1.
INTRODUCTION
TO
TRIGONOMETRY

2.
There is perhaps nothing which so
occupies the middle position of
mathematics as trigonometry.
- J.F. Herbart

3.
INDEX
• Introduction
• Trigonometric ratios
• Trigonometric ratios of some specific angles
• Trigonometric ratios of complementary
angles
• Trigonometric identities

4.
INTRODUCTION
• We will get to know about some ratios of the sides of
a right triangle with respect to its acute angles called
TRIGNOMETRIC RATIOS OF ANGLES.
• We will also define the trigonometric ratios for the
angles of measures 0° and 90°. We will calculate
trigonometric ratios for some specific angles and
establish some identities involving these ratios, called
TRIGNOMETRIC IDENTITIES.

5.
What is TRIGONOMETRY??
• TRIGONOMETRY is that branch of
mathematics which deals with the measurement
of the sides and the angles of a triangle.
• The word TRIGONOMETRY is derived from
Greek words: ‘tri’(meaning three),
‘gon’(meaning sides) and ‘metry’(meaning
measure).

6.
TRIGONOMETRIC RATIOS
• Let us consider a right triangle.
• Here angle A is an acute angle. The position of the side BC with
respect to angle A is the side opposite to angle A. AC is the
hypotenuse of the right angled triangle and the side AB is the
part of angle A. So, we call it as the side adjacent to angle A.
• The trigonometric ratios of angle A in right angle ABC are
defined as follows :

7.
• Sin A= side opposite to angle A = BC
hypotenuse AC
• Cosine A= side adjacent to angle A = AB
hypotenuse AC
• Tangent A= Side opposite to angle A = BC
Side adjacent to angle A AB
• Cosecant A= 1 = AC
Sine A BC
• Secant A= 1 = AC
Cosine A AB
• Cotangent A= 1 = AB
Tangent A BC
• The ratios are abbreviated as sin A, cos A, tan A, cosec A, sec A and cot A respectively.
• The ratios cosec A, sec A and cot A are the reciprocals of the ratios sin A, cos A, and tan A
respectively.
• Also, tan A= sin A and cot A= cos A
cos A sin A
• So, the Trigonometric ratios of an acute angle in a right triangle express the relationship
between the angle and the length of its sides.

8.
Let us take an example..
• If sin A= 3 , find the other trigonometric ratios
5
of the angle A.
Solution:- Let a constant term be ‘a’
sin A= perpendicular = BC = 3a
hypotenuse AC 5a
By Pythagoras Theorem,
h²= p² + b²
(5a) ²= (3a) ² + b²
25a²= 9a² + b²
b²= 25a² – 9a²
b²=16a²
B= 4a

9.
Therefore, cos A= AB = 4a = 4
AC 5a 5
Tan A= BC = 3a = 3
AB 4a 4
Cosec A= AC = 5a = 5
BC 3a 3
Sec A= AC = 5a = 5
AB 4a 4
Cot A= AB = 4a = 4
BC 3a 3

10.
TRIGONOMETRIC RATIOS OF
SOME SPECIFIC ANGLES

11.
TRIGONOMETRIC RATIOS OF
COMPLEMENTARY ANGLES
• As we know that, ABC is a right
Angled triangle, right angled at B,
We know,
Sin A= BC Cos A= AB Tan A= BC
AC AC AB
Cosec A= AC Sec A= AC Cot A= AB ______❶
BC AB BC
• There is one pair of complementary angle in the above triangle.
angleA +angleC = 90
Therefore, C= 90- A
AB is the opposite side and BC is the adjacent side to the angle 90- A.
So, Sin (90-A)= BC Cos (90-A)= AB Tan (90-A)= BC
AC AC AB

12.
Cosec (90-A) = AC Sec (90-A) = AC Cot (90-A) =AB __❷
BC AB BC
• When we compare the ratios in ❶ and ❷, we get:
Sin (90-A)= Cos A Cos (90-A)= Sin A Tan (90-A)= Cot A
Cosec (90-A) = Sec A Sec (90-A) = Sec A Cot (90-A) = Tan A

13.
TRIGONOMETRIC IDENTITIES
• In triangle ABC, right angled at B:
AC²=AB²+BC²(By Pythagoras Theorem)
Dividing each term by AC², we get
AC² = AB² + BC²
AC² AC² AC²
1= (Cos A) ²+(Sin A)² [ Cos A= AB and Sin A= BC ]
1=Cos²A + Sin²A ______❶ AC AC
This is first trigonometric identity.

14.
Again dividing AC²=AB²+BC² BY AB², we get,
AC²=AB²+BC²
AB² AB² AB²
(SecA) ²= 1+(TanA)²
Sec²A= 1+ Tan²A_______❷
This is second trigonometric identity.
Now, dividing AC²=AB²+BC² BY BC², WE GET,
AC²=AB²+BC²
BC² BC² BC²
(CosecA) ²= (TanA)²+1
Cosec²A= Tan²A+1_______❸
This is third trigonometric identity.

15.
• Using these identities, we can express each trigonometric ratio in
terms of other trigonometric ratios, i.e. if any one of the ratios is
known, we can also determine the values of other trigonometric
ratios.
• Let’s see how we can do this using these
identities.
We know that,
Tan A=1 and Cot A=∫3
∫3
And by 2nd identity, we know
Sec²A=1+ Tan²A =1+(1/∫3)²= 1+1/3=4/3
Sec A=∫4/3= 2/∫3
Cos A= ∫3/2
Sin²A=1- Cos²A=1-(∫3/2)²= 1-3/4=1/4
Sin A=∫1/4=1/2
Cosec A=2