How is chess made?
Background
Chess is a classic two person board game. It is played with specially designed pieces on a square board made up of 64 alternating light and dark squares arranged in eight rows and columns. First appearing around A.D. 600, the game steadily evolved into the modern game known today. The earliest methods of production involved carving the board and pieces out of wood or stone. Today, a variety of common modern manufacturing methods such as injection molding and lithographic printing are employed to mass produce thousands of games.
The objective of the modern chess game is to force the opponent's most important piece, the king, into checkmate. This is a position in which the king cannot be moved to avoid capture. The player with the white pieces begins the game by moving a piece to another square following the rules that govern piece movement. The players alternate moves until one player is either checkmated, resigns, or there is a draw. Thousands of books have been published relating to the strategies during the three key stages of chess, including the opening, the middle game, and the end game.
History
While the exact time and place of chess's origin is debated, most scholars believe it was developed sometime around the sixth century A.D. It is a descendant of a game called chaturanga, which was commonly played in India during that time. (Chaturanga is derived from a much older Chinese game.) The name chaturanga is a Sanskrit word that refers to the four divisions of the Indian army, including elephants, chariots, cavalry, and infantry. These pieces became the basis for the four types of pieces in the game. Two of the key similarities between chess and chaturanga is that different pieces have different powers and victory is based on what happens to the king.
During subsequent years, chaturanga spread throughout the Middle East, Asia, and Europe. Chaturanga was introduced to China around A.D. 750 and then to Korea and Japan by the eleventh century. In each of these places, it took on different characteristics. For example, Chinese chess has nine files and 10 ranks. It also has a boundary between the fifth and sixth ranks, which makes it a slower game than the Western version. In Persia, the game was called shtranj and it was in this form that it was introduced to Western Europe when the Moors invaded Spain. By the tenth century, the game was commonly played throughout Europe and Russia..
Shtranj caught the interest of philosophers, kings, poets, and other nobility, and eventually became known as the "royal game." The best players wrote down the moves of each of their games. This practice eventually led to the development of puzzles in which the solver had to find solutions, like finding checkmate in a specific number of moves. During the fifteenth century some significant rule changes were made. For example, castling was introduced, as was the initial two-square pawn advance. One of the most important changes was the transformation of the counselor piece into the queen, the strongest chess piece. These improvements helped make the game popular throughout Europe. Some of the best players during this time—Ruy Lopez and Damiano—put together chess instruction books that also helped to make the game more widely accepted.
The rules and piece design steadily evolved, reaching the current standard during the early nineteenth century. In the twentieth century, chess experienced a tremendous growth in interest resulting in the development of various chess organizations and the crowning of a world champion. The first computer chess program was introduced in 1960. Steady improvements in technologies and algorithms led to the 1996 defeat of the world champion, Garry Kasparov, by a computer called Deep Blue.
Design
Historically, the game's pieces have been both simple and highly decorated. Prior to A.D. 600, the pieces were plain. These were replaced by detailed sets depicting royalty, warriors, and animals. From the ninth to the twelfth centuries, Islamic rules prohibiting the depiction of living creatures resulted in basic pieces made from clay or stone. This change is actually thought to have increased interest in the game at the time because it made sets more widely available and was less distracting to the players. When the game spread to Europe and Russia, highly ornate sets were fashionable.
The standard set for modern chess pieces was introduced by Nathaniel Cook in 1835. His set was patented in 1849 and endorsed by the leading player of the day, Howard Staunton. Staunton's promotion of the set as the standard led to it being known as the Staunton pattern. Today, only Staunton sets are allowed in official international competitions.
A typical chess set has 32 pieces. These are broken down into two sets of 16 pieces each. In each set there are eight pawns, two rooks, two knights, two bishops, one queen, and one king. The different pieces are distinguished by their appearance. The designs vary from simple plastic shapes to intricate, hand-carved statues. While piece size varies depending on the specific set, the tallest piece is typically the king, followed closely in height by the queen. The shortest, least notable pieces are the pawns. The rook has varied considerably over the years, being represented as a ship, castle turret, or a warrior in a chariot.
The chess board is square and made up of 64 alternating light and dark squares arranged in eight rows and columns. The vertical columns extending from one player to the other are known as files. The opposite rows are called ranks.
An important aspect of large scale chess piece manufacture is the process of designing the mold. A mold is a cavity machined from steel. When liquid plastic or molten metal is injected into the mold, it takes on the inverse of the mold's shape when it cools. This results in a finished piece. The mold cavity is highly polished because any flaw can result in a flawed final piece. For making chess game pieces, a two part mold can be used. To make the piece, the two mold sections are joined together and injected with the base raw material. The mold is then opened and the piece drops out. Special release agents and a tapered design help make the parts easier to remove. When molds are designed they are made slightly larger to compensate for the fact that plastic shrinks while it cools.
