Cyclical Functions: Sinusoidal
Nature behaves in a cyclical way that can be model with trigonometric functions. The seasons of the year, how the sea levels increasing every night and decrease during the day, or how much daylight a certain city in the world would get throughout the year, are all examples of sinusoidal behavior. All of these examples are cyclical in nature.
Not only nature have examples of cyclical situation, also in the business world we can find these type of behavior, for example, the swimsuit sales will peak during the summer month and decrease during the winter season, and if you own a restaurant you will notice that during lunch time you will have more clients than during the hours between breakfast and lunch.
There are many disciplines that can be used to model cyclical functions so let’s go ahead and explore cyclical functions with the following example.
For the following activity first we will need some information:
1. What is the Eye of London?
2. How many pods does it have?
3. What is its average speed?
4. What is its overall height?
5. What is its diameter?
The Eye of London as you were able to figure out is a Huge Ferris Wheel located in London, England. It has 32 passenger pods. The Ferris wheel is moving (rotating) at a speed of 26 centimeters per second. It has an overall height is 135 meters, and it is standing on top of a 15 meter high platform, therefore it has a diameter of 120 meters.
Use this information to answer the following question:
1. How high is one pod standing from the ground after 1 minute has elapsed?
To answer the question, first we need to figure out how much time it would take the Ferris wheel to complete one full rotation. So, we know the Eye of London rotates at a constant speed of 26cm/sec and that the diameter of the wheel is 120 meters, thus, we can use the diameter to find the circumference of the wheel using the formula or 377m. To find the time it takes one rotation:noticed that the speed is now in meter per second, to do this you just need to divide 26 cm by 100 to get meters. Now we can solve for x and find the time in seconds.
Since the main question is asking for minutes we need to convert seconds to minutes.
So, 1450 seconds 24.17minutes or approximately 24 minutes. To better understand the question we need to make a graphical representation of the Eye of London. There are different ways to represent this.
One way to find the answer is to use trigonometry!!! We can create a right triangle inside the circle representing the wheel.
Now, write an equation that will help you find the overall height. Compare and share your equation with your peers.
As you might have notice, there are different equations that can be used. Here is one way to do it: height = 60 - x + 15, as you can see we need to find the value of x using trigonometric functions.
Thus, to find x you must use the formula is: which is notice that the hypotenuse is the radius of the circle (120/2).
We are almost there!!! But what is the value of the angle?
Let’s think about this for a second, remember that it takes the wheel 24 minutes to complete one complete (360o) rotation, thus;If we solve for x, we will find out that the angle of rotation for 1 minute is 15o. So; ;
Finally we can use that x value to find the height; height = 60 - 57.95 + 15 = 17.05m
Thus, the height of one pod after 1 minute is 17.05m
Now, use this information to find what would be the height of the pod after 3 minutes, 6 minutes, 10 minutes, 12 minutes, 15 minutes, 22 minutes, and 24 minutes.
Use the table on the page provided.
Remember that for 3 minutes it is 45o. So you will find x by using the equation for Cos 45.
2. Draw a scatterplot with the information from your table and join the points.
3. How would the graph look if you decided to plot two rotations?
The graph you generated is called a cyclical function, to be more specific, it is a sinusoidal function. Using a graphing calculator it is possible to generate the graph and its equation. First, press STAT and Select EDIT, use L1 for the x values and L2 for the y values (use the values of two rotations). Press STAT one more time, now select CALC from the menu on the top and scroll down and select SinReg .
The calculator will generate an equation in the standard form y = asin(bx + c) + d, but to better understand thetransformation of a sinusoidal function it will be necessary to factor b, so your sinusoidal equation will be in the formof:
A = represents the Amplitude of the wave. The amplitude of the wave is the distance between the center of the wave and its maximum or minimum value.
B = represents the Frequency (how many waves per period). The frequency is equal to the quotient of 2p and the period (p = to the distance between cycles, it could be the distance between minimums or maximums).
C= represents the horizontal translation (shiftover the x axis)
D = represent the vertical translation (shift over the y axis)
Definitions:
The period (or wavelength) of fis the length of one complete cycle.
The midline is the horizontal line midway between the function’s minimum and maximum values.
The amplitude is the distance between the function’s maximum (or minimum) value and the midline.
The phase (or horizontal) shift is the number of units that the “start” of the cycle is away from being at the midline.