Trigonometry

Trigonometry ... is all about triangles.

Right Angled Triangle

A right-angled triangle (the right angle is shown by the little box in the corner) has names for each side: Adjacent is adjacent to the angle "θ", Opposite is opposite the angle, and the longest side is the Hypotenuse.

Angles

Angles (such as the angle "θ" above) can be in Degrees or Radians. Here are some examples:

Angle	Degrees	Radians
Right Angle	90°	π/2
__ Straight Angle	180°	π
Full Rotation	360°	2π

"Sine, Cosine and Tangent"

The three most common functions in trigonometry are Sine, Cosine and Tangent. You will use them a lot!

They are simply one side of a triangle divided by another.

For any angle "θ":

Sine Function: sin(θ) = Opposite / Hypotenuse Cosine Function: cos(θ) = Adjacent / Hypotenuse Tangent Function: tan(θ) = Opposite / Adjacent

 

Example: What is the sine of 35°?

Using this triangle (lengths are only to one decimal place): sin(35°) = Opposite / Hypotenuse = 2.8/4.9 = 0.57...

Sine, Cosine and Tangent are often abbreivated to sin, cos and tan.

 

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Try It! Have a try! Drag the corner around to see how different angles affect sine, cosine and tangent And you will also see why trigonometry is also about circles! Notice that the sides can be positive or negative according to the rules ofcartesian coordinates. This makes the sine, cosine and tangent vary between positive and negative also.

Unit Circle What we have just been playing with is the Unit Circle. It is just a circle with a radius of 1 with its center at 0. Because the radius is 1, it is easy to measure sine, cosine and tangent.

Here you can see the sine function being made by the unit circle:

You can see the nice graphs made by sine, cosine and tangent.

Repeating Pattern

Because the angle is rotating around and around the circle the Sine, Cosine and Tangent functions repeat once every full rotation.

When you need to calculate the function for an angle larger than a full rotation of 2π (360°) just subtract as many full rotations as you need to bring it back below 2π (360°):

Example: what is the cosine of 370°?

370° is greater than 360° so let us subtract 360°

370° - 360° = 10°

cos(370°) = cos(10°) = 0.985 (to 3 decimal places)

Likewise if the angle is less than zero, just add full rotations.

Example: what is the sine of -3 radians?

-3 is less than 0 so let us add 2π radians

-3 + 2π = -3 + 6.283 = 3.283 radians

sin(-3) = sin(3.283) = -0.141 (to 3 decimal places)

Solving Triangles

A big part of Trigonometry is Solving Triangles. By "solving" I mean finding missing sides and angles.

Example: Find the Missing Angle "C"

 

It's easy to find angle C by using angles of a triangle add to 180°:

So C = 180° - 76° - 34° = 70°

 

It is also possible to find missing side lengths and more. The general rule is:

If you know any 3 of the sides or angles you can find the other 3

(except for the three angles case)

See Solving Triangles for more details.

Other Functions (Cotangent, Secant, Cosecant)

Similar to Sine, Cosine and Tangent, there are three other trigonometric functions which are made by dividing one side by another:

Cosecant Function: csc(θ) = Hypotenuse / Opposite Secant Function: sec(θ) = Hypotenuse / Adjacent Cotangent Function: cot(θ) = Adjacent / Opposite

 

Trigonometric and Triangle Identities

	The Trigonometric Identities are equations that are true for allright-angled triangles.
	The Triangle Identities are equations that are true for all triangles (they don't have to have a right angle).

 