Raw Materials
Chess sets have been made with a number of raw materials over the years. Materials as diverse as ivory, glass, wood, clay, pewter, stone, and various metals have been used. Today, the most widely available chess sets are made of plastic. Plastic is a mixture of high molecular weight polymers and various fillers. For a plastic to be suitable in chess-piece manufacture it must be easily colored and heat stable, and have good impact strength. The most often used plastics are thennoset plastics such as polymethyl methacrylate (PMMA).
Polymers found in plastics are typically colorless, so colorants are added to make the chess pieces look more appealing. Colorants include soluble dyes or comminuted pigments. Titanium dioxide can be used for white colored pieces. For more ornamental sets, other inorganic materials such as iron oxides can be used to produce yellow, red, black, brown, and tan pieces.
Various filler materials are added to the plastics to produce durable, high quality pieces. For manufacturing ease, plasticizers are often added to the plastic. Plasticizers are nonvolatile solvents that increase the flexibility of the polymer. To improve the overall properties of the plastic, reinforcement materials such as fiberglass may be added. Other additives include ultraviolet (UV) protectors, heat stabilizers, antioxidants, and manufacturing aids.
The Manufacturing Process
The basic steps involved in the creation of a chess game include creating the mold for the pieces, producing the pieces, producing the board, and final assembly. The following manufacturing procedure represents a method that mass producers of the game might use. Some shops still make their sets by hand, a time consuming process that involves carving the pieces from the raw material.
Making the pieces
In the earliest phase of manufacture, designs for the chess set pieces are drawn out on a board and used as a guide in making the molds. Pieces are then handmade, typically starting by making the general outline of the piece with a wire frame. Clay is then molded around the frame and shaped to look exactly like the desired piece.
When the clay model hardens, a plaster mold of it is produced. From this mold, a steel die (or mold) is then machined, which will allow the exact duplication of the clay model. In some cases, a set of steel molds are connected together so that the whole set of chess pieces can be made in a single injection molding step.
With the steel molds made, plastic pellets are transformed into chess game pieces using injection molding. In this process, pellets are put into a hopper connected to the injection molding machine. They are forced through a high-pressure screw and melted. The screw is turned, forcing the melted plastic through a nozzle and into the mold. Just before the plastic is injected, the two halves of the mold are brought together to form the shape of the chess piece. Inside the mold, the plastic is held under pressure for a set amount of time and then allowed to cool. As it cools, the plastic hardens, the mold is opened, and the chess game piece is ejected. The mold then closes again and the process begins again.
Making the board
The construction of the board depends on the starting raw material. Wood and stone sets are cut or carved to specifications. For mass produced sets, the main raw material for the board is cardboard. The cardboard is first cut in a square to the exact dimensions desired, and then it is printed. The printing process involves a printing press fitted with plates. When the press is turned on, the plate passes under a roller and gets coated with water. An ink roller is passed over the plate and ink attaches to the plate in specific printable spots.
Ink is transferred from the plate to a rubber roller. The rubber roller is passed over the cardboard, which causes a transfer of ink. The cardboard is then passed to the next roller assembly where the next color is added by a similar process. The ink is specially formulated so that it dries before it enters the next roller assembly. This process of wetting, inking, and printing allows for continuous manufacture of printed chess boards. After all the printing is done, a special clear polymer coating may be applied to protect it and give it a glossy look.
Final assembly
To finish production of a game set, all the different components are brought to the packaging area. The exact package depends on the final design, however, in most cases the pieces are put into a box along with the board. During this stage, instruction sheets or other booklets are also put in the box. It is then taken by conveyor to a shrink-wrapping machine..
On the shrink-wrap machine, the box is loosely wrapped in a thin plastic film. It is then passed through a heating device that shrinks the film and wraps the box tightly. The boxes are then put into cases and stacked on pallets. They are transferred to trucks that deliver them to local sales outlets.
Quality Control
The quality of the chess game parts are checked during each phase of manufacture. Line inspectors check the plastic parts to ensure they meet size, shape, and consistency specifications. The primary test method is typically visual inspection. When a damaged plastic part is found, it is set aside to be melted again and reformed into a new chess game piece.
The Future
The future of chess sets is likely to involve the improvement of computerized chess sets. Currently, many manufacturers produce single person, computerized games that allow the player to compete against a computer. In the years to come, these computer chess games are likely to become more sophisticated, challenging even the best players in the world. In addition to the current game, variations have been developed. Future chess sets may involve multiple levels in which pieces will be able to attack not only forward and backward, but also up and down. New board shapes have already been introduced making it possible for up to four players to be involved in a game at once.

