Trigonometry
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This document provides information about trigonometry and its origins. It begins by explaining that trigonometry was originally used to define the relationships between the elements (sides and angles) of a triangle. It describes the six trigonometric ratios (sine, cosine, tangent, cosecant, secant, cotangent) and provides the ratios for some special angles like 45°, 30°, 60°, 0°, and 90° degrees. It also describes trigonometric identities and relationships between complementary angles. In closing, it discusses some of the achievements of trigonometry and how it has been used in fields like astronomy, geography, and navigation.
Trigonometry
- 1. MATHS PROJECT To Mrs. Shija mam Shashwat Mishra Hritwik Maurya CLASS-X E ROLL.NO.-42 & 3
- 2. Hipparcus KNOWN AS THE FATHER OF TRIGONOMETRY
- 3. ELEMENTS So originally trigonometry was understood to define relations between elements of a triangle. In a triangle, there are six basic elements: 3 sides and 3 angles. Not any three line segments may serve as the sides of a triangle. They do if they satisfy the triangle inequality, or rather three triangle inequalities. Not any three angles may be the angles of a triangle. In Euclidean geometry, the three angles of a triangle add up to a straight angle. These requirements impose limitations on the manner in which the relations between the elements are defined. In modern trigonometry these relations are extended to arbitrary angles. This can be done, for example, by observing the projections of arotating radius of a circle and a tangent at the end of the radius. One of the oldest surviving fragments of Hipparcus for trigonometry Elements, found from the Histories VIII on Papyrus Oxyrhynchus 2099, early 2nd century AD
- 4. Ratios of trigonometry 1. Sine of <A=side opposite to < A BC hypotenuse AC 2. Cos of <A=side adjacent to <A BA hypotenuse AC 3. Tan of <A= side opposite to <A BC side adjacent to <A BA 4. Cosec of <A= 1 AC sine of <A BC 5. Sec of <A= 1 AC cosine of <A AB 6. Cot of <A= 1 AB tangent of <A BC C B A
- 5. Trigonometric ratios of some special angles 1. Trigonometric ratio of 45˚is: sin 45˚=1/√2 cosec 45˚=√2 cos 45˚=1/√2 sec 45˚=√2 tan 45˚=1 cot 45˚=1
- 6. 2. Trigonomteric ratios of 30˚ and 60˚ are: sin 30˚=1/2 sin 60˚=√3/2 cos 30˚=√3/2 cos 60˚=1/2 tan 30˚=1/√3 tan 60˚=√3 cosec 30˚=2 cosec 60˚=2/√3 sec 30˚=2/√3 sec 60˚=2 cot 30˚=√3 cot 60˚=1/√3
- 7. 3. Trigonometric ratios of 0˚ and 90˚ are: sin 0˚=0 sin 90˚=1 cos O˚=! cos 90˚=0 tan 0˚=0 tan 90˚=! cosec 0˚=! cosec 90˚=1 sec 0˚=1 sec 90˚=! cot 0˚=! cot 90˚=0
- 8. Trigonometric ratios of complementary angles sin(90˚-A)= cos A tan(90˚-A)=cot A sec(90˚-A)=cosec A cos(90˚-A)=sin A cot(90˚-A)=tan A cosec(90˚-A)=sec A for all the values of angle A lying between 0 ˚and 90˚. Check whether this holds for A=˚ or A=90˚
- 10. Trigonometry’s Achievements Trigonometryis not only used in mathematics but it is also used in life aroundus. Trigonometryis oneofthe most ancient subjects studied by the scholarsall over the world. Trigonometrywas invented for it’s need in astronomy.Since then astronomerhave usedit, for instance, tocalculate distances from the Earthto the planets and stars.Trigonometry hasalso spread it’s reachto geography too. It also took navigation along with it. Theknowledge oftrigonometry is usedto construct maps, determine the position ofan island in relation to the longitudes and latitudes.
- 11. THE END 𝑥 + 𝑎 𝑛 = 𝑘=0 𝑛 𝑛 𝑘 𝑥 𝑘 𝑎 𝑛−𝑘 1 + 𝑥 𝑛 = 1 + 𝑛𝑥 1! + 𝑛 𝑛 − 1 𝑥2 2! + ⋯