• football trigonometry

    Posted by Curt Gebhard on 9/17/2015 11:00:00 AM

    http://thesportsquotient.com/nfl/2015/9/16/the-nfl-wining-cycle

     

    The NFL Winning Cycle

     
    Adrian Nelson - The Sports Quotient

    A look at how NFL team success compares to the business cycle.

    There are a few basic concepts that everyone who takes Econ 101 generally walks away with, and these concepts can be applied nearly everywhere — even sports! One is the business cycle. In the most basic sense the business cycle is characterized by the old adage of “what goes up must come down.”

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    The economy goes through periods of growth and decline, and NFL teams are really no different. The NFL has long been called a league of parity. Success is easier to replicate than in a sport like baseball and teams go from worst to first and first to worst seemingly every year.

    The rules and structures in place in the league create the conditions for the league to function as it does. Great teams are composed of great players and great players demand high salaries. The salary cap, along with the lack of max contracts, makes it so that when teams are chock full of a lot of amazing players, it will  be tough to re-sign them all. They will have to let some go, and in doing so, the team will most likely get worse.

    We have already started to see this devaluation with the Seattle Seahawks. After giving hefty extensions to the likes of Richard Sherman, Russell Wilson, and Earl Thomas, the team was forced to let Byron Maxwell walk to Philadelphia and are currently struggling to meet the demands of Kam Chancellor.

    In the MLB, the lack of a salary cap allows the teams that have money to spend it. If a championship team wants to retain its pieces, it can. The NBA is somewhere between the NFL and the MLB. The NBA has a cap, but with Bird Rights, teams frequently are allowed to exceed to maintain their own players, although it does come with a hefty tax.

    In the NFL, there is no such system. The NBA also has max contracts, which artificially keep the prices of stars down, and make it easier for teams to build squads composed of several superstar level players, like the Miami Heat did just a few years back.

    The NFL draft also serves as a way to equalize teams around the league. The worse you are, the better your pick is and the more likely it is that the rookies you draft will be successful in the pros. While this is true of all of the “Big Four,” the salary cap makes the impact of these cheap rookies all the more powerful and valuable.

    All the economic checks and balances are there for NFL teams to function just like the economy, and so I wanted to take a look at the results.

    For each NFL team, I looked at the last 35 years in their history. I didn’t want to go too far back, hence the 30-year time frame, and I looked at the average of sets of four years to make the data a bit smoother. So, for example, the first data point for the Cardinals averaged their records for 2011-2014, the next data point averaged 2010-2013, and so on.

    With this methodology I culled about 30 data points for each NFL team. I then fit the sine curve that was closest to the data. This allowed for me to compare certain attributes, specifically the period of each curve.

    For those of you who don’t remember your high school trigonometry (I know, first I’m throwing economics at you, and then trig), the period of the function is how long it takes for a full cycle to complete. So lower periods mean that over the last 30 years a team has displayed a pattern of cycling quickly between success and failure, like the Bengals.

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    Longer periods mean that a team seems to be going through cycles slowly. It could mean that you’ve been on top for a long time, like the Pittsburgh Steelers …

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    … or they could have been bad for a long time, like the Redskins.

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    The graphs for all of your favorite NFL teams are actually pretty cool, and can be found at the bottom of this article, but don’t scroll down just yet.

    Overall, the analysis showed which franchises have been the most stable over the past few decades and which have been on a more consistent trajectory (whether good or bad).

