Elementary Trigonometry Final Setup

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1 Definitions.(1) Angle :The motion of any revolving line in a plane from its initial position

(initial side) to the final position (terminal side) is called angle. The end point O

about which the line rotates is called the vertex of the angle.

(2) Measure of an angle : The measure of an angle is the amount of

rotation from the initial side to the terminal side.

(3) Sense of an angle : The sense of an angle is

determined by the direction of rotation of the initial side into the

terminal side. The sense of an angle is said to be positive or

negative according as the initial side rotates in anticlockwise or

clockwise direction to get the terminal side.

1. 2.

3.

4. IN

Terminal side

5. 6.

7.

Positive angle

8.

9. 10

Negative angle

11

12


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(4) Right angle : If the revolving ray starting from its initial position to final position describes one

quarter of a circle. Then we say that the measure of the angle formed is a right angle.

(5) Quadrants: Let OXX' and 'YOY be two lines at right angles in the

plane of the paper. These lines divide the plane of paper into four equal

parts. Which are known as quadrants. The lines OXX' and 'YOY are

known as x-axis and y-axis. These two lines taken together are known as

the co-ordinate axes.

(6) Angle in standard position : An angle is said to be in standard position if its vertex conicides

with the origin O and the initial side coincides with OXi.e., the positive direction ofx-axis.

(7) Angle in a quadrant : An angle is said to be in a particular quadrant if the terminal side of the

angle in standard position lies in that quadrant.

(8) Quadrant angle: An angle is said to be a quadrant angle if the terminal side coincides with one of

the axes.

2 System of Measurement of Angles

There are three system for measuring angles

(1) Sexagesimal or English system : Here a right angle is divided into 90 equal parts known as

degrees. Each degree is divided into 60 equal parts called minutes and each minute is further

divided into 60 equal parts called seconds. Therefore, 1 right angle =90 degree )90( o=

60=o minutes )60( '=

60'1 = second )''60(=

(2) Centesimal or French system : It is also known as French system, here a right angle is

divided into 100 equal parts called grades and each grade is divided into 100 equal parts, called

minutes and each minute is further divided into 100 seconds. Therefore,

1 right angle = 100 grades )100( g=

1 grade = 100 minutes )'100(=

1 minute = 100 seconds )''100(=

X

Y

X

IV quadrantIII quadrant

II quadrant I quadrant

O

Y


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(3) Circular system : In this system the unit of measurement is radian. One radian, written

as c , is the measure of 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 rhaving centre at O. LetA be a point on the

circle. Now cut off an arcAP whose length is equal to the radius rof the

circle. Then by the definition the measure of AOP is 1 radian )1( c= .

3 Relation between Three Systems of Measurement of an Angle.LetD be the number of degrees, R be the number of radians and G be the number of grades in an

angle q. Now, o9 = 1 right angle 911 =o right angle

9

DDo = right angles

9

D=q right angles ..(i)

Again, pradians = 2 right angles 1 radianp

2= right angles

R radians

p

R2= right angles anglesright

2

p

qR

= ..(ii)

and 100 grades =1 right angle 1 grade10

1= right angle

G grades100

G= right angles

100

G=q right angles ..(iii)

From (i), (ii) and (iii) we get,

RGD 210090

==

This is the required relation between the three systems of measurement of an angle.

Note :q One radian pp

=o180

radians o180= 1 radian = 57o 5471578.4471 o .

A

P

13.


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4 Relation between an Arc and an Angle.Ifsis the length of an arc of a circle of radius r, then the angle q (in radians) subtended by this

arc at the centre of the circle is given by r

s

=q or r

s = i.e., arc = radius angle in radians

Sectorial area : Let OABbe a sector having central angle Cq and radius r. Then area of the sector

OABis given by r2

2

1.

Important Tips

F The angle between two consecutive digits in a clock is 30o

(= p/6 radians). The hour hand rotates through an angle of30o

in one hour.

F The minute hand rotate through an angle of6o

in one minute.

