The tangent function is a tangent
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Have you ever wondered why the tangent trigonometry function is called a "tangent?" This applet shows where that name comes from. Move the blue point; the length of the red segment is the tangent of the red angle. Also note that the length of segment OC is the secant. |
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Some questions to think about: What happens to the tangent when [math]\theta[/math] is near 90 or 270? Why? When is the tangent positive? Negative? Zero? When is the absolute value of the tangent less than 1? Greater than 1? Equal to 1? What is the period of the tangent function? What is the relationship between the Length of BC and the slope of OC? Why? |
Parallel Parking optimization (US customary units)
In my brief moment of semi-fame, my exploration of parallel parking ended up in the New York Times 10th Annual Year In Ideas magazine [url]http://www.nytimes.com/interactive/2010/12/19/magazine/ideas2010.html#Perfect_Parallel_Parking[/url].[br]Shortly after there was a front page article in the local New Orleans Times-Picayune [url]http://www.nola.com/education/index.ssf/2011/01/offbeat_lusher_charter_school.html[/url][br][br]I certainly have to credit Professor Simon Blackburn, University of London, for his original work which sparked my interest in the subject.[br][url]https://www.ma.rhul.ac.uk/SRBparking[/url][br][br]Even though Geogebra was never mentioned by name in any of the news articles, it was central to my being able to visualize the problem better and test/confirm my equations.[br][br]This Geogebra model is in US customary units. I've also posted a metric model at [url]http://www.geogebratube.org/material/show/id/3023[/url]
Model of my formula:
This exercise is intended to find the minimal distance required for a parallel parking spot (beyond the length of the car itself). Adjust the sliders to see how the various parameters affect the outcome. Note that d_c is the distance you're allowing to exist between the parked car and the parallel curb. Setting d_c to zero would have the car positioned flush against the curb at the end of its backward motion. Setting d_c to a positive value allows the car to pull forward in order to straighten out after initially backing up. Intuitively, the larger you set d_c to be, the less distance you need to fit between the parked cars.[br]Note that the angle alpha is also critical to a good parallel parking job. Check the "optimize" box to use the value of alpha that will minimize the required parking space.[br][br]Note: Setting both d_c and alpha to zero reduces my model to Professor Blackburn's model, but I find this to be overly conservative.[br][br]Below is a model I created of Professor Blackburn's formula, which only accounts for backing into the space (but not pulling forward and ending at some allowable distance from the curb).