trigonometry

Trigonometry

Hypotenuse

Adjacent

Op

posi

te

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TRIGONOMETRY

The word ‘trigonometry’ is derived from the Greek words ‘tri’(meaning three), ‘gon’ (meaning sides) and ‘metron’ (meaning measure).

Trigonometry is the study of relationships between the sides and angles of a triangle.

Early astronomers used it to find out the distances of the stars and planets from the Earth.

Even today, most of the technologically advanced methods used in Engineering and Physical Sciences are based on trigonometrical concepts.

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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.

AC is the hypotenuse The other two sides are

known as legs.

AB and BC are the legs

Trigonometry deals with Right Triangles

A

CB

Hypotenuse



PYTHAGORAS THEOREM

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.

A

CB

Hypotenuse

(Hypotenuse)2 = (Perpendicular)2 + (Base)2

AC2 = BC2 + AB2



PYTHAGORAS THEOREMPythagoras Theorem Proof: Given: Δ ABC is a right angled triangle where

B = 900 And AB = P, BC= b and AC = h.

To Prove: h2 = p2 + b2

Construction : Draw a BD from B to AC , where AD = x and CB = h-x ,

Proof : In Δ ABC and Δ ABD,

Δ ABC Δ ABD --------(AA)In Δ ABC and Δ BDC both are similar

So by these similarity,

p

b

h

A

B C

D

x

(h-x)


PYTHAGORAS THEOREM

Or P2 = x × h And b2 = h (h – x)

Adding both L.H.S. and R.H. S. Then

p2 + b2 = (x × h) + h (h – x)

Or p2 + b2 = xh + h2 – hx

Hence the Pythagoras theorem

p2 + b2 = h2

bxh

hb

px

hp And

p

b

h

A

B C

D

x

(h-x)


TRIGONOMETRIC RATIOS

Let us take a right triangle ABC

Here, ∠ ACB () is an acute angle.

The position of the side AB with respect to angle . We call it the side opposite to angle .

AC is the hypotenuse of the right triangle and the side BC is a part of . So, we call it the side adjacent to angle .

A

CB

Hypotenuse

Sid

e op

posi

te t

o a

ng

le

Side adjacent to angle

‘’




The trigonometric ratios of the angle C in right ABC as follows :

Sine of C =

=

Cosine of C=

=

A

CB

Hypotenuse

Sid

e op

posi

te t

o a

ng

le


‘’

Side opposite to C Hypotenuse

ABAC

Side adjacent to C Hypotenuse

BCAC




Tangent of C =

=

Cosecant of C=

=

Secant of C =

A

CB

Hypotenuse

Sid

e op

posi

te t

o a

ng

le


‘’

Side opposite to CSide adjacent to CAB

BC

Side adjacent to C

Hypotenuse

Side opposite to C

Hypotenuse

ACAB

ACAB

=




Cotangent of C

Above Trigonometric Ratio arbitrates as sin C, cos C, tan C , cosec C , sec C, Cot C .

If the measure of angle C is ‘’ then the ratios are :

sin , cos , tan , cosec , sec and cot

A

CB

Hypotenuse

Sid

e op

posi

te t

o a

ng

le


‘’

Side opposite to C

Side adjacent to C AB

BC= =



RELATION AMONG T-RATIO

Tan = Cosec = 1 / Sin

Sec = 1 / Cos

Cot = Cos / Sin

= 1 / Tan

A

CB

p

b

h


cossin


TABLE

1. Sin = p / h

2. Cos = b / h

3. Tan = p / b

4. Cosec = h / p

5. Sec = h / b

6. Cot = b / p

A

CB

p

b

h



TRIGONOMETRIC RATIOS OF SOME SPECIFIC ANGLES

Trigonometric Ratios of 45°In Δ ABC, right-angled at B, if one angle is 45°, thenthe other angle is also 45°, i.e., ∠ A = ∠ C = 45°So, BC = ABNow, Suppose BC = AB = a.Then by Pythagoras Theorem,

