INTRODUCTION TO TRIGNOMETRY

MADE BY : KRISHNARAJ MISHRA

SUBJECT :MATHS

SUBMITTED TO : NM GIRI S IR

1) Introduction to trignometry

2) History

3) Trignometric ratios

4) Values of trignometric function

5) Trignometric ratios of some specific angles

6) Trignometric ratios of some complementry angles

7) Trignometric identities

8) Conclusion

INTRODUCTION

• The distances or heights can be

found by using some

mathematical techniques , which

come under a branch of

mathematics called

‘trignometry’.

• The word ‘trignometry’ is

derived from Greek words ‘tri’

(meaning three),’gon’(meaning

sides) and metron(meaning

measure).

INTRODUCTION

Trigonometry is the branch of mathematics which

deals with triangles, particularly triangles in a plane

where one angle of the triangle is 90 degrees

Triangles on a sphere are also studied, in spherical

trigonometry.

Trigonometry specifically deals with the relationships

between the sides and the angles of triangles, that is, on

the trigonometric functions, and with calculations

based on these functions.

History

The origins of trigonometry can be traced to the civilizations of ancient Egypt, Mesopotamia and the Indus Valley, more than 4000 years ago.

Some experts believe that trigonometry was originally invented to calculate sundials, a traditional exercise in the oldest books

The first recorded use of trigonometry came from the Hellenistic mathematician Hipparchus circa 150 BC, who compiled a trigonometric table using the sine for solving triangles.

The Sulba Sutras written in India, between 800 BC and 500 BC, correctly compute the sine of π/4 (45°) as 1/√2 in a procedure for circling the square (the opposite of squaring the circle).

Many ancient mathematicians like Aryabhata, Brahmagupta,Ibn Yunus and Al- Kashi made significant contributions in this field(trigonometry).

Right Triangle

A triangle in which one angle is equal to 90 is called right triangle.

The side opposite to the right angle is known as hypotenuse.

AB is the hypotenuse

The other two sides are known as legs.

AC and BC are the legs

Trigonometry deals with Right Triangles

In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of areas of the squares whose sides are the two legs.

In the figure

AB2 = BC2 + AC2

TRIGONOMETRIC RATIOS

Sine(sin) opposite side/hypotenuse

Cosine(cos) adjacent side/hypotenuse

Tangent(tan) opposite side/adjacent side

Cosecant(cosec) hypotenuse/opposite side

Secant(sec) hypotenuse/adjacent side

Cotangent(cot) adjacent side/opposite side

sin = a/c

cos = b/c

tan = a/b

cosec = c/a

sec = c/b

cot = b/a

• In Δ ABC, right-angled at B, if one angle is 45°, then the other angle is also 45°, i.e., ∠ A = ∠ C = 45° .

• Suppose BC = AB = a.

• Then by Pythagoras Theorem, AC2 = AB2 + BC2

= a2 + a2 = 2a2,

Therefore, AC = 2 a

Trigonometric Ratios of 45° sin 45° = side opposite to angle 45° / hypotenuse

=BC/AC = a/a√2 = 1/ √2

cos 45° = side adjacent to angle 45°/ hypotenuse=AB/AC = a/a √2 = 1/ √2

tan 45° =side opposite to angle 45°/ side adjacent to angle 45°

=BC/AB = a/a = 1

cosec 45°=1/sin 45°= √2

sec 45°=1/cos 45°= √2

cot 45°=1/tan 45°= 1

cosec 30°=1/sin 30° = 2

sec 30°=1/cos 30° = 2/√3

cot 30°=1/tan 30° = √3

sin 60°= a√3/2a = √3/2

cos 60°= ½

tan 60°= √3

cos 60°= 2/√3

sec 60°= 2

cot 60°= 1/√3

Trigonometric Ratios of 0°And 90°

Sin 0⁰ = 0

cos 0⁰ = 1

Sin 90⁰ = 1

Cos 90⁰ =0

VALUES OF TRIGONOMETRIC

FUNCTION

0 30 45 60 90

Sine 0 0.5 1/2 3/2 1

Cosine 1 3/2 1/2 0.5 0

Tangent 0 1/ 3 1 3 Not defined

Cosecant Not defined 2 2 2/ 3 1

Secant 1 2/ 3 2 2 Not defined

Cotangent Not defined 3 1 1/ 3 0

RELATION BETWEEN DIFFERENT

TRIGONOMETRIC IDENTITIES

Sine

Cosine

Tangent

Cosecant

Secant

Cotangent

sin (90⁰-A) = cos A

tan (90⁰-A) = cot A

sec (90⁰-A) = cosec A

cos (90⁰-A) = sin A

cot (90⁰-A) = tan A

cosec (90⁰-A) = sec A

Trigonometric identities

Osin2A + cos2A = 1

O1 + tan2A = sec2A

O1 + cot2A = cosec2A

Osin(A+B) = sinAcosB + cosAsin B

Ocos(A+B) = cosAcosB – sinAsinB

O tan(A+B) = (tanA+tanB)/(1 – tanAtan B)

Osin(A-B) = sinAcosB – cosAsinB

Ocos(A-B)=cosAcosB+sinAsinB

O tan(A-B)=(tanA-tanB)(1+tanAtanB)

sin2A =2sinAcosA

cos2A=cos2A - sin2A

tan2A=2tanA/(1-tan2A)

sin(A/2) = ±{(1-cosA)/2}

Cos(A/2)= ±{(1+cosA)/2}

Tan(A/2)= ±{(1-cosA)/(1+cosA)}

Trigonometric identities

Conclusion

Trigonometry is a branch of Mathematics with

several important and useful applications.

Hence it attracts more and more research with

several theories published year after year.

Thank You……..