NSTP

First Aid Emergency Information
During Your RV Travelling

There are two kinds of first aid: the emergency, life-saving measures you must take to aid a seriously ill or injured person before you can get medical help, and the home treatment of minor injuries. The objectives of first aid are to save life, prevent further injury, and limit infection. However, first aid isn't a substitute for proper medical treatment. Accidents and emergencies are always unexpected and frightening for most of us and when you're travelling in your recreational vehicle RV you have to ask yourself: Are you prepared? The more you know about first aid, the more effective you will be. Learn simple and efficient ways to handle some of the more common accidents, injuries or other common situations that could arise while in your recreational vehicle. How to perform first aid? The following are basic guidelines for how to perform first aid: The most important rule is not to panic. Many people learn first aid and are then too frightened to use it when it becomes necessary. STOP - Stop, Think, Observe and Plan It is important that you calmly take in what you see and form a plan based on the available information. CHECK, CALL, CARE First, the scene must be checked for safety, and then the victim must be checked for signs and symptoms. Next, professionals must be called to help, and then first aid is given as much as it is practical. FIRST AID KIT RESCUE BREATHING CPR

Allergic shock Animal Bites Black Eye Bleeding Bruises Burns Choking Cuts, Scrapes & Punctures Dehydration Diarrhea Dislocation Drowning Electric Shock Eye Injuries Fainting Fever Food Poisoning Fractures Frostbites Heatstroke Hyperventilation Hypothermia Insects Stings Nosebleeds Poison Ivy Snake Bites Splinters Sprains Strains Stomach Pain Sunburn Swallowed Objects Unconsciousness

Bookkeeping

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Bookkeeping (also book-keeping or book keeping) is the recording of all financial transactions undertaken by an individual or organization. The organization may be a business, a charitable organization or even a local sports club. Bookkeeping is "keeping records of what is bought, sold, owed, and owned; what money comes in, what goes out, and what is left." ^[1] A financial transaction is any event that involves money.

Individual and family bookkeeping involves keeping track of income and expenses in a cash account record, checking account register, or savings account passbook. Individuals who borrow or lend money also track how much they owe to others or are owed from others.

Bookkeeping may be performed using paper and a pen or pencil. With increasing complexity in tax regulations and to minimize calculation errors, many organizations use accounting software to assist in bookkeeping.

Two common bookkeeping methods used by businesses and other organizations are the single-entry bookkeeping system and the double-entry bookkeeping system. Single-entry bookkeeping uses only income and expense accounts, recorded primarily in a "Revenue and Expense Journal". Single-entry bookkeeping is adequate for many small businesses. Double-entry bookkeeping requires posting (recording) each transaction twice, using debits and credits.^[2]

A bookkeeper (or book-keeper), sometimes called an accounting clerk in the United States, is a person who records the day-to-day financial transactions of an organization.^[3] A bookkeeper is usually responsible for writing up the "daybooks". The daybooks consist of purchase, sales, receipts and payments. The bookkeeper is responsible for ensuring that all transactions are recorded in the correct daybook, suppliers ledger, customer ledger and general ledger. The bookkeeper brings the books to the trial balance stage. An accountant may prepare the profit and loss statement and balance sheet using the trial balance and ledgers prepared by the bookkeeper.

Contents [hide] 1 Single account bookkeeping 2 Single-entry bookkeeping 3 Double-entry bookkeeping 4 Computerised bookkeeping 4.1 Online bookkeeping 5 Notes and references

[edit] Single account bookkeeping

Simple bookkeeping for individuals and families involves recording income, expenses and current balance in a cash record book or a checking account register.