AC2 = BC2 + AB2 = a2 + a2

AC2 = 2a2 , or AC = a2

A

CB

450

a

a

450



TRIGONOMETRIC RATIOS OF 45°

Sin 450 = = = = 1/2

Cos 450 = = = = 1/2

Tan 450 = = = = 1

Cosec 450 = 1 / sin 450 = 1 / 1/2 = 2

Sec 450 = 1 / cos 450 = 1 / 1/2 = 2

Cot 450 = 1 / tan 450 = 1 / 1 = 1

Side opposite to 450Hypotenuse

ABAC

aa2

Side adjacent to 450Hypotenuse

BCAC

a

Side opposite to 450Side adjacent to 450

ABBC

aa

a2



TRIGONOMETRIC RATIOS OF 30° AND 60°

Consider an equilateral triangle ABC. Since each angle in an equilateral triangle is 60°, therefore,

∠ A = ∠ B = ∠ C = 60°.Draw the perpendicular AD from A to the side BC,

Now Δ ABD Δ ACD ≅ --------- (S. A. S)Therefore, BD = DCand ∠ BAD = ∠ CAD -----------(CPCT)Now observe that:Δ ABD is a right triangle, right-angled at D with ∠

BAD = 30° and ∠ ABD = 60°

600

300

A

B D C




As you know, for finding the trigonometric ratios, we need to know the lengths of the sides of the triangle.

So, let us suppose that AB = 2a.BD = ½ BC = a

AD2 = AB2 – BD2 = (2a)2 - (a)2 = 3a2

AD = a3

Now Trigonometric ratios

Sin 300 = =

= = ½

600

300

A

B D C

2a

2a 2a

a aSide opposite to 300Hypotenuse

BDAB

a2a




Cos 300 = = = 3 / 2

Tan 300 = = = 1 / 3

Cosec 300 = 1 / sin 300 = 1 / ½ = 2

Sec 300 = 1 / cos 300 = 1 / 3/2 = 2 / 3

Cot 300 = 1 / tan 300 = 1 / 1/3 = 3

Now trigonometric ratios of 600

ADAB

a32a

BDAD

aa3

300

A

B D C

2a

2a 2a

a a




Sin 600 = = = 3 / 2

Cos 600 = = = ½

Tan 600 = = = 3

Cosec 600 = 1 / Sin 600 = 1 /3 / 2 = 2 / 3

Sec 600 = 1 / cos 600 = 1 / ½ = 2

Cot 600 = 1 / tan 600 = 1 /3

ADAB

a32a

BDAB

a2a

ADBD

a3a

600

A

B D C

2a

2a 2a

a a



VALUES OF TRIGONOMETRIC FUNCTION

T. Ratios 0 30 45 60 90

Sine 0 ½ 1/2 3/2 1

Cosine 1 3/2 1/2 ½ 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 OF ANGLES WITH T. RATIOS

Relation of with Sin when 00 900 The greater the value of ‘’, the greater is the

value of Sin.Smallest value of Sin = 0Greatest value of Sin = 1

Relation of with Cos when 00 900 The greater the value of ‘’, the smaller is the

value of Cos.Smallest value of Cos = 0Greatest value of Cos = 1



RELATION OF ANGLES WITH T. RATIOS

Relation of with tan when 00 900 Tan increases as ‘’ increases But ,tan is not defined at ‘’ = 900 Smallest value of tan = 0



SOME MORE FORMULAE If 00 900

1. Sin(900- ) = Cos 2. Cos(900- ) = Sin

If 00< 900

1. Tan(900- ) = Cot 2. Sec(900- ) = Cosec

If 00 < 900

1. Cot(900- )= Tan 2. Cosec(900- ) = Sec

A

CB

p

b

h



SOME MORE FORMULAE

Sin2 +Cos2 = 1

Sec2 -Tan2 = 1

Cosec2 - Cot2 = 1



The End

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