Sinusoidal Function Projects

 Apply the Sinusoidal Function

The objective of this short project is to use apply knowledge and skills of trigonometric functions to a real life context.

There are four areas of inquiry suggested in the table below. Each area of inquiry will yield a graph that is periodic and has very similar properties to the sine (or cosine) curve. Therefore an equation of the form y=a\sin (b(x-c)^{\circ})+d will be a good choice of curve to model the data.

Choose one area of inquiry from the four listed below, or pursue your own. Using the values in the data set to calculate parameters a and d, and using the period of the curve to calculate b and if necessary c, find an equation of the form that models the data well.

Once your model is made, what can it be used for? In what way or ways does the real life situation differ from your model? Can your model be improved in any way?

This is an exercise in calculating the parameters a, b, c and d.  I may be interesting to see how your curve compares to one generated by geogebra (‘fitsin’ command); or desmos (regression).

Note that data can often be imported directly by copy, paste to geogebra spreadsheet/desmos table, or via an excel spreadsheet/google sheet.

Graphing on geogebra classic.  Graphing on desmos.

Title Description
Visible area of the moon For your time zone, copy two months worth of data. Find the parameters of a sine curve that model the data. How can you use your curve to make predictions?

How much moon will be visible on your birthday this year, according to your model? According to the data table?

The London Eye The London Eye is 135 m tall, has a diameter of 120 m and takes 30 minutes to rotate. Suppose you begin your journey at 1300 hrs. Plot points to show your height above the ground at 1305 hrs, 1310 hrs etc. Find an equation to model the motion of your capsule. Is there any reason your model is not 100% accurate?
Daylight Hours Find your GPS coordinates for your home town. Now generate a table of daylight hours for one year. Take at least two entries from the table per month, and plot daylight hours over one year. Find the parameters of a sine curve to model the data. How can you use your curve to make predictions?

How will this curve change if your chosen location is further North? South? East? West?

Engine Piston to Crankshaft Consider the motion of a piston in an engine cylinder.

See an animation of the action of a piston here. A good intro on how an internal combustion engine works can be found in this series of you tube videos – the narrative is aimed at kids, but the diagrams are great – part 1; part 2; part 3. Suppose the cylinder is 82 mm in diameter, and the height displaced is 75 mm, and it rotates the crankshaft 6000 times per minute. Plot at least 8 points showing the volume of the chamber at time t, for two revolutions of the piston. Find an equation to model the points.


End of Unit!

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