# AIEEE Concepts®

## A Complete Coverage Over AIEEE Exam

### Trigonometry

Trigonometry

Angle and Its Measurement

Angle:

an angle may be defined as a measure of the rotation of a half ray about its origin. An angle XOP is positive if it is traced by a ray revolving in the anticlockwise direction and negative if

it is traced by a ray revolving in the clockwise direction.

Measurement of an Angle:

There are three systems of measurement of an angle:

i) Sexagesimal System or English System

ii) Centesimal system or French System

iii) Circular System

Sexagesimal System (degree measure):

In this system an angle is measured in degrees, minutes and seconds. One complete revolution is 3600 where one degree is written as 10. Further. Thus 10 = 60 minutes (or 60') and 1'

= 60 seconds (or 60''). An angle of 900 is also called a right angle.

In this system an angle is measured in grades, minutes and seconds.

Here 1 right angle = 100 grades, written as 100g,

1 grade = 100 minutes, written as 100',

and 1 minute = 100 seconds, written as 100''.

In this system an angle is measured in radians. A radian is an angle subtended at the centre of a circle by an arc whose length is equal to its radius.

AOB = 1 radian (written as 1C).

Since the whole circle subtends an angle of 3600 (4 right angles) at the centre and the angles at the centre of a circle are in the ratio of substending arcs,

so that

AOB =

1 radian = × 4 right angles = × right angle

p radians = 2 right angles = 1800 = 200 g

1 radian = degree = = 57016'22'' nearly.

Trigonometric Ratios or Circular Functions:

In the above figures, Let OM = x, MP = y and OP = r > 0. The circular functions are defined as:

(i). = sin

(ii). = cos

(iii). = tan , x 0

(iv). = cot , y 0

(v). = cosec , y 0

(vi). = sec , x 0

Trigonometric ratios (or functions) may also be defined with respect to a triangle.

In a right angled triangle ABC, CAB = A and BCA = 90o = /2. With reference to angle A, the six trigonometric ratios are:

is called the sine of A, and written as sinA.

is called the cosine of A, and written as cosA.

is called the tangnet of A, and written as tanA.

Obviously,. The reciprocals of sine, cosine and tangent are called the cosecant, secant and cotangent of A respectively. We write these as cosecA, secA, cotA

respectively. Since the hypotenuse is the greatest side in a right angle triangle, sinA and cosA can never be greater than unity and cosecA and secA can never be less than unity. Hence

sin A 1, cos A 1,cosec A 1, sec A1, while tan A and cot A may take any numerical value.

Note:

All the six trigonometric functions have got a very important property in common that is of periodicity.

Remember that the trigonometrical ratios are real numbers and remain same as long as angle A is real.

Signs of Trigonometric Ratios:

The following table describes the signs of various trigonometric ratios:

Trigonometrical Identities:

An identity is a relation which is true for all values of the independent variable. There are three fundamental identities involving trigonometrical ratios:

i) sin2 + cos2 = 1

ii) 1 + tan2 = sec2

iii) 1 + cot2 = cosec2

Trigonometric Ratios of Standard Angles

 Angle Ratios Sin Cos Tan Cosec Sec Cot 0o 0 1 0 Not defined 1 Not defined 30o 2 45o 1 1 60o 2 90o 1 0 Not defined 1 Not defined 0

Allied Angles

Table - I
 equals sin cos tan cot sec cosec - - sin cos -tan - cot sec -cosec 90o - cos sin cot tan cosec sec 90o + cos - sin -cot - tan -cosec sec 180o- sin - cos - tan - cot - sec cosec 180o+ - sin - cos tan cot - sec -cosec 360o- - sin cos - tan - cot sec -cosec 360o+ sin cos tan cot sec cosec

Note:

Angle

and 90
o
-

are complementary angles,

and 180
o
- q are supplementary angles

sin(np + (- 1)
n
) = sin

, n

I

cos(2np

) = cos

, n

I

tan(np +

) = tan

, n

I

i.e. sine of general angle of the form np + (-1)
n
will have same sign as that of sine of angle

and so on. The same is true for the respective reciprocal functions also.