Sample checking account register (United States, 2003)^[4]

¤AD-Automatic Deposit ¤AP-Automatic Payment ¤ATM-Teller Machine ¤DC-Debit Card
NUMBER OR CODE	DATE	TRANSACTION DESCRIPTION	PAYMENT AMOUNT		/	FEE	DEPOSIT AMOUNT		BALANCE
		balance forward							1331	85
AD	3/15	paycheck					1823	56	3155	41
AP	3/26	electricity	104	31					3051	10
704	3/26	car registration	58	50					2992	60
ATM	3/30	cash withdrawal	100	00		1.00			2891	60
DC	4/2	groceries	127	35					2764	25

[edit] Single-entry bookkeeping

The primary bookkeeping record in single-entry bookkeeping is the Revenue and Expense Journal, which is similar to a checking account register but allocates the income and expenses to various income and expense accounts. Separate account records are maintained for petty cash, accounts payable and receivable, and other relevant transactions such as inventory and travel expenses.

Sample revenue and expense journal for single-entry bookkeeping^[5]

No.	Date	Description	Revenue	Expense	Sales	Sales Tax	Services	Inventory	Advert.	Freight	Office Suppl	Misc
	7/13	Balance forward	1,826.00	835.00	1,218.00	98.00	510.00	295.00	245.00	150.00	83.50	61.50
1041	7/13	Printer- Advert flyers		450.00					450.00
1042	7/13	Wholesaler - inventory		380.00				380.00
1043	7/16	office supplies		92.50							92.50
--	7/17	bank deposit	1,232.00
		- Taxable sales			400.00	32.00
		- Out-of-state sales			165.00
		- Resales			370.00
		- Service sales					265.00
bank	7/19	bank charge		23.40								23.40
1044	7/19	petty cash		100.00								100.00
		TOTALS	3058.00	1,880.90	2,153.00	130.00	775.00	675.00	695.00	150.00	176.00	184.90

[edit] Double-entry bookkeeping

Main article: double-entry bookkeeping system

[edit] Computerised bookkeeping

Computerised bookkeeping removes many of the "books" that are used to record transactions and enforces double entry bookkeeping. Computer software increases the speed at which bookkeeping can be performed.

[edit] Online bookkeeping

Online bookkeeping allows source documents and data to reside in web-based applications which allow remote access for bookkeepers and accountants. Typically, a company scans its business documents and uploads them to a secure location or into an online bookkeeping application on a regular basis. This allows the bookkeeper to work remotely with these documents to update the books. Users of this technology include

Word processing

History of word processing

Word processing developed as specialised programs on mainframe computers during the 1970s as online computing with the use of personal terminal devices having keyboards and display screens became more common. These programs evolved from text based editors used by programmers and computer professionals. Microprocessors and, in the late 1970s, the ability to place intelligent devices on the desks of workers at reasonable cost including cheaper and smaller printers, led to the introduction of machines dedicated to "word processing". These were primarily aimed at typists, particularly those in centralised typing pools where other workers sent handwritten notes or dictaphone tapes to be transcribed into documents that could be printed and returned for reviewing. Considerable time saving economies were achieved by word processing operators. This resulted from:

• the faster typing speeds achieved by as a result of electronic keyboards
• the assistance of the word processing software for functions like layout and spell checking....

Commercial evolution

This evolution from typing using mechanical devices, to electronic word processing systems, to do-it-yourself PC based packages provided commercial opportunities as well as pitfalls. Companies rose, grew strong, and then declined and even disappeared as a result of the fast changes that occurred. Perhaps the best example of a company that became very successful due to specialised word processing systems was Wang Computers. Wang collapsed when it lost its revenue from word processing systems and was not able to substitute newer forms of computing quickly enough. Olivetti is another company that struggled to migrate from mechanical typewriter devices to word processing systems and then to PC computing.

The Parts of Speech

Traditional grammar classifies words based on eight parts of speech: the verb, the noun, the pronoun, the adjective, the adverb, the preposition, the conjunction, and the interjection.

Each part of speech explains not what the word is, but how the word is used. In fact, the same word can be a noun in one sentence and a verb or adjective in the next. The next few examples show how a word's part of speech can change from one sentence to the next, and following them is a series of sections on the individual parts of speech, followed by an exercise.

Books are made of ink, paper, and glue.

In this sentence, "books" is a noun, the subject of the sentence.

Deborah waits patiently while Bridget books the tickets.

Here "books" is a verb, and its subject is "Bridget."

We walk down the street.

In this sentence, "walk" is a verb, and its subject is the pronoun "we".

The mail carrier stood on the walk.

In this example, "walk" is a noun, which is part of a prepositional phrase describing where the mail carrier stood.

The town decided to build a new jail.

Here "jail" is a noun, which is the object of the infinitive phrase "to build."

The sheriff told us that if we did not leave town immediately he would jail us.

