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Trigonometry Identities

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• TRIGONOMETRIC RATIOS & IDENTITIES

1. INTRODUCTION

The word 'Trigonometry' is derived from two Greek words

(1) Trigonon and(2) MetronThe word trigonon means a triangle and the word metron means a measurement. Hence trigonometry means thescience of measuring triangles.

2. ANGLE

Consider a ray OAuuur

. If this ray rotates about its end point O and takes the position OB , then the angle AOB has

been generated.

Vertex O

= angle

Initial side A

B

Termin

al side

An angle is considered as the figure obtained by rotating a given ray about its end - point.

The initial position OA is called the initial side and the final position OB is called terminal side of the angle. The endpoint O about which the ray rotates is called the vertex of the angle.

3. SENSE OF AN ANGLE

The sense of an angle is said to be positive or negative according as the initial side rotates in anticlockwise orclockwise direction to get to the terminal side.

O = +ve

Anticlockwise directionA

B O = ve

Clockwise directionA

B

4. RIGHT ANGLE

When two lines intersect at a point in such a way that two adjacent angles made by them are equal, then eachangle is called a right angle.

9090

OX' X

A

5. A CONSTANT NUMBER pipipipipiThe ratio of the circumference to the diameter of a circle is always equal to a constant and this constant is denotedby the Greek letter pi

TRIGONOMETRIC RATRIGONOMETRIC RATRIGONOMETRIC RATRIGONOMETRIC RATRIGONOMETRIC RATIOS & IDENTITIESTIOS & IDENTITIESTIOS & IDENTITIESTIOS & IDENTITIESTIOS & IDENTITIES

• TRIGONOMETRIC RATIOS & IDENTITIES

i.e. Circumference of a circle

Diameter of the circle= pi

(constant)

The constant pi is an irrational number and its approximate value is taken as 227 . The more accurate value to six

decimals places is taken as 355113 .

6. SYSTEMS OF MEASUREMENT OF AN ANGLESThere are three systems for measuring angles

6.1 Sexagesimal or English system6.2 Centesimal or French system6.3 Circular system

6.1 Sexagesimal system : The principal unit in this system is degree (). One right angle is divided into 90 equalpart and each part is called one degree (1) . One degree is divided into 60 equal parts and each part is calledone minute. Minute is denoted by (1'). One minute is equally divided into 60 equal parts and each part is calledone second (1").In Mathematical form :

One right angle = 90 (Read as 90 degrees )1 = 60' (Read as 60 minutes )1' = 60" (Read as 60 seconds )

Ex.1 40 30' is equal to

(1) o41

2

(2) 81 (3) o81

2

(4) None of these

Sol. We know that , 30' = o1

2

; 40 + o1

2

=

o812

6.2 Centesimal system : The principal unit in system is grade and is denoted by (g). One right angle is dividedinto 100 equal parts, called grades, and each grade is subdivided into 100 minutes, and each minute into 100seconds.

In Mathematical form :

Ex.2 25' is equal to -Sol. 100' is equal to 1g

so is equal to g g1 125

100 4

=

Relation between Sexagesimal and Centesimal systems :One right angle = 90 (degree system) ..... (1)

• TRIGONOMETRIC RATIOS & IDENTITIES

One right angle = 100g (grade system) ..... (2)by (1) and (2)

90 = 100g or , D G90 100

=

then we can say, ; 1 = g10

9

, 1g = o9

10

Ex.3 80g is equal to

Sol. We know that 1g = o9

10

then, 80g = o9 80

10

80g = 72

6.3 Circular system : In circular system the unit of measurement is radian. One radian, written as 1C, is the measureof an angle subtended at the centre of a circle by an arc of length equal to the radius of the circle.Consider a circle of radius r having centre of O. Let A be a point on the circle. Now cut off an arc AB whoselength is equal to the radius r of the circle.

B O rIc

rr

C

A

In the adjacent figure OA = OC = arc AC = r = radius of circle, then measurement of AOC is one radianand denoted by 1c. Thus AOC = 1c .

6.3.1 Some Important conversion

pi 6pi

4pi

23pi

76pi

• TRIGONOMETRIC RATIOS & IDENTITIES

Ex.4 240 is equal to

[1] C4

3

p [2] C3

4

p [3] '4

3p [4]

'34p

Sol. We know that180 = pi

240 = C

x240180

p

=

C43

p

Ans. [1]

Ex.5 The difference between two acute angle of a right angle triangle is 9p . Then the angles in degree are -

[1] 50, 30 [2] 25, 45 [3] 20, 40 [4] 35, 55Sol. In triangle ABC let C = 90

So A B =

9p = 20 .......... (i)

and sum of all the angles in ABC

A + B + C = 180

C = 90

A + B = 90 ......... (ii)Solving (i) & (ii) A = 55, B = = 35 Ans. [4]

