands of job applications per week) in your county can be modeled by the function
where t is time in years since January, 1995. Calculate the amplitude, the vertical offset, the phase shift, the angular frequency, and the period, and interpret the results.
SolutionTo calculate these constants, first subtract $2π$ from the argument (this does not effect the value of the sine function) so that it has the form $4.3\sin(0.82t - n) + 7.3:$
|$4.3\sin(0.82t + 0.3) + 7.3 = 4.3\sin(0.82t + 0.3 - 2π) + 7.3$|
|$\approx 4.3\sin(0.82t - 5.983) + 7.3$|
(rounding to two significant digits; notice that all the terms were given to two digits.) Finally, we get the period using the formula
We can interpret these answers in the form of the following little report:
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