Here "jail" is part of the compound verb "would jail."

They heard high pitched cries in the middle of the night.

In this sentence, "cries" is a noun acting as the direct object of the verb "heard."

The baby cries all night long and all day long.

But here "cries" is a verb that describes the actions of the subject of the sentence, the baby.

The next few sections explain each of the parts of speech in detail. When you have finished, you might want to test yourself by trying the exercise.

Written by Heather MacFadyen

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A Brief History of the Philippines from a Filipino Perspective

Pre-Colonial Period

The oldest human fossil remains are found in Palawan, on the western fringe of the archipelago. These remains are about 30,000 years old, suggesting that the first human migrations to the islands took palce during the last Ice Age, when land bridiges connected the archipelago to mainland Asia and Borneo.

The islands were eventually inhabited by different groups that developed domestic trade linkages. The archaelogical evidence shows a rich pre- colonial culture that included skills in weaving, ship-building, mining and goldsmithing. Contact with Asian neighbors date back to at least 500 B.C. Trade linkages existed with the powerful Hindu empires in Java and Sumatra. These linkages were venues for exchanges with Indian culture, including the adoption of syllabic scripts which are still used by indigenous groups in Palawan and Mindoro. Trade ties with China were extensive by the 10th centuray A.D. while contact with Arab traders reached its peak about the 12th century. By the time the Spaniards arrived, Islam was well established in Mindanao and had started to influence groups as far north as Luzon.

Many existing health beliefs and practices in the Philippines are rooted back in the pre-colonial period. This includes magico-religious elements, such as beliefs in spirits and sorcery as causes of illness, as well as empirical aspects such as the use of medicinal plants. Archaelogical sites in the Philippines have yielded skeletal remains showing intricate ornamental dental work and the use of trephination (boring a hole into the skull as a magical healing ritual).

Today's traditional medicinal practitioners can trace their origins back to the pre-colonial period - the psychic surgeons, with their flair for drama, parallel the pre-hispanic religious practicioners (babaylan and catalonan) who also played roles as healers.

---

The Spanish Occupation

When the Spaniards arrived in the Philippines, the indios (natives) had reached different levels of political development, including simple communal groups, debt peonage (often erroneously described as slavery) and proto-feudal confederations.

The Spaniards imposed a feudal system, concentrating populations under their control into towns and estates. During the first two centuries of their occupation, the Spaniards used the Philippines mainly as a connecting point for their China-Acapulco (Mexico) trade. The country's economic backwardness was reinforced by Roman Catholicism, which was practiced in a form that retained many pre-colonial elements such as animism while incorporating feudal aspects of the colonizers' religion such as dogmatism, authoritarianism and patriarchial oppression. The Spaniards wer never able to consolidate political control over the entire archipelago, with Muslims and indigenous resisting the colonizers most effectively. Among the groups that were subjugated, there were numerous localized revolts throughout the Spanish occupation.

In the 19th century, the Philippines was opened to world trade, allowing the limited entry of liberal ideas. By the late 19th century, there was a distinct Filipino nationalist movement which erupted into a revolution in 1896, culminating with the establishment of Asia's first republican government in 1898.

Spain laid the foundation for a feudal health care system. The religious orders built charity hospitals, often next to churches, dispensing services to the indio. Medical education was not extended to the indio until late in the 19th century, through the University of Santo Tomas. This feudal system of the rich extending charity to the poor persists to this day among many church-run as well as non-sectarian institutions.

---

The U.S. Occupation (1898-1946)

The first Philippine Republic was short-lived. Spain had lost a war with the United States. The Philippines was illegally ceded to the United States at the Treaty of Paris for US$20 million, together with Cuba and Puerto Rico.

A Filipino-American War broke out as the United States attempted to establish control over the islands. The war lasted for more than 10 years, resulting in the death of more than 600,000 Filipinos. The little-known war has been described by historians as the "first Vietnam", where US troops first used tactics such as strategic hamleting and scorched-earth policy to "pacify" the natives.

The United States established an economic system giving the colonizers full rights to the country's resources. The Spanish feudal system was not dismantled; in fact, through the system of land registration that favored the upper Filipino classes, tenancy became more widespread during the US occupation. A native elite, including physicians trained in the United States, was groomed to manage the economic and political system of the country. The U.S. also introduced western modells of educational and health-care systems which reinforced elitism and a colonial mentality that persists to this day, mixed with the Spanish feudal patron-client relationship.

