Trigonometric Formula

Useful trig formulas and some very helpful links for learning trigonometric concepts.

Triangle ABC is any triangle with side lengths a,b,c Law of Cosines Law of Sines Reciprocal identities Pythagorean Identities Quotient Identities Co-Function Identities Even-Odd Identities Sum-Difference Formulas Double Angle Formulas Power-Reducing/Half Angle Formulas Sum-to-Product Formulas Product-to-Sum Formulas  Circular functions In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. Above--- Trigonometric function sinθ for selected angles θ, π − θ, π + θ, and 2π − θ in the four quadrants. (Bottom) Graph of sine function versus angle. Angles from the top panel are identified.

Mathematica: You guys must see this guy "The Human Calculator"....

Mathematica: You guys must see this guy "The Human Calculator"....: "Rüdiger Gamm Rüdiger Gamm (born July 10, 1971) is a German ' mental calculator '. He attained the ability to mentally evaluate large ari..."

You guys must see this guy "The Human Calculator".

Rüdiger Gamm

Rüdiger Gamm (born July 10, 1971) is a German "mental calculator". He attained the ability to mentally evaluate large arithmetic expressions at the age of 21. He can also speak backwards, and calculate calendars. Featured on the Discovery Channel program The Real Superhumans, he was examined by Allan Snyder, an expert on savants, who concluded that Gamm's ability was not a result of savant syndrome but connected to genetics.

In terms of mental calculations, Gamm's most notable talent is the ability to memorise large powers. In the 2008 Mental Calculation World Cup in Leipzig, he recited 81¹⁰⁰, which took approximately 2 minutes and 30 seconds. In the tournament itself, he performed strongly, finishing in 5th position overall.

-----From Wikipedia, the free encyclopedia

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think cool ???

The Indian Mathematical genius -- Sir Srinivasa Ramanujan.

Srinivasa Ramanujan was one of India's greatest mathematical geniuses. He made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series.

Ramanujan was born in his grandmother's house in Erode, a small village about 400 km southwest of Madras. When Ramanujan was a year old his mother took him to the town of Kumbakonam, about 160 km nearer Madras. His father worked in Kumbakonam as a clerk in a cloth merchant's shop. In December 1889 he contracted smallpox.

When he was nearly five years old, Ramanujan entered the primary school in Kumbakonam although he would attend several different primary schools before entering the Town High School in Kumbakonam in January 1898. At the Town High School, Ramanujan was to do well in all his school subjects and showed himself an able all round scholar. In 1900 he began to work on his own on mathematics summing geometric and arithmetic series.

Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic. The following year, not knowing that the quintic could not be solved by radicals, he tried (and of course failed) to solve the quintic.

It was in the Town High School that Ramanujan came across a mathematics book by G S Carr called Synopsis of elementary results in pure mathematics. This book, with its very concise style, allowed Ramanujan to teach himself mathematics, but the style of the book was to have a rather unfortunate effect on the way Ramanujan was later to write down mathematics since it provided the only model that he had of written mathematical arguments. The book contained theorems, formulae and short proofs. It also contained an index to papers on pure mathematics which had been published in the European Journals of Learned Societies during the first half of the 19^th century. The book, published in 1856, was of course well out of date by the time Ramanujan used it.

By 1904 Ramanujan had begun to undertake deep research. He investigated the series ∑(1/n) and calculated Euler's constant to 15 decimal places. He began to study the Bernoulli numbers, although this was entirely his own independent discovery.

Ramanujan, on the strength of his good school work, was given a scholarship to the Government College in Kumbakonam which he entered in 1904. However the following year his scholarship was not renewed because Ramanujan devoted more and more of his time to mathematics and neglected his other subjects. Without money he was soon in difficulties and, without telling his parents, he ran away to the town of Vizagapatnam about 650 km north of Madras. He continued his mathematical work, however, and at this time he worked on hypergeometric series and investigated relations between integrals and series. He was to discover later that he had been studying elliptic functions.