Valence Electrons

Valence Electrons The valence electrons are the electrons in the last shell or energy level of an atom. They do show a repeating or periodic pattern. The valence electrons increase in number as you go across a period. Then when you start the new period, the number drops back down to one and starts increasing again. For example, when you go across the table from carbon to nitrogen to oxygen, the number of valence electrons increases from 4 to 5 to 6. As we go from fluorine to neon to sodium, the number of valence electrons increases from 7 to 8 and then drops down to 1 when we start the new period with sodium. Within a group--starting with carbon and going down to silicon and germanium--the number of valence electrons stays the same.       C 4 N 5 O 6 F 7 Ne 8 Na 1     Si 4               Ge 4         A quick way to determine the number of valence electrons for a representative element is to look at which group is it in.  Elements in group Ia have 1 valence electron.  Elements in group IIa have 2 valence electrons.  Can you guess how many valence electrons elements in group VIa have?  If you guessed 6 valence electrons, then you are correct!  The only group of representative elements that this method doesn't work for is group 0.  Those elements certainly have more than 0 valence electrons; in fact, all of them except for helium have 8 valence electrons.  Why doesn't helium have 8 valence electrons?  Think for a moment about how many electrons helium has - it has a total of only two electrons, so helium only has 2 valence electrons.  So generally speaking, the number of valence electrons stays the same as you go up or down a group, but they increase as you go from left to right across the periodic table. The preceding statement works very well for the representative elements, but it comes a bit short of the truth when you start talking about the transition elements.   Electrons going into the d sublevels of the transition metals complicate this pattern. In some ways these electrons behave like valence electrons. In some other ways they behave like shielding electrons, which are discussed in the next section. The first electrons into a d sublevel seem to behave more like valence electrons but the last ones seem to act more like shielding electrons, with variations along the way. Switching the order from 4s3d to 3d4s is one way to represent this.   Sc Ti V Cr Mn Fe Co Ni Cu Zn outer configuration 4s23d1 4s23d2 4s23d3 4s13d5 4s23d5 4s2 3d6 4s2 3d7 4s2 3d8 3d104s1 3d104s2 apparent valence electrons 3 2-4 2-5 2-6 2-7 2 or 3 2 or 3 2 or 3 1 or 2 2 As it turns out, the idea of valence electrons is not very useful for transition metals, at least not in a reliable, predictable way. Electron Dot Diagrams For a chemist, the valence electrons are quite possibly the most important electrons an atom has.  "Why the valence electrons?", you might ask.  Well, since the valence electrons are the electrons in the highest energy level, they are the most exposed of all the electrons ... and, consequently, they are the electrons that get most involved in chemical reactions.   Chemists use a notation called electron dot diagrams, also known as Lewis diagrams, to show how many valence electrons a particular element has.  An electron dot diagram consists of the element's symbol surrounded by dots that represent the valence electrons.  Typically the dots are drawn as if there is a square surrounding the element symbol with up to two dots per side.  (An element will never have more than eight valence electrons.) As we discussed above, you can determine how many valence electrons an element has by determining which group it is in.  What would the dot diagram for helium look like?  It has 2 valence electrons, so it should have 2 dots like this:    .He.    or    He: Example 6 in your workbook has a few more dot diagrams to study, then try your hand at the ones in Example 7. Answers for Example 7: K is in group Ia, so it has 1 valence electron (1 dot).   Al is in group IIIa, so it has 3 valence electrons (dots).  As is in group Va, so it has 5 valence electrons (dots). F is in group VIIa, so it has 7 valence electrons (dots).

Cheese rolling!!!

WELCOME to  

www.cheese-rolling.co.uk 

The Official Site Of

Cheese-Rolling At Cooper's Hill In Gloucestershire.

Cheese-Rolling at Cooper's Hill 2013

Picture Courtesy of Adam Wood

Did you compete? Did you come in the first three places? Please let us have your details for the record of the event.

Cheese Ban

News media around the world are now reporting an attempt to prevent Diana Smart from selling her Double Gloucester Cheeses which have traditionally been used in the Cheese-Rolling at Cooper's Hill.

Diana Smart will surely appreciate the publicity for her now unique product!

Rob Smart talks about police warnings against supplying the cheese.

Many More To Come?

Who could wish for more or better advertising than this just 3 days before the event?

Diana Smart at work producing her famous, Traditional, Double Gloucester Cheeses

If they can't get to buy the traditional Double Gloucester, why not try a traditional Parmesan Cheese?

The average Parmigiano-Reggiano wheel is about 18–24 cm (7.1–9.4 in) high, 40–45 cm (16–18 in) in diameter and weighs 38 kg (84 lb).

Not THAT'S some Cheese - Imagine the damage that could do!

 

Cheese-Rolling at Cooper's Hill 2013

In 2013 the Cheese-Rolling At Copper's Hill Took Place As Usual Commencing At Midday.

Richard Jefferies: 

Richard's voice was familiar to thousands of fans as the definitive voice of the Cheese Rolling at Cooper's Hill.

26th February 1940 - 19th April 2012

Seen here commentating at the 2009 Cheese Rolling at Cooper's Hill.

An enthusiastic and tireless supporter of the Cheese Rolling, his was the voice heard in the many radio, television & film interviews over the years, this in addition to being the commentator for the event each year.

Richard will be sadly missed by all who had the pleasure and honor to have known him.

---

Making the traditional Double Gloucester Cheese 

Since 1988, the cheeses have been hand-made by Mrs. Diana Smart of Churcham, using milk from her herd of Brown Swiss, Holstein and Gloucester cows.  