     

    Team Period (yrs) Team Period Team Period Team Period
    Atlanta Falcons 6.8 Arizona Cardinals 13.4 New Orleans Saints 20.7 Tampa Bay Buccaneers 31.7
    Cincinnati Bengals 8.4 San Diego Chargers 13.9 Chicago Bears 22.3 Indianapolis Colts 33.8
    Baltimore Ravens 9.0 St. Louis Rams 15.7 New York Jets 23.4 Minnesota Vikings 36.1
    Jacksonville Jaguars 9.0 Denver Broncos 15.9 Detroit Lions 24.2 Cleveland Browns 43.6
    Tennessee Titans 9.0 Seattle Seahawks 15.9 Kansas City Chiefs 25.1 Oakland Raiders 43.6
    Carolina Panthers 11.1 Green Bay Packers 16.8 Buffalo Bills 26.4 Pittsburgh Steelers 45.5
    Philadelphia Eagles 13.0 Houston Texans 16.9 San Francisco 49ers 28.3 Washington Redskins 51.5
    Dallas Cowboys 13.0 New York Giants 19.6 New England Patriots 31.3 Miami Dolphins 61.6

     

    The analysis also showed that the median cycle length for an NFL team is 20.2 seasons. So if your team has been bad for about the last ten years or so, look for the corner to be turned really soon.

    Graphs for each NFL team can be found below.

    Arizona Cardinals

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    Atlanta Falcons

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    Baltimore Ravens

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    Buffalo Bills

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    Carolina Panthers

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    Cincinnati Bengals

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    Cleveland Browns

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    Dallas Cowboys

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    Denver Broncos

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    Detroit Lions

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    Green Bay Packers

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    Houston Texans

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    Indianapolis Colts

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    Jacksonville Jaguars

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    Kansas City Chiefs

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    Miami Dolphins

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    Minnesota Vikings

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    New England Patriots

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    New Orleans Saints

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    New York GiantsImage title

    New York Jets

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    Oakland Raiders

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    Philadelphia Eagles

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    Pittsburgh Steelers

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    San Diego Chargers

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    San Francisco 49ers

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    Seattle Seahawks

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    St. Louis Rams

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    Tampa Bay Buccaneers

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    Tennessee Titans

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    Washington Redskins

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    Comments (-1)
  • sine term

    Posted by Curt Gebhard on 1/26/2015 8:00:00 AM
    http://indianexpress.com/article/opinion/columns/for-the-record-in-the-word-sine-we-see-the-interconnection-of-three-mathematical-traditions-indian-arabic-and-european-says-amartya-sen/2/
     
     

    of three mathematical traditions – Indian, Arabic and European, says Amartya Sen

    Test Memory Skills
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    Written by Amartya Sen | Posted: January 13, 2015 12:00 am | Updated: January 13, 2015 11:08 am

    a huge role of international and interregional exchange of ideas. Indian research was deeply influenced by the knowledge of foreign works on the subject, and in turn, Indian maths influenced mathematical work even in those countries, including Greece and Rome and Baghdad, from where Indians themselves had learned many things.

    Let me end with an example. The history of the term “sine” in trigonometry illustrates how we learn from each other. That trigonometric idea was well developed by Aryabhata, who called it jya-ardha, and sometimes shortened it to jya. Arab mathematicians, using Aryabhata’s idea, called it “jiba”, which is phonetically close. Jiba is a meaningless sound in Arabic, but jaib, which has the same consonants, is a good Arabic word, and since the Arabic script does not specify vowels, later generation of Arab mathematicians used the term jaib, which means a bay or a cove. Then, in 1150, when Italian mathematician Gherardo of Cremona translated the word into Latin, he used the Latin word “sinus”, which means a bay or a cove in Latin. And it is from this — the Latin sinus — that the modern trigonometric term “sine” is derived. In this one word we see the interconnection of three mathematical traditions: Indian, Arabic and European.

    - See more at: http://indianexpress.com/article/opinion/columns/for-the-record-in-the-word-sine-we-see-the-interconnection-of-three-mathematical-traditions-indian-arabic-and-european-says-amartya-sen/2/#sthash.AOF6fzsZ.dpuf

    In the word “sine”, we see interconnection of three mathematical traditions – Indian, Arabic and European, says Amartya Sen

    Test Memory Skills
    Free online assessment. Get fast,   reliable online results. Start Now. www.learningrx.com
    Ads by Google
    Written by Amartya Sen | Posted: January 13, 2015 12:00 am | Updated: January 13, 2015 11:08 am

    a huge role of international and interregional exchange of ideas. Indian research was deeply influenced by the knowledge of foreign works on the subject, and in turn, Indian maths influenced mathematical work even in those countries, including Greece and Rome and Baghdad, from where Indians themselves had learned many things.