In 1906 Ramanujan went to Madras where he entered Pachaiyappa's College. His aim was to pass the First Arts examination which would allow him to be admitted to the University of Madras. He attended lectures at Pachaiyappa's College but became ill after three months study. He took the First Arts examination after having left the course. He passed in mathematics but failed all his other subjects and therefore failed the examination. This meant that he could not enter the University of Madras. In the following years he worked on mathematics developing his own ideas without any help and without any real idea of the then current research topics other than that provided by Carr's book.

Continuing his mathematical work Ramanujan studied continued fractions and divergent series in 1908. At this stage he became seriously ill again and underwent an operation in April 1909 after which he took him some considerable time to recover. He married on 14 July 1909 when his mother arranged for him to marry a ten year old girl S Janaki Ammal. Ramanujan did not live with his wife, however, until she was twelve years old.

Ramanujan continued to develop his mathematical ideas and began to pose problems and solve problems in the Journal of the Indian Mathematical Society. He devoloped relations between elliptic modular equations in 1910. After publication of a brilliant research paper on Bernoulli numbers in 1911 in the Journal of the Indian Mathematical Society he gained recognition for his work. Despite his lack of a university education, he was becoming well known in the Madras area as a mathematical genius.

A comment about Maths .

 Galileo Galilee 

"Mathematics is the language with which God has written the universe." --- Every successful scientists and researcher must know maths. Think the universe as a book and mathematics as a alphabets  with out mathematics (alphabets) you cannot read the universe (The book).

^^^^Mathematics the language of the Universe ^^^^

As governments around the world prepare to slash science research budgets, it is worth remembering how essential a role mathematics plays in science.
Mathematics can often appear arcane, esoteric, unworldly and irrelevant. Its blue-skies status could easily make it a target for the harsh cuts to science budgets that governments around the world are contemplating. But before the axe falls, it is worth remembering what the 17th-century scientist Galileo Galilei once declared:

The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. 
It is written in mathematical language and the letters are triangles, circles and other geometrical figures, without which means 

it is humanly impossible to comprehend a single word

The scientists at Cern will certainly agree with Galileo. Their ability to make predictions about the particles they are expecting to see inside the Large Hadron Collider is entirely down to mathematics. Rather than triangles and circles, strange, symmetrical objects in multidimensional space - shapes that exist only in the mathematician's mind's eye - are helping us to navigate the strange menagerie of particles that we see in these high-energy collisions.

Biochemists trying to understand the three-dimensional shapes of protein strings will also sympathise with Galileo's sentiments. Each protein consists of a string of chemicals, which are like letters in a book. But to read this language and predict how the one-dimensional string will fold up in three-dimensional space, you need a mathematical dictionary. Studying protein folding is key to understanding several neurodegenerative diseases that are a result of the incorrect folding of certain protein strings.

It is striking, however, how much of my subject remains a mystery: how many mathematical stories are still without endings, or read like texts that have yet to be deciphered.

Take the atoms of mathematics, the primes. Indivisible numbers such as seven and 17 are the building blocks of all numbers because every number is built by multiplying these primes together. They are like the hydrogen and oxygen of the world of mathematics. Dmitri Mendeleev used the mathematical patterns he had discovered in the chemical elements to create the periodic table, the most fundamental tool in chemistry. So powerful were these patterns that they helped chemists to predict unknown elements that were missing from the table. But mathematicians are still to have their Mendeleev moment. The pattern behind the primes that might help us to predict their location has yet to be uncovered. Reading through a list of primes is like staring at hieroglyphs. We have made progress and have unearthed something resembling the Rosetta Stone of prime numbers - but the ability to decode the stone still eludes us.

Mathematicians have been wrestling with the mystery of the primes for 2,000 years, but some of the biggest problems in maths are far more recent. There was a flurry of excitement this summer when the Hewlett-Packard engineer Vinay Deolalikar claimed to have cracked the "P v NP" problem. First posed in the 1970s, this is a problem about complexity. There are many ways to state it, but the classic formulation is the "travelling salesman problem".

An example is the following challenge: you are a salesman who needs to visit ten clients, each located in a different town. The towns are connected by roads, as shown on the following map, but you have only enough fuel for a journey of 238 miles

The distance between towns is given by the number on the road joining them. Can you find a journey that lets you visit all ten clients, stopping in each town only once, and then return home without running out of fuel? (The solution appears at the end of the article.) The big mathematical question is whether there is a general algorithm or computer program that will produce the shortest path for any map you feed in, which would be significantly quicker than getting the computer to carry out an exhaustive search. The number of possible journeys grows exponentially as you increase the number of towns, so an exhaustive search soon becomes practically impossible.

The general feeling among mathematicians is that problems of this sort have an inbuilt complexity, which means that there won't be any clever way of finding the solution. But proving that something doesn't exist is always a tough task. The recent excitement that this problem had been cracked has since evaporated and it remains one of the toughest on the mathematical books.

But the recent solution of another challenging problem, the Poincaré conjecture, gives us hope that even the most elusive problems can be conquered. The Poincaré conjecture is a fundamental problem about the nature of shape. It challenges mathematicians to list the possible shapes into which three-dimensional space can be wrapped up. In 2003, the Russian mathematician Grigori Perelman succeeded in producing a periodic table of shapes from which all other such shapes can be built.

These fundamental mathematical questions are not just esoteric puzzles of interest solely to mathematicians. Given that we live in a world with three spatial dimensions, the Poincaré conjecture tells us ultimately what shape our universe could be. Many questions of biology and chemistry can be reduced to versions of the travelling salesman problem, where the challenge is to find the most efficient solution among a whole host of possibilities. A resolution of the P v NP problem could therefore have significant real-world repercussions.

Modern internet codes rely on properties of prime numbers. So any announcement of a breakthrough on the primes is likely to spike the interest not just of pure mathematicians, but of e-businesses and national security agencies. At a time when blue-skies research without obvious commercial benefits could be under threat from sweeping cuts, it is worth remembering how Galileo concluded his statement about the language of mathematics: without it, we will all be wandering around, lost in a dark  labyrinth. --------- from--
                                              read more-- http://www.newstatesman.com/education/2010/10/problem-science-mathematics.

Isaac Newton and Mathematics #

INTRO-                                                      Newton, Sir Isaac (1642-1727), mathematician and physicist, one of the foremost scientific intellects of all time. Born at Woolsthorpe, near Grantham in Lincolnshire, where he attended school, he entered Cambridge University in 1661; he was elected a Fellow of Trinity College in 1667, and Lucasian Professor of Mathematics in 1669. He remained at the university, lecturing in most years, until 1696. Of these Cambridge years, in which Newton was at the height of his creative power, he singled out 1665-1666 (spent largely in Lincolnshire because of plague in Cambridge) as "the prime of my age for invention". During two to three years of intense mental effort he prepared Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) commonly known as the Principia, although this was not published until 1687.


Life as Mathematician -                        In mathematics too, early brilliance appeared in Newton's student notes. He may have learnt geometry at school, though he always spoke of himself as self-taught; certainly he advanced through studying the writings of his compatriots William Oughtred and John Wallis, and of Descartes and the Dutch school. Newton made contributions to all branches of mathematics then studied, but is especially famous for his solutions to the contemporary problems in analytical geometry of drawing tangents to curves (differentiation) and defining areas bounded by curves (integration). Not only did Newton discover that these problems were inverse to each other, but he discovered general methods of resolving problems of curvature, embraced in his "method of fluxions" and "inverse method of fluxions", respectively equivalent to Leibniz's later differential and integral calculus. Newton used the term "fluxion" (from Latin meaning "flow") because he imagined a quantity "flowing" from one magnitude to another. Fluxions were expressed algebraically, as Leibniz's differentials were, but Newton made extensive use also (especially in thePrincipia) of analogous geometrical arguments. Late in life, Newton expressed regret for the algebraic style of recent mathematical progress, preferring the geometrical method of the Classical Greeks, which he regarded as clearer and more rigorous.