6.3.2 Relation between systems of measurement of angles

D G 2C90 100

= =

pi

Ex.6 The length of an arc of a circle of radius 5 cm subtending a central angle measuring 15 is -

[1] 312p

cm [2] 712p

cm [3] 512p [4] None of these

Sol. Let s be the length of the arc subtending an angle at the centre of a circle of radius r.

then, = s

r

Here, r = 5 cm, and = 15 = C

15x180

p

= C

12

p

= s

r = 12

p =

s

5

= 512

pcm

• TRIGONOMETRIC RATIOS & IDENTITIES

7. TRIGONOMETRICAL RATIOS OR FUNCTIONSLet a line OA makes angle with a fixed line OX and AM is perpendicular from A on OX. Then in right-angledtriangle AMO, trigonometrical ratios (functions) with respect to are defined as follows :

sin = perpendicular(P)hypotenuse(H)

cos = base(B)

hypotenuse(H)

tan = perpendicular (P)

Base (B)

cosec = HP . sec =

HB , cot =

BP

Note :

(i) Since t-ratios are ratio between two sides of a right angled triangle with respect to an angle, so they arereal numbers.(ii) may be acute angle or obtuse angle or right angle.

8. RELATIONS BETWEEN TRIGONOMETRICAL RATIOS

(i) 1 1 1cosec , sec ,cotsin cos tan

= = = (ii)

sintancos

=

(iii) coscotsin

= (iv) sin

2 + cos2 = 1

(v) 1 + tan2 = sec2 (vi) 1 + cot2 = cosec2Ex.7 If cosec A + cot A = 11/2, then tan A is equal to

[1] 21/12 [2] 15/16 [3] 44/117 [4] 117/43Sol. Cosec A + cot A = 11/2

1cosecA cot A+ =

211 ....... (1)

cosec A cot A = 211 ..... (2)

(1) (2) = 2 cot A = 11 22 11- = 11722

= tan A = 44

117 Ans [3]

O B

M

H P

Y

X

A

• TRIGONOMETRIC RATIOS & IDENTITIES

Ex.8cos

1 tanq

q- + sin

1 cotq

q- is equal to

[1] sin cos [2] sin + cos [3] tan + cot [4] tan cot

Sol.cos

1 tanq

q- + sin

1 cotq

q-

=

cos sinsin cos1- 1cos sin

q qq qq q

+

-

=

2 2cos sincos sin cos sin

q qq q q q

-

- -

=

2 2cos sincos sin

q qq q

-

-

= cos + sin Ans [2]Ex.9 tan2 sec2 (cot2 cos2 ) equals

[1] 0 [2] 1 [3] 1 [4] 2Sol. tan2 sec2 (cot2 cos2 )

= sec2 (tan2 cot2 tan2 cos2 )

= sec2 2

22

sin1 coscos

q qq

- = sec2 (1sin2)

= sec2. cos2 = 1 Ans. [3]

9. SIGN OF TRIGONOMETRIC RATIOS(i) All ratios sin, cos, tan cot, sec and cosec are positive

in Ist quadrant.(ii) sin( or cosec) positive in IInd quadrant, rest are negative.(iii) tan( or cot) positive in IIIrd quadrant, rest are negative.(iv) cos( or sec) positive in IVth quadrant, rest are negative.

Ex.10 The value of sin and tan if cos = 1213

-

and lies in the third quadrant is -

[1] 513- and 5

12 [2] 5

12 and 5

13- [3] 1213- and

513

- [4] none of these

Sol. We have cos2 + sin2 = 1

sin = 21 cos q -

x-axis

y-axis

• TRIGONOMETRIC RATIOS & IDENTITIES

In the third quadrant sin is negaitve, therefore

sin = 21 cos q- sin = 2121

13

- = 5

13

then, tan = sincos

qq tan =

5 13x

13 12-

-

=

512 Ans.[1]

Ex.11 If 2p

< < pi, then 1 sin1 sin

qq

-

+ +

1 sin1 sin

qq

+

-

is equal to

[1] 2 cosec [2] 2 cosec [3] 2 sec [4] 2 sec

Sol. Exp. = 2(1 sin ) (1 sin )

1 sinq q

q

- + +

- =

2cosq

-

= 2 sec Ans.[4]

10. DOMAIN AND RANGE OF A TRIGONOMETRICAL FUNCTIONIf f : X Y is a function, defined on the set X, then the domain of the function f, written as Domain is theset of all independent variables x, for which the image f(x) is well defined element of Y, called the co-domainof f.Range of f : X Y is the set of all images f(x) which belongs to Y , i.e.Range f = {f(x) Y:x X} Y The domain and range of trigonometrical functions are tabu