    Let me end with an example. The history of the term “sine” in trigonometry illustrates how we learn from each other. That trigonometric idea was well developed by Aryabhata, who called it jya-ardha, and sometimes shortened it to jya. Arab mathematicians, using Aryabhata’s idea, called it “jiba”, which is phonetically close. Jiba is a meaningless sound in Arabic, but jaib, which has the same consonants, is a good Arabic word, and since the Arabic script does not specify vowels, later generation of Arab mathematicians used the term jaib, which means a bay or a cove. Then, in 1150, when Italian mathematician Gherardo of Cremona translated the word into Latin, he used the Latin word “sinus”, which means a bay or a cove in Latin. And it is from this — the Latin sinus — that the modern trigonometric term “sine” is derived. In this one word we see the interconnection of three mathematical traditions: Indian, Arabic and European.

    - See more at: http://indianexpress.com/article/opinion/columns/for-the-record-in-the-word-sine-we-see-the-interconnection-of-three-mathematical-traditions-indian-arabic-and-european-says-amartya-sen/2/#sthash.AOF6fzsZ.dpuf

    of three mathematical traditions – Indian, Arabic and European, says Amartya Sen

    Test Memory Skills
    Free online assessment. Get fast,   reliable online results. Start Now. www.learningrx.com
    Ads by Google
    Written by Amartya Sen | Posted: January 13, 2015 12:00 am | Updated: January 13, 2015 11:08 am

    a huge role of international and interregional exchange of ideas. Indian research was deeply influenced by the knowledge of foreign works on the subject, and in turn, Indian maths influenced mathematical work even in those countries, including Greece and Rome and Baghdad, from where Indians themselves had learned many things.

    Let me end with an example. The history of the term “sine” in trigonometry illustrates how we learn from each other. That trigonometric idea was well developed by Aryabhata, who called it jya-ardha, and sometimes shortened it to jya. Arab mathematicians, using Aryabhata’s idea, called it “jiba”, which is phonetically close. Jiba is a meaningless sound in Arabic, but jaib, which has the same consonants, is a good Arabic word, and since the Arabic script does not specify vowels, later generation of Arab mathematicians used the term jaib, which means a bay or a cove. Then, in 1150, when Italian mathematician Gherardo of Cremona translated the word into Latin, he used the Latin word “sinus”, which means a bay or a cove in Latin. And it is from this — the Latin sinus — that the modern trigonometric term “sine” is derived. In this one word we see the interconnection of three mathematical traditions: Indian, Arabic and European.

    - See more at: http://indianexpress.com/article/opinion/columns/for-the-record-in-the-word-sine-we-see-the-interconnection-of-three-mathematical-traditions-indian-arabic-and-european-says-amartya-sen/2/#sthash.AOF6fzsZ.dpuf
     
    Comments (-1)
  • Astronomy uses trigonometry -- yes, trig is a thing

    Posted by Curt Gebhard on 12/10/2014 9:30:00 AM
    Comments (-1)
  • When am I ever going to use trigonometry? Farming and hot air ballooning.

    Posted by Curt Gebhard on 10/26/2014
    Comments (-1)
  • Interactive Trig website

    Posted by Curt Gebhard on 10/13/2014
     
    This site has interactive demonstrations. 
     
     
    sine 30 60 90
    Check out the angles app:
     
     coterminal  reference angle
    angle - coterminal = 360 degrees                           reference angle of of 145° is 35°
     
    and the graphing app:
    graph the sine  
    Comments (-1)
  • Unit Circle Hand Trick

    Posted by Curt Gebhard on 9/21/2014
    Trigonometry Blog
     
    Here's a neat trick for remembering the (x, y) coordinates of the unit circle using just your hands.
     
     hand trick
     
     
    hand 60  
    Comments (-1)
Last Modified on September 17, 2015