Militant peasant and workers' groups were formed during the U.S. occupation despite the repressive situation. A movement for Philippine independence, involving diverse groups, continued throughout the occupation. A Commonwealth government was established in 1935 to allow limited self-rule but this was interrupted by the Second World War and the Japanese occupation. The guerilla movement against Japanese fascism was led mainly by socialists and communists, known by their acronym, HUKS.

Shortly after the end of the Second World War, flag independence was regained although the U.S. imposed certain conditions, including the disenfranchisement of progressive political parties, the retention of U.S. military bases and the signing of economic agreements allowing the U.S. continued control over the Philippine economy.

---

The Philippine Republic (1946 - )

The political system of the Philippines was basically pattered after the U.S., with a bicameral legislature and a president elected every four years, limited to one re-election. Philippine democracy remained elitist with two political parties taking turns at the leadership. In 1972, Ferdinand Marcos declared martial law as his second term was about to end, amid a resurgence of a nationalist movement that was questioning treaties on the US military bases and the U.S. economic "parity" rights.

Political repression reached its height under Marcos. His preferential treatment for foreign investors further contributed to the deterioration of the Philippine economy, particularly with the use of government funds and foreign loans for the Marcos family and their cronies. Until the 1960s, the Philippines was economically among the most developed countries in Southeast Asia; today (1991 when this was written - Ken), it is the second poorest country in the region.

In the early years after the declaration of martial law, opposition against Marcos was spearheaded by the Left. A new Communist Party was established in 1968, followed by the New People's Army (NPA) in 1969. After Marcos's declaration of martial law in 19782, a broader political grouping called the National Democratic Front (NDF) was established with an anti-imperialist, anti-feudal and anti-fascist line. In the southern Philippines, the Muslim fought for secession through the Moro National Liberation Front (MNLF).

The assassination of Senator Benigno Aquino Jr. in 1983 precipitated an economic and political crisis that further broadened the ranks of those opposed to Marcos. Strapped for funds, the Marcos regime agreed to a "stabilization plan" from the International Monetary Fund (IMF) that plunged the economy back to 1975 levels. In February 1986, after holding blatantly fraudulent presidential elections, Marcos was overthrown by a civilian uprising supported by the military. Marcos's rival in the election, Corazon Aquino, became the new president.

The economic and political crisis in the country continues even the the restoration of formal democratic processes including the ratification of a new Constitution and the election of a Congress. The new Congress remains dominated by the elite, including former officials during the Marcos dictatorship. Economic policies remain essentially conservative with an Omnibus Investments Code that favors foreign investors and a limited land reform law. The new government has pledged to pay the entire foreign debt of US$28 billion, much of which had been incurred by Marcos under anomalous conditions. In 1990, the government agreed to another IMF stabilization plan that includes cutbacks on government budgets; reduction or elimination of subsidies and increased taxes. Graft and corruption remains endemic and has eroded support from the middle class.

The new government is essentially a fractious coalition of conservative forces representing traditional interests as exemplified by their policies on land reform, labor, foreign investments and their antagonism toward progressive groups. The perennial attempted coups by right-wing elements in the military are manifestations of power struggles among the members of the conservative elites, who ride on continuing discontent among the people brought about by the slow pace of economic and political change. Independent and progressive groups that work with peasants, workers, students and other sectors have sustained the struggle for more substantial social changes but face increasing repression, particulalrly from paramilitary (vigilante) groups formed with the tacit support of the government.

Serious questions about the dominant models of development, including those used in health care with its hospital- and doctor-centered orientation, have spurred new initiatives in health care among altlernative organizations. Community-based health programs are part of the popular movements that seek to democratize health care even as the struggle goes on for other structural reforms.

Trigonometry

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Wikibooks has a book on the topic of

Trigonometry

All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.

Trigonometry (from Greek trigōnon "triangle" + metron "measure"^[1]), informally called trig, is a branch of mathematics that deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right angled triangles). It specifically deals with the relationships between the sides and the angles of triangles; the trigonometric functions, and calculations based upon them. The insights of trigonometry permeate other branches of geometry, such as the study of spheres using spherical trigonometry. Trigonometry has important applications in many branches of pure mathematics as well as of applied mathematics and, consequently remains applicable in natural sciences. Trigonometry is usually taught in secondary schools, often in a precalculus course.

The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. Calculating the final position of the astronaut at the end of the arm requires repeated use of the trigonometric functions of those angles.

