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mathematics symposium

mathematics symposium

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Hari love sachin Hari love sachin Presentation Transcript

  • WELCOME
  • HARILAL KRISHNA VISHNU K N AMITH KUMAR SANOOJ SAURAV S K Trigonometry
  • WHICH SO OCCUPIES THERE IS PERHAPS NOTHING THE MIDDLE POSITION OF MATHEMATICS AS TRIGONOMETRY - J.F.HERBART
  • TRIGONOMETRY
    • Trigonometry is a field of mathematics first compiled by 2nd century BCE by the Greek mathematician Hipparchus . The history of trigonometry and of trigonometric functions follows the general lines of the history of mathematics .
    • Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics ( Rhind Mathematical Papyrus ) and Babylonian mathematics . Systematic study of trigonometric functions begins in Hellenistic mathematics , reaching India as part of Hellenistic astronomy . In Indian astronomy , the study of trigonometric functions flowers in the Gupta period , especially due to Aryabhata (6th century). During the Middle Ages, the study of trigonometry is continued in Islamic mathematics , whence it is adopted as a separate subject in the Latin West beginning in the Renaissance with Regiomontanus . The development of modern trigonometry then takes place in the western Age of Enlightenment , beginning with 17th century mathematics ( Isaac Newton , James Stirling ) and reaching its modern form with Leonhard Euler (1748).
  • History Of Trigonometry
    • Trigonometry was originally created by the Greeks to help in the study of astronomy
    • Hipparchus of Bithynia (190-120 B.C.) used trig ratios to calculate planets’ positions
  • EARLY TRIGONOMETRY
    • The ancient Egyptians and Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries. But pre-Hellenic societies lacked the concept of an angle measure and consequently, the sides of triangles were studied instead, a field that would be better called "trilaterometry“ .The Babylonian astronomers kept detailed records on the rising and setting of stars , the motion of the planets , and the solar and lunar eclipses , all of which required familiarity with angular distances measured on the celestial sphere . Based on one interpretation of the Plimpton 322 cuneiform tablet (c. 1900 BC), some have even asserted that the ancient Babylonians had a table of secants. There is, however, much debate as to whether it is a table of Pythagorean triples , a solution of quadratic equations, or a trigonometric table .
    • The Egyptians, on the other hand, used a primitive form of trigonometry for building pyramids in the 2nd millennium BC.
    • The first trigonometric table was apparently compiled by Hipparchus of Nicaea (180 – 125 BC), who is now consequently known as "the father of trigonometry." Hipparchus was the first to tabulate the corresponding values of arc and chord for a series of angles.
    • Although it is not known when the systematic use of the 360° circle came into mathematics, it is known that the systematic introduction of the 360° circle came a little after Aristarchus of Samos composed On the Sizes and Distances of the Sun and Moon (ca. 260 BC), since he measured an angle in terms of a fraction of a quadrant.It seems that the systematic use of the 360° circle is largely due to Hipparchus and his table of chords . Hipparchus may have taken the idea of this division from Hypsicles who had earlier divided the day into 360 parts, a division of the day that may have been suggested by Babylonian astronomy.In ancient astronomy, the zodiac had been divided into twelve "signs" or thirty-six "decans". A seasonal cycle of roughly 360 days could have corresponded to the signs and decans of the zodiac by dividing each sign into thirty parts and each decan into ten parts. It is due to the Babylonian sexagesimal numeral system that each degree is divided into sixty minutes and each minute is divided into sixty seconds.
  • Hipparchus of Bithynia 190-120 B.C.
    • Born in Nicaea in Bithynia
    • The 1 st work on trig functions was related to the chords of circles
    • Hipparchus made the 1 st known table of chords of circles in about 140 BC.
    • Although these tables have not survived, it is claimed that twelve books of tables of chords were written by Hipparchus.
    • This makes Hipparchus the founder of trigonometry.
  • ETYMOLOGY
    • The complex origins of trigonometry are embedded in the history of the simple word "sine," a mistranslation of an Arabic transliteration of a Sanskrit mathematical term! The complex etymology of "sine" reveals trigonometry's roots in Babylonian, Greek, Hellenistic, Indian, and Arabic mathematics and astronomy.
    • Although trigonometry now is usually taught beginning with plane triangles, its origins lie in the world of astronomy and spherical triangles. Before the sixteenth century, astronomy was based on the notion that the earth stood at the center of a series of nested spheres. To calculate the positions of stars or planets, one needed to use concepts we now refer to as trigonometry.
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  • Trigonometric functions
    • The primary trigonometric functions are the sine and cosine of an angle. These are sometimes abbreviated sin( θ ) and cos( θ ), respectively, where θ is the angle, but the parentheses around the angle are often omitted, e.g., sin  θ and cos  θ .
    • The tangent (tan) of an angle is the ratio of the sine to the cosine:
    • Finally, the reciprocal functions secant (sec), cosecant (csc), and cotangent (cot) are the reciprocals of the cosine, sine, and tangent:
  • Pythagorean identity
    • The basic relationship between the sine and the and the cosine is the Pythagorean trigonometric identity :
    • where sin2 θ means (sin( θ ))2.
    • This can be viewed as a version of the Pythagorean theorem , and follows from the equation x 2 + y 2 = 1 for the unit circle . This equation can be solved for either the sine or the cosine:
  • TRIGONOMETRIC RATIOS
    • Ancient Greek and Hellenistic mathematicians made use of the chord . Given a circle and an arc on the circle, the chord is the line that subtends the arc. A chord's perpendicular bisector passes through the center of the circle and bisects the angle. One half of the bisected chord is the sine of the bisected angle, that is,
    • and consequently the sine function is also known as the "half-chord". Due to this relationship, a number of trigonometric identities and theorems that are known today were also known to Hellenistic mathematicians, but in their equivalent chord form
  • TRIGONOMETRIC RATIOS
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    • The chord of an angle subtends the arc of the angle.
  • Opposite side: the side opposite the angle angle angle angle opposite opposite opposite opposite angle
  • Adjacent side: the side beside the angle adjacent adjacent adjacent adjacent angle angle angle angle
  • Hypotenuse: the longest side hypotenuse hypotenuse hypotenuse hypotenuse
  • Tangent of angle A A B 90 o C a c b tan( A ) = opposite adjacent a b =
  • Tangent of angle B A B 90 o C a c b tan( B ) = opposite adjacent b a =
  • Sine of angle A A B 90 o C a c b sin( A ) = = a c opposite hypotenuse
  • Sine of angle B A B 90 o C a c b sin( B ) = = b c opposite hypotenuse
  • Cosine of angle A A B 90 o C a c b cos( A ) = = b c adjacent hypotenuse
  • Cosine of angle B A B 90 o C a c b cos( B ) = = a c adjacent hypotenuse
  • Example 1 Find all the angles x, where 0 < x < 4 π where sin x = 0.5 Principle value: Using symmetry: Using periodicity: