Both the sine and cosine functions satisfy the differential equation i.e. each is the additive inverse of its own second derivative. Within the 2-dimensional vector space V consisting of all solutions of this equation, the sine function is the unique solution satisfying the initial conditions y(0) = 0 and y′(0) = 1, and the cosine function is the unique solution satisfying the initial conditions y(0) = 1 and y′(0) = 0. Since the sine and cosine functions are linearly independ ...

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Trigonometric function, Trigonometric function - List of trigonometric functions, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

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The six trigonometric functions can also be defined in terms of the unit circle, the circle of radius one centered at the origin. The unit circle definition provides little in the way of practical calculation; indeed it relies on right triangles for most angles. The unit circle definition does, however, permit the definition of the trig functions for all positive and negative arguments, not just for angles between 0 and π/2 radians. It also provides a single visual picture that encapsulates at once all the important triangles used ...

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Trigonometric function, Trigonometric function - List of trigonometric functions, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

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Please note: Here, and generally in calculus, all angles are measured in radians. (See also below). Using only geometry and properties of limits, it can be shown that the derivative of sine is cosine and the derivative of cosine is the opposite of sine. One can then use the theory of Taylor series to show that the following identities hold for all real numbers x: These identities are often taken as the definitions of the sine and cosine function. They are often used ...

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Trigonometric function, Trigonometric function - List of trigonometric functions, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

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This theorem may have a greater variety of known proofs than any other (the law of quadratic reciprocity being also a contender for that distinction); the book Pythagorean Proposition, by Elisha Scott Loomis, contains over 250 different proofs. James Garfield, who later became a President of the United States, devised an original proof of the Pythagorean theorem in 1876. The external links below provide a sampling of the many different proofs of the Pythagorean theorem. Pythagorean theorem - Geometrical proof. Like many of the proofs of the Pythagorean theorem, this one is based on the proporti ...

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A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths. Evidence from megalithic monuments on the British Isles shows that such triples were known before the discovery of writing. Such a triple is commonly written (a, b,  ...

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Please note: Here, and generally in calculus, all angles are measured in radians. (See also below). Using only geometry and properties of limits, it can be shown that the derivative of sine is cosine and the derivative of cosine is the negative of sine. One can then use the theory of Taylor series to show that the following identities hold for all real numbers x: These identities are often taken as the definitions of the sine and cosine function. They are often used ...

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Trigonometric function, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

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Both the sine and cosine functions satisfy the differential equation i.e. each is the additive inverse of its own second derivative. Within the 2-dimensional vector space V consisting of all solutions of this equation, the sine function is the unique solution satisfying the initial conditions y(0) = 0 and y′(0) = 1, and the cosine function is the unique solution satisfying the initial conditions y(0) = 1 and y′(0) = 0. Since the sine and cosine functions are linearly independ ...

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Trigonometric function, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

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The six trigonometric functions can also be defined in terms of the unit circle, the circle of radius one centered at the origin. The unit circle definition provides little in the way of practical calculation; indeed it relies on right triangles for most angles. The unit circle definition does, however, permit the definition of the trig functions for all positive and negative arguments, not just for angles between 0 and π/2 radians. It also provides a single visual picture that encapsulates at once all the important triangles used ...

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Trigonometric function, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

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In order to define the trigonometric functions for the angle A, start with an arbitrary right triangle that contains the angle A: We use the following names for the sides of the triangle: The hypotenuse is the side opposite the right angle, or defined as the longest side of a right-angled triangle, in this case h. The opposite side is the side opposite to the angle we are interested in, in this case a. The adjacent side is the side that is in contact with the angle we are interested in and the right angl ...

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Trigonometric function, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

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The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, the Euclidean form of the Pythagorean theorem given above does not hold in non-Euclidean geometry. For example, in spherical geometry, all three sides of the right triangle bounding an octant of the unit sphere have length equal to π / 2; this violates the Euclidean Pythagorean theorem because . This means that in non-Euclidean geometry, the Pythagorean theorem must necessarily take a different form from the Euclidean t ...

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In heraldry, the Pythagorean theorem appears as a charge in the arms of Seissenegger. The theorem is referenced in an episode of The Simpsons. After finding a pair of glasses at the Nuclear Power Plant, Homer puts them on and in an attempt to sound smart, comments "the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side." A man in a nearby toilet stall then yells out "That's a right triangle, you idiot!" (This was a homage to The Wizard of Oz. When the Scarecrow receives his diploma from the Wizar ...

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Pythagorean theorem, Pythagorean theorem - History, Pythagorean theorem - Proofs, Pythagorean theorem - Geometrical proof, Pythagorean theorem - A visual proof, Pythagorean theorem - Converse of the theorem, Pythagorean theorem - Algebraic Proof, Pythagorean theorem - Pythagorean triples, Pythagorean theorem - Generalizations, Pythagorean theorem - The Pythagorean theorem in non-Euclidean geometry, Pythagorean theorem - Other facts, Pythagorean theorem - Notes

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In order to define the trigonometric functions for the angle A, start with an arbitrary right triangle that contains the angle A: We use the following names for the sides of the triangle: The hypotenuse is the side opposite the right angle, or defined as the longest side of a right-angled triangle, in this case h. The opposite side is the side opposite to the angle we are interested in, in this case a. The adjacent side is the side that is in contact with the angle we are interested in and the right angl ...

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Trigonometric function, Trigonometric function - List of trigonometric functions, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

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A few other functions were common historically (and appeared in the earliest tables), but are now little-used, such as: versed sine (versin = 1 − cos) exsecant (exsec = sec − 1). Many more relations between these functions are listed in the article about trigonometric identities. ...