Contents [hide] 1 History 2 Overview 2.1 Extending the definitions 2.2 Mnemonics 2.3 Rule of quarters 2.4 Calculating trigonometric functions 3 Applications of trigonometry 4 Common formulae 4.1 Trigonometric identities 4.1.1 Pythagorean identities 4.1.2 Sum and product identities 4.1.2.1 Sum to product: 4.1.2.2 Product to sum: 4.1.2.3 Sine, cosine and tangent of a sum 4.1.3 Double-angle identities 4.1.4 Half-angle identities 4.2 Triangle identities 4.2.1 Law of sines 4.2.2 Law of cosines 4.2.3 Law of tangents 5 See also 6 References 7 External links

[edit] History

Table of Trigonometry, 1728 Cyclopaedia

Main article: History of trigonometric functions

Trigonometry was probably invented for use in sailing as a navigation method used with astronomy.^[2] The origins of trigonometry can be traced to the civilizations of ancient Egypt, Mesopotamia and the Indus Valley, more than 4000 years ago.^{[citation needed]} The common practice of measuring angles in degrees, minutes and seconds comes from the Babylonian's base sixty system of numeration. The Sulba Sutras written in India, between 800 BC and 500 BC, correctly computes the sine of π/4 (45°) as 1/√2 in a procedure for circling the square (the opposite of squaring the circle).

The first recorded use of trigonometry came from the Hellenistic mathematician Hipparchus[1] circa 150 BC, who compiled a trigonometric table using the sine for solving triangles. Ptolemy further developed trigonometric calculations circa 100 AD.

The ancient Sinhalese in Sri Lanka, when constructing reservoirs in the Anuradhapura kingdom, used trigonometry to calculate the gradient of the water flow. Archeological research also provides evidence of trigonometry used in other unique hydrological structures dating back to 4 BC.^[3]

The Indian mathematician Aryabhata in 499, gave tables of half chords which are now known as sine tables, along with cosine tables. He used zya for sine, kotizya for cosine, and otkram zya for inverse sine, and also introduced the versine. Another Indian mathematician, Brahmagupta in 628, used an interpolation formula to compute values of sines, up to the second order of the Newton-Stirling interpolation formula.

In the 10th century, the Persian mathematician and astronomer Abul Wáfa introduced the tangent function and improved methods of calculating trigonometry tables. He established the angle addition identities, e.g. sin (a + b), and discovered the sine formula for spherical geometry:

Also in the late 10th and early 11th centuries, the Egyptian astronomer Ibn Yunus performed many careful trigonometric calculations and demonstrated the formula

.

Indian mathematicians were the pioneers of variable computations algebra for use in astronomical calculations along with trigonometry. Lagadha (circa 1350-1200 BC) is the first person thought to have used geometry and trigonometry for astronomy, in his Vedanga Jyotisha.

Persian mathematician Omar Khayyám (1048-1131) combined trigonometry and approximation theory to provide methods of solving algebraic equations by geometrical means. Khayyam solved the cubic equation x³ + 200x = 20x² + 2000 and found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle. An approximate numerical solution was then found by interpolation in trigonometric tables.

Detailed methods for constructing a table of sines for any angle were given by the Indian mathematician Bhaskara in 1150, along with some sine and cosine formulae. Bhaskara also developed spherical trigonometry.

The 13th century Persian mathematician Nasir al-Din Tusi, along with Bhaskara, was probably the first to treat trigonometry as a distinct mathematical discipline. Nasir al-Din Tusi in his Treatise on the Quadrilateral was the first to list the six distinct cases of a right angled triangle in spherical trigonometry.

In the 14th century, Persian mathematician al-Kashi and Timurid mathematician Ulugh Beg (grandson of Timur) produced tables of trigonometric functions as part of their studies of astronomy.

The mathematician Bartholemaeus Pitiscus published an influential work on trigonometry in 1595 which may have coined the word "trigonometry".

[edit] Overview

In this right triangle: sin(A) = a/c; cos(A) = b/c; tan(A) = a/b.

By definition, one angle of a right triangle is 90 degrees. If one of the other angles is known, the third can be calculated since all three angles of any triangle must add up to 180 degrees. The shape of a right triangle is completely determined, up to similarity, by the angles. This means that once one of the other angles is known, the ratios of the various sides are always the same regardless of the overall size of the triangle. These ratios are given by the following trigonometric functions of the known angle:

• The sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.

• The cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.

• The tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.

The adjacent leg is the side that is adjacent to the angle but not the hypotenuse. The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively.

