MATHEMATICS POWERPOINT PRESENTATION

TOPIC : TRIGONOMET RY BASICS

CREATED BY : YOGIE GUPTACLASS : 10TH -- ‘ A ’

Right Triangle Trigonometry

Topics Checklist1) 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.

There is perhaps nothing which so occupies the middle position of Mathematics as Trigonometry

- Sir J.F.Herbert (1890)

What is Trigonometry ???....

LET’S

iNVESTIGAT

E

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

Trigonometry is a branch of math'swhich deals with the measurement of

the sides and angles of a right .

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 .

Do you know ?

Hipparchus, credited the first Trigonometry Table, he is known

as “The Father of Trigonometry”

Angle B

AOθ

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 sideAn angle is considered as a figure obtained

by rotating a given ray about its endpoint. Measure of an

AngleThe measure of an angle is the amount of rotation from the initial side to the terminal side.

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:

oppo

siteHere’s the

right angle hypotenuse

adjacent

Here’s the anglewe 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.

TRIGONOMETRIC RATIOS 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.Hy

poten

use

Side adjacent to angle A

Side opposite to angle A

A B

C

θ

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° = ypotenusepposite The sin function specifies

these two sides of the triangle, and they must be arranged as shown in the Figure.

hypotenuse

opposite

x°

S-O-H

so

h

Cosine Ratio The next trig function you need

to know is the cosine function (cos):

os x° = ypotenusedjacent

hypotenuse

adjacent

x°

c

C- A- H

ah

Tangent Ratio The next trig function you need to

know is the tangent function (tan):an x° =

djacentpposite

adjacent

opposite

x°

t oa

T- O- A

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.

MNEMONICS FROM WIKIQUOTE

SOH- CAH- TOA

Another method is to expand the letters such as“ Saints On High Can Always Have Tea Or Alcohol.

COSECANT RATIO The next trig function you need

to know is the Cosecant function (cosec):

cosec x° = oppositehypotenuse

x°

hypotenuse opposite

SECANT FUNCTION The next trig. function you need

to know is the secant function (sec):

sec x° = adjacenthypotesuse

hypotenuse

adjacent

x°

COTANGENT FUNCTION

The next trig function you need to know is the tangent function (tan):

cot x° = oppositeadjacent

x°

adjacent

opposite

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 =

DID YOU OBSERVED !!!!

sin Acos Acos Asin ANOTE sin A is an abbreviation for sine

of angle A

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/ H2) Cosine= cos= B/H3) Tangent= tan= P/B4) Cosecant=cosec= H/P5) Secant= sec= H/B6) Cotangent= cot= B/H

RATIOS

PBPHHB

Hypotenus

e

Perp

endi

cula

rBase

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.

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’.

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.

θA

Y

X

PQ

M N

PROOF

RTP

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.

θA M

PQ

N

Y

X

QNAQ

PMAP

Thus, the two corresponding angles of triangles AMP and ANQ are equal and, therefore by AA similarity criterion, we have

=AP AQ =PM

QNAMAN

PMAP

QNAQ

The values of the Trigonometric ratios of an angle do not vary with the lengths of the sides of the triangle, if the angle remains the same.

HENCE :

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

Since, the Hypotenuse is the longest side in a

Right triangle, the value of Sin A or Cos A is always less then 1 (or, in particular, equal

to 1).

REMARK :-

TRIGONOMETRIC RATIOS OF SOME SPECIFIC ANGLES

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

x0

= Not Defined, where x is a real number

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

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

Trigonometric Ratios of 30° & 60°

A

B CDa a

2a

2a

30°30°

60° 60°

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 aaB60° 60° C

A

BDAB

√3a

= a . 2a

=12

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° =

ADAB

√3a 2a

√3 2

==

TRIGONOMETRIC RATIO OF 0° Let XAY = θ be an Acute angle

and let P be a point on its Terminal side AY.

Draw Perpendicular PM from P on AX.

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.

PMAP

sin 0° = PMAP = 0

.AP

=0

A

P

M x

y

θ

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

MθA

Py

x

TRIGONOMETRIC RATIO OF 90°

AM = 0 and AP = PM

Therefore, sin 90° =

PMAP

= PMPM

= 1

THE FOLLOWING TABLE GIVES THE VALUES OF SIN RATIOS0°, 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 - -

NOW, AS WE HAVE ALREADY STUDIED THE RELATION THAT

Therefore, the Table below shows the

tan θ =

PMAM

= PM AP .

AMAP

= sin θ cos θ

NOW, AS WE HAVE ALREADY STUDIED THE RELATION THAT

cosec θ =

1 .sin

sec θ = 1 .cos

cot θ = 1 .sin

CONGRATULATIONS !!!!!

You have Successfully learned the Basic Concept of TRIGONOMETRY !!!!

GREAT JOB !!!