5 Trigonometrical Ratios or Functions.

In the right angled triangle OMP, we have base = OM = x,

perpendicular =PM =y and

hypotenues = OP =r. We define the following trigonometric ratio

which are also known as trigonometric function.

r

y==

Hypotenues

larPerpendicusinq

r

x==

Hypotenues

Basecosq

y==

Bas

larPerpendicutanq

y

x==

larPerpendicu

Basecotq ,

r==

Bas

Hypotenuessecq

y

r==

larPerpendicu

Hypotenuescosecq

(1) Relation between trigonometric ratio (function)

(i) 1sec.sin =qqco (ii) 1cot.tan =qq

(iii) 1sec.cos =qq (iv)q

qq

cos

sintan = (v)

q

qq

sin

coscot =

(2) Fundamental trigonometric identities

(i) 1cossin 22 =+ qq (ii) qq 22 sectan1 =+ (iii) qq 22 coseccot1 =+

A

B

14.15.

s

r

Y

P(x,y)

MX

Oq

A

x

r


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Important Tips

F If qsec=x + qtan , then qq tansec1

-=x

.

F If ,cotcoesc qq +=x then qq cotcosec1

-=

x

.

(3) Sign of trigonometrical ratios or functions : Their signs depends on the quadrant in which the

terminal side of the angle lies.

(i) In first quadrant : ,0tan,0cos,0sin0,0 >=>=>=>>y

r

x

r

yyx qqq 0cosec >=

y

rq ,

0sec >=r

q and 0cot >=

y

xq . Thus, in the first quadrant all trigonometric functions are positive.

(ii) In second quadrant : ,0cosec,0tan,0cos,0sin0,0 >=+ .

3. Show that the equation sin(x + a ) = a sin 2x + b has four roots whose sum is equal to (2n + 1)p;where nz.

4. Find the values of a and b such that 0 < a , b

0) such that the length of the longest interval in which the funct ion

f(x) = sin1

|sin kx| + cos1

(cos kx) is constant is4

p.

6. In an isosceles triangle ABC with AB = AC, the bisector of angle B meets the side AC at D. Prove

that BC = AB + AD, if and only if triangle ABC is right angled triangle with A =2p .

7. In a triangle ABC if r1 + r2 + r3 = s, then prove thata b c

2 33

+ + D and hence show thatA B C 1

tan tan tan2 2 2 27 .

8. f(x, y, z) = cosx + cosy cosz, if x + y + z = p, then prove that 1 < f(x, y, z) < 3.

9. If in a triangle ABC, line joining the circumcentre and orthocentre is parallel to the side AC, thenprove that tan A, tan B and tan C are in arithmetic progression.

10. In a triangle the area D and the semiperimeters are fixed. Prove that the local extremum of a side isa root of the equation sx

2(x s) + 4D2 = 0.


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TrigonometrySl.No Name of the Topic Estimated Days

1 Elementary Trigonometry 8

2 General Solutions 6

3 Inverse Trigonometry 8

4 Solutions of Triangles 12

2D - Geometry

1 Straight Lines 182 Pair of Straight Lines 6

Differential Calculus1 Functions 8

2 Limits, Continuity & Differentiability 14

3 Differentiation 5

4 Application of Differentiation 8

5 Mean Value Theorems 36 Monotonocity & Maxima & Minima of

Functions

10

3D Geometry1 Co-ordinate System, DCs; DRs 5

2 The Plane 8

3 The Line 8

4 Line & Plane 6

Vector Algebra1 L.C., L.I., L.D. 6

2 Product of 2 Vectors 6

3 Triple Product 6

4 Application of Vectors 4

Total Number of Estimated Classes


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Coordinate GeometrySl.No Name of the Topic Estimated Days

1 Circles & System of Circles 12

2 Parabola 10

3 Ellipse 10

4 Hyperbola 10

Integral Calculus1 Indefinite Integrals 12

2 Definite Integrals 10

3 Areas 8

4 Differential Equations 8

Algebra

1 Quadratic Equations & Expressions 122 Matrices & Determinants 12

3 Progressions 8

4 Binomial Theorem 8

5 Permutations & Combinations 12

6 Probability 12

7 Complex Numbers 10

Total Number of Estimated Classes

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Elementary Trigonometry Final Setup