Einstein and Math

Einstein is well known for his work in physics, but what about math? Did he brought new ideas to mathematics, or was he influenced by mathematical ideas? Before answering these questions, lets look on a short summary of Einstein achievements - from Wikipedia:

Albert Einstein was a German-born theoretical physicist. He is best known for his theory of relativity and specifically mass–energy equivalence, E = mc^2. Einstein received the 1921 Nobel Prize in Physics "for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect."

Einstein's many contributions to physics include his special theory of relativity, which reconciled mechanics with electromagnetism, and his general theory of relativity, which extended the principle of relativity to non-uniform motion, creating a new theory of gravitation. His other contributions include relativistic cosmology, capillary action, critical opalescence, classical problems of statistical mechanics and their application to quantum theory, an explanation of the Brownian movement of molecules, atomic transition probabilities, the quantum theory of a monatomic gas, thermal properties of light with low radiation density (which laid the foundation for the photon theory), a theory of radiation including stimulated emission, the conception of a unified field theory, and the geometrization of physics.

As you can see he contributed to a lot of different topics in physics. Math is mentioned nowhere in this short summary and for a good reason - he was after all a physicist. However to do anything in any of the fields mentioned above, a very good understanding of math is required.

In my opinion, Einstein relationship with math can be divided into two periods. The first period was in childhood when he first heard about mathematics:
Einstein was introduced to math when he was ten years old. In 1889, family friend Max Talmud, introduced him to mathematics, and philosophy texts, including Kant's Critique of Pure Reason and Euclid's Elements (Einstein called it the "holy little geometry book"). From Euclid, Einstein began to understand deductive reasoning, and by the age of twelve, he had learned Euclidean geometry. Soon thereafter he began to investigate calculus.
In 1895 Einstein tried to apply to Swiss Federal Institute of Technology in Zürich, Switzerland. He was required to pass an entrance exam which he failed - but he got exceptional marks in mathematics and physics. In 1896 he finally enrolled in the mathematics program at ETH. He graduated in 1900 from there with a degree in physics as a teacher of mathematics and physics.

His graduation marks the beginning of a second period - now he was no longer studying math. He was now looking on it as a tool to solve problems that he encountered. However math refused to be only a tool for him:
By mid 1901 he had a temporary job as a teacher, teaching mathematics at the Technical High School in Winterthur, and after this in a private school in Schaffhausen. These positions were all temporal - he had a desire to get to an university. It took a lot of time and effort but in 1912, after publishing a lot of groundbreaking articles (in physics), Einstein returned to Switzerland to accept a professorship at the ETH. There he met mathematician Marcel Grossmann who introduced him toRiemannian geometry, and at the recommendation of Italian mathematician Tullio Levi-Civita, Einstein began exploring the usefulness of general covariance (essentially the use of tensors) for his gravitational theory. Ultimately, the ideas and approaches of Riemann geometry provided him what he needed to develop his general relativity theory - the distortion of space-time under the influence of gravity. While it is usually thought that physics "push" mathematics in this particular moment it was mathematics that contributed to advance of physics.


The above is only a short summary - it is possible to say more, but the answers to the questions I started with are already obvious: While there are no evidence to suggest that Einstein had any direct influence on the development of math, he and his ideas were greatly effected by it. On the other hand, as it is always with physics, his theories put forward new problems that required development of new mathematical methods to be solved.

Quotes about math by Einstein:
"God does not care about our mathematical difficulties. He integrates empirically."

"Do not worry about your difficulties in Mathematics. I can assure you mine are still greater."

"As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality."

"Yes, we have to divide up our time like that, between our politics and our equations. But to me our equations are far more important, for politics are only a matter of present concern. A mathematical equation stands forever."

"Pure mathematics is, in its way, the poetry of logical ideas."

"How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?"

"One reason why mathematics enjoys special esteem, above all other sciences, is that its laws are absolutely certain and indisputable, while those of other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts."

Don't worry

Do you guys hate maths?
& think you are no good at it ? don't worry ! The greatest mathematicians and geniuses who have been with us all felt the same way at first ! and the way to be a successful maths-genius you don't want to study maths but understand it ! never study understand!