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.



Centesimal System (grade measure):

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

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

1 grade = 100 minutes, written as 100',

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



Circular System (radian measure):

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 1^C).

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 = 90^o = /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:

Ist quadrant	IInd quadrant	IIIrd quadrant	IVth quadrant
All positive	sin , cosec positive	tan , cot positive	sec , cos positive

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:

The Addition 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) =



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

.