The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec) and cotangent (cot), respectively. The inverse functions are called the arcsine, arccosine, and arctangent, respectively. There are arithmetic relations between these functions, which are known as trigonometric identities.

With these functions one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and an angle or two angles and a side or three sides are known. These laws are useful in all branches of geometry, since every polygon may be described as a finite combination of triangles.

[edit] Extending the definitions

Graphs of the functions sin(x) and cos(x), where the angle x is measured in radians.

The above definitions apply to angles between 0 and 90 degrees (0 and π/2 radians) only. Using the unit circle, one may extend them to all positive and negative arguments (see trigonometric function). The trigonometric functions are periodic, with a period of 360 degrees or 2π radians. That means their values repeat at those intervals.

The trigonometric functions can be defined in other ways besides the geometrical definitions above, using tools from calculus and infinite series. With these definitions the trigonometric functions can be defined for complex numbers. The complex function cis is particularly useful

See Euler's and De Moivre's formulas.

[edit] Mnemonics

Students often make use of mnemonics to remember the relationships and facts in trigonometry. For example, the sine, cosine and tangent ratios in right triangles can be remembered by representing all three ratios at once as a string of letters; SOH CAH TOA (sine-opposite-hypotenuse ::: cosine-adjacent-hypotenuse ::: tangent-opposite-adjacent), which can be pronounced as a single word. In addition, many remember similar letter sequences by creating sentences that consist of words that begin with the letters to be remembered, so that they are remembered in the correct order. For example, to remember Tan = Opposite/Adjacent, the letters TOA must be remembered in order. Any memorable phrase constructed of words beginning with the letters 'T, O, A' will serve, and often sentences are constructed to remember all three ratios at once. Other types of mnemonic describe facts in a simple, memorable way, such as "Plus to the right, minus to the left, positive height, negative depth" when referring to the trigonometric functions of a revolving line.

[edit] Rule of quarters

The rule of quarters makes it easy to remember the sine function of special angles:

[edit] Calculating trigonometric functions

Main article: Generating trigonometric tables

Trigonometric functions were among the earliest uses for mathematical tables. Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy. Slide rules had special scales for trigonometric functions.

Today scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan and sometimes cis) and their inverses. Most allow a choice of angle measurement methods, degrees, radians and, sometimes, Grad. Most computer programming languages provide function libraries that include the trigonometric functions. The floating point unit hardware incorporated into the microprocessor chips used in most personal computers have built in instructions for calculating trigonometric functions.

[edit] Applications of trigonometry

Marine sextants like this are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a marine chronometer, the position of the ship can then be determined from several such measurements.

Main article: Uses of trigonometry

There are an enormous number of applications of trigonometry and trigonometric functions. For instance, the technique of triangulation is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. The sine and cosine functions are fundamental to the theory of periodic functions such as those that describe sound and light waves.

Fields which make use of trigonometry or trigonometric functions include astronomy (especially, for locating the apparent positions of celestial objects, in which spherical trigonometry is essential) and hence navigation (on the oceans, in aircraft, and in space), music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development.

[edit] Common formulae

Main article: Trigonometric identity

Main article: Trigonometric function

Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. Many express important geometric relationships. For example, the Pythagorean identities are an expression of the Pythagorean Theorem. Here are some of the more commonly used identities, as well as the most important formulae connecting angles and sides of an arbitrary triangle. For more identities see trigonometric identity.

[edit] Trigonometric identities

Trigonometry

History Usage Functions Inverse functions Further reading

Reference

List of identities Exact constants Generating trigonometric tables CORDIC

Euclidean theory

Law of sines Law of cosines Law of tangents Pythagorean theorem

Calculus

The Trigonometric integral Trigonometric substitution Integrals of functions Integrals of inverses

[edit] Pythagorean identities

[edit] Sum and product identities

[edit] Sum to product:

[edit] Product to sum:

[edit] Sine, cosine and tangent of a sum

[edit] Double-angle identities

[edit] Half-angle identities

Note that is correct, it means it may be either one, depending on the value of A/2.

[edit] Triangle identities

Laws of Sines and Cosines

In the following identities, A, B and C are the angles of a triangle and a, b and c are the lengths of sides of the triangle opposite the respective angles.

[edit] Law of sines

The law of sines (also know as the "sine rule") for an arbitrary triangle states:

where R is the radius of the circumcircle of the triangle.

[edit] Law of cosines

The law of cosines (also known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles:

or equivalently:

[edit] Law of tangents

The law of tangents: