TRIGONOMETRY : TRIGONOMETRY Piyush Kumar Singh CLASS : XI – A Airforce School, Bamrauli. ARYABHATTA AND
WHAT IS TRIGONOMETRY?: WHAT IS TRIGONOMETRY? Trigonometry in basic words is the mathematics of triangles and trigonometric functions. The word “Trigonometry” comes from the Greek words: ‘Trigonon’ meaning ‘triangle’ and ‘metron’ meaning a ‘measure’. In a broader sense, trigonometry is that branch if mathematics which deals with the measurement of the sides and the angles of a triangle and the problems allied with angles.
ORIGIN OF ‘SINE’: ORIGIN OF ‘SINE’ The first use of the idea of ‘sine’ in the way we use it today was in the work Aryabhatiyam by Aryabhata in A.D. 500. Aryabhata used the word ‘ ardha-jya ’ for the half chord which came to be known as ‘ jiva ’ in due course. Later, ‘ jiva ’ came to be known as ‘ sinus ’ and later as ‘ sine ’ . An English Professor Edmund Gunter (1581-1626) first used the abbreviated notation ‘ sin ’ . “Trigonometry is not the work of any one person or nation. Its history spans thousands of years and has touched every major civilization .” Aryabhata A.D. 476-550
Slide 4: The origin of the terms ‘ cosine’ and ‘tangent’ was much later. The cosine function arose from the need to compute the sine of the complementary angle. Aryabhata called ‘kotijya’. The name cosinus originated with Edmund Gunter. In 1674, the English Mathematician Sir Jonas Moore first used the abbreviated notation ‘cos’ COSINE AND TANGENT Edmund Gunter (1581 –1626)
THE TRIGONOMETRIC RATIOS: THE TRIGONOMETRIC RATIOS Tangent tan Opposite Adjacent Cotangent cot Adjacent Opposite Secant sec Hypotenuse Adjacent Cosecant cosec Hypotenuse Opposite Function Abbr. Description Sine sin Opposite Hypotenuse Cosine cos Adjacent Hypotenuse Note: The formulas provided are in respect to the picture. The Cosecant, Secant, and Cotangent are the Reciprocals of the Sine, Cosine,and Tangent respectively .
THE TRIGONOMETRIC VALUES: THE TRIGONOMETRIC VALUES Angle A 0 o 30 o 45 o 60 o 90 o sin A 0 1 2 1 √2 √3 2 1 cos A 1 √3 2 1 √2 1 2 0 tan A 0 1 √3 1 √3 Not Defined cosec A Not Defined 2 √2 2 √3 1 sec A 1 2 √3 √2 2 Not Defined cot A Not Defined √3 1 1 √3 0
SOME MORE WORKS OF ARYABHATTA: SOME MORE WORKS OF ARYABHATTA Place value system and zero The place-value system, first seen in the 3rd century Bakhshali Manuscript , was clearly in place in his work . While he did not use a symbol for zero , the French mathematician Georges Ifrah explains that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten with null coefficients .
WORKS OF ARYABHATTA: WORKS OF ARYABHATTA Approximation of π Aryabhata worked on the approximation for pi (π) , and may have come to the conclusion that π is irrational . In the second part of the Aryabhatiyam ( gaṇitapāda 10 ) , he writes : caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ . " Add four to 100, multiply by eight, and then add 62,000 . By this rule the circumference of a circle with a diameter of 20,000 can be approached .“ This implies that the ratio of the circumference to the diameter is ((4 + 100) × 8 + 62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significant figures .
Slide 9: THANK YOU