The PowerPoint presentation is on the "BASICS OF TRIGONOMETRY". It includes the -- 1) Definition of Trigonometry, 2) History of Trigonometry and its Etymology, 3) Angles of a Right Triangle, 4) About different Trigonometric Ratios, 5) Some useful Mnemonics to remember the Trig. ratios, 6) Theorem, which states that -- "Trigonometric Ratios are same for the same angles" 7) Trigonometric Ratios for some specific/ standard angles.

1. TOPIC : TRIGONOMETRY BASICS
YOGIEGUPTA

2. 1) Definition of Trigonometry.
3) Angles of Right Triangles.
4) About different Trigonometric Ratios.
5) Some useful Mnemonics to remember the Trig. Ratios.
2) History of Trigonometry { Etymology}
6) Theorem {Trig. Ratios are same for same Angles }
7) Trigonometric Ratios of Some Specific Angles.

3. - Sir J.F.Herbert (1890)

4. THE WORD ‘TRIGONOMETRY’ IS DERIVED
FROM THE FROM THE GREEK WORDS --
‘TRI’=THREE ,
‘GON’= ANGLES AND
‘METRON’=MEASURE.
SO, ‘TRIGONOMETRY’= SCIENCE OF
MEASURING SIDES & ANGLES OF TRIANGLES

5. HISTORY OF TRIGONOMETRY
 Early study of Triangle can be traced to the 2nd
millennium ( a period of 1000 years) BC, in
Egyptian and Babylonian Mathematics.
 Systematic study of trigonometric functions
began in Hellenistic Mathematics .
Doyouknow?

6. Angle B
AO
θ
Consider a ray OA. If it rotates about its end points o
and takes the position ob, then we say that the angle
aob has been generated.
Terminal side /
generating line
Initial side
Measure of an Angle
The measure of an angle is the amount of rotation from the initial side to the
terminal side.

8. RIGHT TRIANGLES
 We will only talk about right triangles
 A right triangle is one in which one of the angles is 90°
 Here’s a right triangle:
opposite
Here’s the
right angle
adjacent
Here’s the angle
we are looking at
 We call the longest side the hypotenuse.
 We pick one of the other angles--not the right angle.
 We name the other two sides relative to that angle.

9.  Some ratios of the sides of a triangle with respect to its acute angles
used to find the remaining sides and angles of a when some of its
sides and angles are given.
 Let us take a right ABC,
here angle CAB is acute ,
BC= the side opposite to
angle A, AC= hypotenuse of the right ,
AB= side adjacent to angle A.
NOTE (i) The position of sides changes when
you
consider angle C in place of A.
(ii) The Greek letter θ(theta) is also
used to denote an angle.
Side adjacent to
angle A
Sideoppositeto
angleA
A
B
C
θ

10. Sine Ratio
 When you talk about the sin of an angle, that means you
are working with the opposite side, and the hypotenuse
of a right triangle.
 Given a right triangle, and reference angle A:
in x° =
ypotenuse
pposite
The sin function specifies
these two sides of the
triangle, and they must be
arranged as shown in the
Figure.
opposite
x°
s
o
h

11. Cosine Ratio
 The next trig function you need to
know is the cosine function (cos):
os x° = ypotenuse
djacent
adjacent
x°
c
a
h

12. Tangent Ratio
 The next trig function you need to know
is the tangent function (tan):
an x° =
djacent
pposite
adjacent
opposite
x°
t
o
a

13.  The Sine, Cosine and Tangent ratios in a Right
Triangle can be remembered by representing them
and their corresponding sides as strings of letters.
For instance, a mnemonics
Sine = Opposite ÷ Hypotenuse
Cosine = Adjacent÷ Hypotenuse
Tangent = Opposite ÷ Adjacent.
SOH- CAH- TOA
Another method is to expand the letters such as
“ Saints On High Can Always Have Tea Or Alcohol.

14. COSECANT RATIO
 The next trig function you need to
know is the Cosecant function
(cosec):
cosec x° =
opposite
hypotenuse
x°
opposite

15. SECANT FUNCTION
 The next trig. function you need to
know is the secant function (sec):
sec x° = adjacent
hypotesuse
adjacent
x°

16. COTANGENT FUNCTION
 The next trig function you need to know
is the tangent function (tan):
cot x° =
opposite
adjacent
x°
adjacent
opposite

17. THE RELATIONSHIP BETWEEN TRIG. RATIOS
 The ratios cosec A, sec A and cot A are
respectively, the reciprocals of the ratios
sin A, cos A and tan A.
 Also, observe that tan A=
similarly, cot A =
sin A
cos A
cos A
sin A
NOTE sin A is an abbreviation for sine of angle A

18.  Since a Triangle has three sides, so there are
six ways to divide the lengths of the sides.
 Memorize the Mnemonic-
here P= perpendicular,
B= base, and H= hypotenuse.
 Each of the Six Ratios are-
1) Sine = sin= P/ H
2) Cosine= cos= B/H
3) Tangent= tan= P/B
4) Cosecant=cosec= H/P
5) Secant= sec= H/B
6) Cotangent= cot= B/H
Perpendicular
Base

19. ETYMOLOGY
{AN ACCOUNT OF WORD’S ORIGIN AND
DEVELOPMENT}
 Our modern word “sine” is derived from the
Latin word “sinus” which means “ bay/
bosom or fold”.
 The first use of the idea of ‘sine’ in the way
we use it today was in the work
“Aryabhatiyam” by Aryabhata, in A.D. 500.

20.  Aryabhata used ‘jiva’ for Half-cord ,
when Aryabhatiyam was translated into Arabic
and Latin .
Soon the word jiva was translated into ‘sinus’ which
means ‘curve’, then from ‘sinus’ to ‘sine’ which
became common in Mathematical texts.
 The origin of the terms ‘cosine’ and ‘tangent’ was
much later.
The cosine function arose from the need to
compute the sine of complementary angle.
 Aryabhata called cosine as ‘kotijya’, then used
abbreviation notation ‘cos’.

21. THEOREM:
THE TRIGONOMETRIC RATIOS ARE SAME
FOR THE SAME ANGLE
 AX= initial side , AY= terminal side ,
P and Q be two points on AY.
PM and QN are perpendiculars from P and Q
respectively on AX.
Trigonometric ratios of angle θ are same in both
the AMP and ANQ.
In AMP and ANQ, we have
MAP= XAY= NAQ
and, AMP= ANQ= One right angle.
θ
A
Y
X
P
Q
M N
PROOF
RTP

22.  In AMP, we have
sinθ =
also, in ▲ANQ sinθ =
 This shows that the value of sinθ is
independent of the position of point P.
 Similarly, it can be proved that other
Trigonometric ratios are independent of the
position of point P.
θ
A M
P
Q
N
Y
X
QN
AQ
PM
AP
Thus, the two corresponding angles of triangles AMP and ANQ are
equal and, therefore by AA similarity criterion, we have
=AP
AQ
=PM
QN
AM
AN
PM
AP
QN
AQ
HENCE :

23.  If any one of Trigonometric ratio is given
the we can easily find out all the other ratio’s
also.

24.  Now we shall find the Sine ratios of some Standard Acute
Angles i.e. 0°, 30°, 45°, 60° and 90°.
 We will find the ratios by using some elementary knowledge
of Geometry.
 Please note that,
0
x
= 0, where x is a real number
x
0
= Not Defined, where x is a real number

25. TRIGONOMETRIC RATIO OF 45°
 In▲ABC, right-angled at B, if one angle is 45°,
then the other angle is also 45°, i.e. A= C= 45°.
So, BC = AB
Now, suppose BC= AB= a.
Then by Pythagoras Theorem,
AC²= AB² + BC² = a² + a² = 2a²,
and, therefore, AC = a√2.
 Using the definition of the Trigonometric ratio, we
have :
sin 45°=
A B
C
Side opposite to angle 45°
hypotenuse
=
BC
AC
=
a .
a√2
=
1 .
√2
45°
45°
a
a

26.  Consider an Equilateral Triangle ABC with
each side of length 2a. Now, each angle of
ABC is of 60°.
 Let AD be perpendicular from A on BC.
Therefore, AD is the bisector of A and D is
the mid-point of BC.
 BD = DC = a and BAD = 30°
Thus, in ABD, D is a right angle,
hypotenuse
AB = 2a and BD = a
 So, by Pythagoras Theorem, we have
AB² = AD² + BD²
(2a)² = AD² + a²
AD² = 4a² - a²
AD = √3a
A
B C
Da a
2a 2a
30°30°
60° 60°

27. TRIGONOMETRIC RATIO OF 30°
In right triangle ADB, we have
Base = AD = √3a,
Perpendicular = BD = a,
Hypotenuse = AB = 2a
and DAB = 30°
Therefore ,
sin 30° =
30°30°
2a 2a
D aaB
60° 60°
C
A
BD
AB √3a
=
a .
2a
=
1
2
Trigonometric ratios of 60°
In right angle ADB, we have
Base = BD = a, Perpendicular = AD = √3a, Hypotenuse = AB = 2a
and ABD = 60°
Therefore,
sin 60° =
AD
AB
√3a
2a
√3
2
==

28. TRIGONOMETRIC RATIO OF 0°
 Let XAY = θ be an Acute angle and let P be
a point on its Terminal side AY.
 Draw PerpendicularPM fromP onAX.
 In ▲AMP, we have
sin θ =
 It is evident from▲AMP that as θ becomes
smaller and smaller, line segment PM also
becomes smaller and smaller; and finally
when θ become 0°; the point P coincides
with M.
Consequently, we have PM = 0 and
AP = AM.
PM
AP
sin 0° =
PM
AP
=
0 .
AP
= 0
A
P
M
x
y
θ

29.  Now from ▲AMP, it is evident
that as θ increase, line segment
AM becomes smaller and smaller
and finally when θ becomes 90°
the point M will coincide with A.
Consequently, we have
M
θ
A
P
y
x
TRIGONOMETRIC RATIO OF 90°
AM = 0 and AP = PM
Therefore, sin 90° =
PM
AP
= PM
PM
= 1

30. THE FOLLOWING TABLE GIVES THE VALUES OF SIN RATIOS
0°, 30°, 45°, 60°AND 90° FOR READY REFERENCE.
You would be amazed to know that ratios of cos
θ for some specific angle is just reverse of sin θ.
That is - -

31. NOW,AS WE HAVE ALREADY STUDIED THE RELATION THAT
 Therefore, the Table below shows the
tan θ = PM
AM
=
PM
AP .
AM
AP
=
sin θ
cos θ

32. NOW,AS WE HAVE ALREADY STUDIED THE RELATION THAT
cosec θ
=
1 .
sin
sec θ =
1 .
cos
cot θ =
1 .
sin

33. CONGRATULATIONS
!!!!!