TRIGONOMETRY

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GOYAL

NIMIT ARORA

TRIGONOMETRYTrigonometry (from Greek trigōnon "triangle"

+ metron "measure") is a branch of mathematics that

studies triangles and the relationships between the

lengths of their sides and the angles between those

sides.

Trigonometry defines the trigonometric

functions, which describe those

relationships and have applicability

to cyclical phenomena, such as waves.

The field evolved during the third

century BC as a branch

of geometry used extensively for

astronomical studies . It is also the

foundation of the practical art

of surveying.

HISTORY Of TRIGONOMETRY

Classical Greek

mathematicians (such

as Euclid and Archimed

es) studied the

properties

of chords and inscribed

angles in circles, and

proved theorems that

are equivalent to

modern trigonometric

formulae, although they

presented them

geometrically rather

than algebraically.

The modern sine

function was first

defined in the Surya

Siddhanta, and its

properties were

further documented

by the 5th

century Indian

mathematician and

astronomer Aryabh

ata.

These Greek and Indian works were translated

and expanded by medieval Islamic

mathematicians. By the 10th century, Islamic

mathematicians were using all six

trigonometric functions, had tabulated their

values, and were applying them to problems

in spherical geometry.

The Father of Trigonometry

The

first trigonometri

c table was

apparently

compiled

by Hipparchus,

who is now

consequently

known as "the

father of

trigonometry.

RIGHT TRIANGLE

A right triangle or right-

angled triangle is

a triangle in which

one angle is a right

angle (that is, a 90-

degree angle). The

relation between the

sides and angles of a

right triangle is the

basis for trigonometry.

The side opposite the right

angle is called

the hypotenuse (side c in

the figure above). The

sides adjacent to the right

angle are called legs.

Side a may be identified as

the side adjacent to angle

B and opposed

to (or opposite) angle A,

while side b is the

side adjacent to angle

A and opposed to angle B.

PYTHAGORAS THEOREM

The Pythagorean theorem:

The sum of the areas of the

two squares on the legs (a

and b) equals the area of the

square on the hypotenuse

(c).

If one angle of a triangle is 90 degrees

and one of the other angles is known, the

third is thereby fixed, because the three

angles of any triangle add up to 180

degrees. The two acute angles therefore

add up to 90 degrees: they

are complementary angles. The shape of

a triangle is completely determined,

except for similarity, by the angles.

TRIGONOMETRIC RATIOS

Once the angles are known,

the ratios of the sides are

determined, regardless of the

overall size of the triangle. If the

length of one of the sides is

known, the other two are

determined. These ratios are given

by the following trigonometric

functions of the known angle A,

where a, b and c refer to the

lengths of the sides in the

accompanying figure:

The hypotenuse is the side opposite to the 90 degree angle in a

right triangle; it is the longest side of the triangle, and one of the

two sides adjacent to angle A. The adjacent leg is the other side

that is adjacent to angle A. The opposite side is the side that is

opposite to angle A. The terms perpendicular and base are

sometimes used for the opposite and adjacent sides respectively.

The reciprocals of these functions are named the cosecant (csc or

cosec), secant (sec), and cotangent (cot), respectively:

TRIGONOMETRIC FUNCTIONS

STANDARD IDENTITIES

Identities are those equations that hold true for any value

REDUCTION FORMULA

Sin (90-A) =Cos A

Tan (90-A)= Cot

A

Cosec (90-A)= Sec

A

Calculator1) This calculates the value of

trigonometric functions of

different angles.

2) First enter whether you

want enter the angle in

radian or in degree.

3) Then enter the required

trigonometric function in

the format given below:

4) Enter 1 for Sin

5) Enter 2 for Cosine

6) Enter 3 for tangent

7) Enter 4 for Cosecant

8) Enter 5 for Secant

9) Enter 6 for cotangent

10)Then enter the magnitude

of angle.

Applications of trigonometry

Fields that use trigonometry or trigonometric functions

include astronomy (especially for locating apparent positions of

celestial objects, in which spherical trigonometry is essential)

and hence navigation (on the oceans, in aircraft, and in

space), music theory, audio synthesis, acoustics, optics,

analysis of financial markets, electronics, probability

theory, statistics, biology, medical imaging (CAT

scans and ultrasound)

Application of trigonometry in

Astronomy

1) Since ancient times trigonometry was used in

astronomy

2) The technique triangulation is used to measure the to

nearby stars .

3) In 240 B.C, a mathematician named Eratosthenes

discovered the radius of the earth using trigonometry

and geometry

4) In 2001 , a group of European astronomers did an

experiment that started in 1997 about the distance of

Venus from the sun.

conclusionTRIGONOMETRY IS A BRANCH OF

MATHEMATICS WITH SEVERAL

IMPORTANT AND USEFUL

APPLICATIONS.HENCE IT

ATTRACTSMORE AND MORE

RESEARCHWITH SEVERAL THEORIES

PUBLISHED YEAR AFTER YEAR.

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