Applications of Sinusoidal Functions
Applications of Sinusoidal Functions
The equation of a sinusoidal function can be determined by examining the properties of the graph. Consider the following:
• The amplitude ‘a’ is half of the distance between the maximum and minimum: [pic]
It is the distance from equilibrium to the peak.
• The value for ‘k’ can be determined by observing the period (one cycle) from peak to peak: [pic]
• The value ‘c’ is the horizontal displacement to the line/axis of equilibrium; ‘c’ is positive if it is moved up and negative if it is moved down. We can also use the equation [pic]
• There are multiple options for the phase shift ‘d’. Look to see
how far a starting point of the sine/cosine curve on the
equilibrium line has been moved to the right. Hint:
1. For applications where the function starts from an extrema (maximum or minimum), use a cosine curve.
2. For applications where the function starts from a point in the middle (at equilibrium), use a sine curve.
Example 1
Determine an equation for each sinusoidal graph below:
a) b)
Example 2
The first Ferris wheel built in 1893 had a diameter of 80 m. The base of the wheel was 5 m above the ground. It took 20 minutes to do 2 full revolutions. If Milton boarded the bottom of the wheel at 9:07, how high from the ground would he be at 9:20?
[pic]
Example 3
The top of a building designed by Fractals Incorporated sways left (-10 m) then right (10 m) of its stable equilibrium position. One full swing from equilibrium to the left, to the right then back to equilibrium takes 4 seconds.
At t= 1s, the building is at equilibrium and about to swing left. What is the displacement at the top of the building at 3.5 seconds?
HMWK: pg398 # 1, 2, 4, 5, 11, 13
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[pic]
[pic]
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