MATHS PROJECT

To Mrs. Shija mam

Shashwat MishraHritwik Maurya

CLASS-X EROLL.NO.-42 & 3

Hipparcus

KNOWN AS THE FATHER OF

TRIGONOMETRY

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

Ratios of trigonometry 1. Sine of <A=side opposite to < A

BC hypotenuse

AC2. Cos of <A=side adjacent to <A BA hypotenuse

AC3. Tan of <A= side opposite to <A BC side adjacent to <A BA4. Cosec of <A= 1 AC sine of <A BC

5. Sec of <A= 1 AC cosine of <A AB6. Cot of <A= 1 AB tangent of <A BCC B

A

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

2. Trigonomteric ratios of 30˚ and 60˚ are:

sin 30˚=1/2 sin 60˚=√3/2cos 30˚=√3/2 cos 60˚=1/2tan 30˚=1/√3 tan 60˚=√3cosec 30˚=2 cosec 60˚=2/√3sec 30˚=2/√3 sec 60˚=2cot 30˚=√3 cot 60˚=1/√3

3. Trigonometric ratios of 0˚ and 90˚ are:

sin 0˚=0 sin 90˚=1cos O˚=! cos 90˚=0tan 0˚=0 tan 90˚=! cosec 0˚=! cosec 90˚=1sec 0˚=1 sec 90˚=!cot 0˚=! cot 90˚=0

Trigonometric ratios of

complementary angles

sin(90˚-A)= cos Atan(90˚-A)=cot Asec(90˚-A)=cosec Acos(90˚-A)=sin Acot(90˚-A)=tan Acosec(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˚

Trigonometric Identities

cos2A+sin2A=1

1+ tan2A=sec2A

cot2A+1=cosec2AC B

A

Trigonometry’sAchievementsTrigonometry is not only used in mathematics

but it is also used in life around us. Trigonometry is one of the most ancient subjects studied by the scholars all over the world. Trigonometry was invented for it’s need in astronomy. Since then astronomer have used it, for instance, to calculate distances from the Earth to the planets and stars. Trigonometry has also spread it’s reach to geography too. It also took navigation along with it. The knowledge of trigonometry is used to construct maps, determine the position of an island in relation to the longitudes and latitudes.

THE

END

(𝑥+𝑎 )𝑛=∑𝑘= 0

𝑛

(𝑛𝑘)𝑥𝑘𝑎𝑛−𝑘

(1+𝑥 )𝑛=1+𝑛𝑥1 !

+𝑛 (𝑛−1 ) 𝑥2

2 !+…