Trigonometric
Ratios
Malvika Kaushik
X–B
Roll no. 25
What Is Trigonometry ?
It is that branch of mathematics that
deals with the measurement of angles
and the problems allied with angles.
The word “Trigonometry” is derived
from Greek words ‘tri’ (meaning
three), ‘gon’ (meaning sides) and
‘metron’ (meaning measure).
It is used to find Distances and Heights
of objects in real life.
WHAT IS TRIGONOMETRIC
RATIOS ?
A TRIGON OM ETRIC RATIO is a rat io
of t he lengt hs of t wo sides in a right
t riangle.
We define the ratios in trigonometry as
“TRIGONOMETRIC RATIOS”.
Trigonom et ric rat ios are also known as
“ T-RATIOS” .
HOW TO USE
TRIGONOMETRY ?
A
Take the right-angled
triangle ABC. Hy
p ot
en
Notice that A and C are us
Base
e
acute angles and B is
right angle.
Acute angles are taken B C
Perpendicular
under consideration
for finding T-ratios.
FIRST, LABEL THE SIDES
Let’stake angle A
under A
consideration.
Hy
p ot
en
Each side is given a us
Base
e
label in relation
to angle ‘A’.
There is the B C
hypotenuse, Perpendicular
perpendicular
and base side.
SO, NOW THE
TRIGONOMETRY
As previously A
mentioned, the
Hy
trigonometric po
te
nu
functions: sine, cosine se
,tangent , cosecant ,
Base
secant and cotangent
C
are ratios of the sides B
Perpendicular
in relation to angle
‘A’.
THE RATIOS ARE AS
FOLLOWS
Sine A
Also written as sin A
A
Cosine A
Also written as cos A
Tangent A
Also written as tan A
Cosecant A
Also written as cosec A
B C
Secant A
Also written as sec A
Cotangent A
Also written as cot A
FIRSTLY, Sin A
The rat io sine is A
short ened t o “ sin” . Hy
p ot
en
per us
sin A =
e
Base
hyp
B C
Perpendicular
Possible values of sine A
are bet ween 0 and 1.
SECONDLY, Cos A
The rat io cosine is A
short ened t o “ cos” . Hy
p ot
en
us
base e
Base
cos A =
hyp
B C
Perpendicular
As wit h sine A, possible
values of cosine A are
bet ween 0 and 1.
THIRDLY, Tan A
The rat io t angent is A
short ened t o “ t an” . Hy
p ot
en
us
per e
Base
tan A =
base
B C
Perpendicular
Any value for t angent A
is possible, bot h
posit ive and negat ive.
FORTHLY, Cot A
A
Hy
The ratio cotangent is p ot
en
shortened to “cot”. us
e
Base
base B C
cot A =
Perpendicular
per
FIFTHLY, Sec A
A
Hy
The ratio secant is p ot
en
shortened to “sec”. us
e
Base
C
hyp B
Perpendicular
sec A =
base
AND FINALLY Cosec A
A
Hy
The ratio cosecant is p ot
en
shortened to “cosec”. us
e
Base
hyp B
Perpendicular
C
cos ecA =
per
HOW TO REMEMBER THEM?
Here is a very simple way of learning them!
(Pandit Badri Prasad)
P B P = sin cos
tan
H H B cosec sec
cot
(Har Har Bhole)
sin A = P/H cos A = B/H tan A = P/B
cot A = B/P sec A = H/B cosec A =
Cosec A = 1
Sin A
RECI
PROC
AL
RELA
TION
FEW
POIN Sec A = 1
TS
ON
Cos A
Tan A = sin A
TRIG TIENT
QUO
cos A
ONO RELA
MET ON
TION
Cot A = cos A
T-
RIC RATIO
S
Cot A = sin
1 A
RATI
OS Tan A
Tan A x Cot A = 1
FOR EXAMPLE :
per 8 4
sin A = = =
hyp 10 5
C
base 6 =3
cos A = =
hyp 10 5
per 8 4
tan A = = = (H) (P)
base 6 3
base 10 8
cot A = 6 3
per = =
8 4
hyp 10 5
sec A = = = A 6 (B)
B
base 6 3
hyp 10 5
cos ecA = = =
per 8 4
EXAMPLE 1 :
Finding the Value of Trigonometric
Ratio
sin A, cos A, tan A, cosec A, sec A and
cot A.
A
17
8
C B
15
SOLUTION :
Sin A =
P 15 H 17
= cos ecA = =
H 17 P 15
B 8 H 17
Cos A = = sec A = =
H 17 B 8
P 15 B 8
Tan A = = cot A = =
B 8 P 15
EXAMPLE 2:
Find the values of all the trigonometric ratios of .
Pythagoras Theorem:
?
4 (3)² + (4)² = c²
θ 5=c
3
SOLUTION :
P 4 B 3
sin θ = = cotθ = =
H 5 P 5
B 3 H 5
cosθ = = secθ = =
P 5 B 3
P 4 H 5
tanθ = = cosecθ = =
B 3 P 4
EXAMPLE 3:
If tan A=4/3 then calculate sinA.cosA
SOLUTION
: A
P BC 4
TanA = = =
B AB 3 3k
Therefore BC=4k,AB=3k, where k is any
positive number.
Now, by using Pythagoras theorem we have B C
4k
AC2 = AB2 + BC2
=(3K)2 +(4K)2
=9K2 + 16K2
AC2 =25K2
AC=5K
SOLUTION :
P BC 4k 4
sin A = = = = A
B AC 5k 5
B AB 3k 3
cos A = = = = 3k 5K
H AC 5k 5
B C
4k
4 3 12
sin A. cos A = × =
5 5 25
Answer= 12/25
COMMON ERROR :
ALERT!!!!!
Students tend to make simple mistakes by
mislabeling the perpendicular and base
To overcom e t his m ist ake first det erm ine t he
Hypot enuse.
The side opposit e t o acut e angle under
Considerat ion is t he perpendicular.
“There is perhaps nothing which so
occupies the
middle position of mathematics as
trigonometry”