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Trigonometric function, Trigonometric function - List of trigonometric functions, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

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See the main article at astrology and numerology Some astrologers believe that each number from 0 to 9 is ruled by a celestial body in our solar system -- the layout below is the most widely accepted system amongst modern astrologers but there are other conflicting systems as well. ...

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Numerology, Numerology - Esoteric significance of numbers, Numerology - One, Numerology - Two, Numerology - Three, Numerology - Four, Numerology - Five, Numerology - Six, Numerology - Seven, Numerology - Eight, Numerology - Nine, Numerology - Ten, Numerology - Eleven, Numerology - Twelve, Numerology - Thirteen, Numerology - Twenty-two, Numerology - Zero, Numerology - Alphabetic harmonics, Numerology - Numerological divination, Numerology - Numerology in science, Numerology - Postmodern critique, Numerology - Numerology and astrology, Numerology - In popular culture, Numerology - Days of Birth reveal your character, Numerology - Sunday born, Numerology - Monday born, Numerology - Tuesday born, Numerology - Wednesday born, Numerology - Thursday born, Numerology - Friday born, Numerology - Saturday born

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In the movie π, the protagonist is searching for hidden numerical patterns in the stock market and the Torah. Each Hebrew letter corresponds to a number. The true name of God is said to correspond to a 216 digit number. British goth band Inkubus Sukkubus changed their name from 'Incubus Succubus' on the advice of a friend who said that the numerology of their first name was bringing them bad luck. The TV series Lost ...

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Numerology, Numerology - Esoteric significance of numbers, Numerology - One, Numerology - Two, Numerology - Three, Numerology - Four, Numerology - Five, Numerology - Six, Numerology - Seven, Numerology - Eight, Numerology - Nine, Numerology - Ten, Numerology - Eleven, Numerology - Twelve, Numerology - Thirteen, Numerology - Twenty-two, Numerology - Zero, Numerology - Alphabetic harmonics, Numerology - Numerological divination, Numerology - Numerology in science, Numerology - Postmodern critique, Numerology - Numerology and astrology, Numerology - In popular culture, Numerology - Days of Birth reveal your character, Numerology - Sunday born, Numerology - Monday born, Numerology - Tuesday born, Numerology - Wednesday born, Numerology - Thursday born, Numerology - Friday born, Numerology - Saturday born

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There is also a serious postmodern critique of Number and the actual cognitive, linguistic, and political meaning of numbers. John Zerzan and George Lakoff are among the best known of these theorists. A common argument in such circles is that the Greek and Roman worlds elevated Number to a god, in part for its power to predict timing of natural phenomena, and engineer reliable infrastructure. At the core of such claims is that primates have an intuitive ability to "count up to four" using their own senses, and that retaining the counted item ...

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Numerology, Numerology - Esoteric significance of numbers, Numerology - One, Numerology - Two, Numerology - Three, Numerology - Four, Numerology - Five, Numerology - Six, Numerology - Seven, Numerology - Eight, Numerology - Nine, Numerology - Ten, Numerology - Eleven, Numerology - Twelve, Numerology - Thirteen, Numerology - Twenty-two, Numerology - Zero, Numerology - Alphabetic harmonics, Numerology - Numerological divination, Numerology - Numerology in science, Numerology - Postmodern critique, Numerology - Numerology and astrology, Numerology - In popular culture, Numerology - Days of Birth reveal your character, Numerology - Sunday born, Numerology - Monday born, Numerology - Tuesday born, Numerology - Wednesday born, Numerology - Thursday born, Numerology - Friday born, Numerology - Saturday born

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In contrast to the verbal society we currently live in, mythical ancient civilisations such as Atlantis and Lemuria are believed by some to have had a slightly different means of communication which was composed on an elaborate system, including art forms which conveyed special messages to the observer. In our current age of known history, early man recorded events which transpired by using pictorial representations which told elaborate stories. As time went on these pictures were abbreviated to form hieroglyphics with each symbol depicting a word. As more time elapsed the glyphs w ...

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Numerology, Numerology - Esoteric significance of numbers, Numerology - One, Numerology - Two, Numerology - Three, Numerology - Four, Numerology - Five, Numerology - Six, Numerology - Seven, Numerology - Eight, Numerology - Nine, Numerology - Ten, Numerology - Eleven, Numerology - Twelve, Numerology - Thirteen, Numerology - Twenty-two, Numerology - Zero, Numerology - Alphabetic harmonics, Numerology - Numerological divination, Numerology - Numerology in science, Numerology - Postmodern critique, Numerology - Numerology and astrology, Numerology - In popular culture, Numerology - Days of Birth reveal your character, Numerology - Sunday born, Numerology - Monday born, Numerology - Tuesday born, Numerology - Wednesday born, Numerology - Thursday born, Numerology - Friday born, Numerology - Saturday born

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In numerological divination, a student of the field will use the name, birthdate and birthtime of an individual to analyze and define something of the personality and propensities of that individual. Specific numbers are also assigned to the letters of the alphabet. One such system (for the English alphabet) is represented here: The basis of the belief that dates and times have numerologic significance appears to be that underlying vibrations of the universe as a whole occur in regular cycles and that things created or changed ...

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Numerology, Numerology - Esoteric significance of numbers, Numerology - One, Numerology - Two, Numerology - Three, Numerology - Four, Numerology - Five, Numerology - Six, Numerology - Seven, Numerology - Eight, Numerology - Nine, Numerology - Ten, Numerology - Eleven, Numerology - Twelve, Numerology - Thirteen, Numerology - Twenty-two, Numerology - Zero, Numerology - Alphabetic harmonics, Numerology - Numerological divination, Numerology - Numerology in science, Numerology - Postmodern critique, Numerology - Numerology and astrology,