This is the original and Only Authoritative Site dedicated to the historic event of Cheese Rolling At Cooper's Hill In Gloucestershire.

   

This site is totally independent and exists in support of the Cheese-Rolling event and its many thousands of loyal fans.

Here you will find all the latest up-to-the-minute information you need to know about past, present and possible future events.

Have you won the event in the past? - Do we have your details?

Just email your details by clicking this link - Thank you. 

   

Press and media from time to time, ask for competitor details to enable radio and television interviews with current and past winners, would you like us to have your contact details to pass on to interested organisations?

Cooper's Hill, as viewed from the bottom and from the top. Will YOU be here this year? - Who will win?  - Will there be a new record? Cooper's Hill is just 300 Meters up this lane from the top of the car park, a very easy walk!

'Cheese Rolling in Gloucestershire - the e-book' is available to purchase and download. (Also available on CD by post)

Warren Buffett struck again in February, announcing a deal with Brazilian billionaire Jorge Paulo Lemann's 3G Capital to snap up iconic ketchup producer H.J. Heinz Co. for $23.2 billion. It wasn't the only deal for Buffett's Berkshire Hathaway lately. In November 2012 Berkshire acquired Omaha-based party supplier Oriental Trading Co. and in December 2012 the company purchased $1.2 billion of its own

Warren Buffett

Net Worth

 

$53.5 B As of March 2013

At a Glance

• CEO, Berkshire Hathaway
• Age: 83
• Source of Wealth: Berkshire Hathaway, self-made
• Residence: Omaha, NE
• Country of Citizenship: United States
• Education: Master of Science, Columbia University; Bachelor of Arts / Science, University of Nebraska Lincoln
• Marital Status: Widowed, Remarried
• Children: 3

Forbes Lists

#4 Billionaires

• #2 in United States
• #3 in 2012

#15 Powerful People

#2 Forbes 400

Steave Jobs

Steve Jobs

Deceased

At a Glance

Data from September 2011

• Source of Wealth: Apple, Pixar, self-made
• Country of Citizenship: United States
• Education: Drop Out, Reed College
• Children: 4

• Steve Jobs

  Steve Jobs was the co-founder and CEO of Apple and formerly Pixar.

  Steve Jobs was born in San Francisco, California to Joanne Simpson and a Syrian father. Paul and Clara Jobs of Mountain View, California then adopted him. In 1972, Jobs graduated from Homestead High School in Cupertino, California and enrolled in Reed College in Portland, Oregon. One semester later, he had dropped out, later taking up the study of philosophy and foreign cultures.

  Steve Jobs had a deep-seated interest in technology, so he took up a job at Atari Inc, then a leading manufacturer of video games. He struck a friendship with fellow designer Steve Wozniak and attended meetings of the “Homebrew Computer Club” with him.

  After saving up some money, Jobs took off for India in the search of enlightenment. Once he returned, he convinced Wozniak to quit his job at Hewlett Packard to join him in his venture that concerned personal computers. They sold items like scientific calculators to raise the seed capital.

  In 1976, Jobs and Wozniak founded Apple Computer in the Jobs family garage. The first personal computer was sold for $666.66. By 1980, Apple had already released three improved versions of the personal computer. It had a wildly successful IPO, which made both founders millionaires many times over.

  A tiff with the Apple’s Board of Directors and John Scully led to Jobs’ resignation. Steve Jobs decided that he wanted to change the hardware industry. The company was called NeXTStep, which produced the NeXT Computer. The machine was a commercial washout but helped with future work in object-oriented programming, PostScript, and magneto-optical devices. Jobs returned to his original company after Apple acquired NeXT in 1996.

  Steve Jobs also started Pixar, which has produced multiple blockbuster films, including Toy Story (1995); A Bug’s Life (1998); Toy Story 2 (1999); Monsters, Inc. (2001); Finding Nemo (2003); and The Incredibles (2004).

  In 2004, Jobs was diagnosed with a malignant tumor in his pancreas, which was successfully treated.

  Jobs resigned as CEO of Apple on August 24, 2011 and subsequently assumed the role of Chairman of the Board.

  On October 5, 2011, Jobs passed away.

  
  
  
  
  Read more: http://www.crunchbase.com/person/steve-jobs#ixzz2e2UI5A9S 
  
  

Forbes Lists

#39 Forbes 400 (2011)

• #42 in 2010

#110 Billionaires (2011)

#17 Powerful People (2010)

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