Trigonometric Ratios of Compound Angles:

(i). sin (A + B) = sin A cos B + cos A sin B

(ii). cos (A + B) = cos A cos B - sin A sin B

(iii). tan (A + B) =

Subtraction Formulae:
(i). sin (A - B) = sin A cos B - cos A sin B

(ii). cos (A - B) = cos A cos B + sin A sin B

(iii). tan (A - B) =

Some Important Deductions:
(i). sin (A + B) sin (A - B) = sin
2
A - sin
2
B

(ii). cos (A + B) cos (A - B) = cos
2
A - sin
2
B

(iii). cot (A + B) =

(iv). cot (A - B) =

Transformation Formulae:
(a) Transformation of products into sums or differences:

(i). 2 sin A cos B = sin (A + B) + sin (A - B)

(ii). 2 cos A sin B = sin (A + B) - sin (A - B)

(iii). 2 cos A cos B = cos (A + B) + cos (A - B)

(iv). 2 sin A sin B = cos (A - B) - cos (A + B)

The above formulae can be easily derived by taking the sum and the difference of the addition and subtraction formulae.

(b) Transformation of sums or differences into products:

(i). sin (A + B) + sin (A - B) = 2 sin A cos B

or, sin C + sin D = 2 sin

cos

(ii). sin (A + B) - sin (A - B) = 2 cos A sin B

or, sin C - sin D = 2 cos

sin

(iii). cos (A + B) + cos (A - B) = 2 cos A cos B

or, cos C + cos D = 2 cos

cos

(iv). cos (A - B) - cos (A + B) = 2 sin A sin B

or, cos C - cos D = 2 sin

sin

Here A - B = C and A + B = D

A =

and B =

.

(v). tan A + tan B =

.

Trigonometric Ratios of Multiple Angles:
(i). sin 2A = 2 sin A cos A

(ii). cos 2A = cos
2
A - sin
2
A = 2 cos
2
A - 1 = 1 - 2 sin
2
A

(iii). tan 2A =

(iv). sin 3A = 3 sin A - 4 sin
3
A = 4 sin (60
0
- A) sin A sin (60
0
+ A)

(v). cos 3A = 4 cos
3
A - 3 cos A = 4 cos (60
0
- A) cos A cos (60
0
+ A)

(vi). tan 3A =

= tan (60
0
- A) tan A tan (60
0
- A).

Trigonometric Ratios of Submultiple of an Angle:
(i). sin A = 2 sin

cos

(ii). cos A = cos
2
- sin
2
= 2 cos
2
- 1 = 1 - 2 sin
2
(iii). tan A =

(iv). sin A = 3 sin

- 4 sin
3
(v). cos A = 4 cos
3
- 3 cos

(vi). tan A =

or sin

+cos

=

or sin

- cos

=

tan

=

The ambiguities of signs are removed by locating the quadrant in which

lies.

| a cosA + b sinA |

Also cosA

sinA =

Notes:

Any formula that gives the value of sin

in terms of sinA shall also give the value of sine of

.

Any formula that gives the value of cos

in terms of cosA shall also give the value of cos of

.

Any formula that gives the value of tan

in terms of tanA shall also give the value of tan of

(a) For any angles A, B, C:
sin (A + B +C) = sinA cosB cosC + cosA sinB cosC + cosA cosB sinC - sinA sinB sinC

cos (A + B +C) = cosA cosB cosC- cosA sinB sinC - sinA cosB sinC - sinA sinB cosC

;

(b) If A, B, C are the angles of a triangle ( or A + B + C =

), then

sinA cosB cosC + cosA sinB cosC + cosA cosB sinC = sinA sinB sinC

cosA sinB sinC + sinA cosB sinC + sinA sinB cosC = 1 + cosA cosB cosC

tanA + tanB + tanC = tanA tanB tanC

cotB cotC + cotC cotA + cotA cotB = 1

sin2A + sin2B + sin2C = 4sinA sinB sinC

cos2A + cos2B + cos2C = -1-4cosA cosB cosC

cos
2
A + cos
2
B + cos
2
C = 1 - 2cosA cosB cosC

Graphs of Trigonometric Functions:
We have seen that all trigonometric function are periodic. Since sin (2

+ x) = sin x, cos (2

+ x) = cos x, and tan (

+ x) = tan x, the period of sine and cosine functions is 2p where as

the period of tangent function is

. Moreover, period of sin ax or cos ax is

and that of tan ax is

.