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What is trigonometry ?What is trigonometry ?

Trigonometry is a branch of Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles gbetween these sides.

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What is trigonometry ?What is trigonometry ?The word Trigonometry comes f t G k d from two Greek words,

trigonon = triangle, and trigonon triangle, and

metron = measure.

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Origin of Trigonometry

The method was originated in the second

Origin of Trigonometry

century B.C. by Hipparchus and other early Greek astronomers in their attempts to solve certain spherical triangles. The term trigonometry was not used until the close of the sixteenth century.

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Labeling Right Angled TrianglesThe most important skill you need The most important skill you need right now is the ability to correctly l b l h d f h l d label the sides of a right angled triangle.The names of the sides are:

the hypotenusethe hypotenusethe opposite sidethe adjacent side

Vikasana – Bridge Course 2012the adjacent side

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Labeling Right Triangles

The hypotenuse is easy to locate because it is side opposite to the right pp gangle.

Since this side is across from the

Here is the right angle

across from the right angle, this must be the h tright angle... hypotenuse.

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Before you label the other two Before you label the other two sides you must have a reference angle selectedangle selected.

It can be either of the two acute anglesangles.

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In the triangle below, let’s pick angle B In the triangle below, let s pick angle B as the reference angle.

B This will be our reference angle...

A C

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Remember, angle B is our reference gangle.The hypotenuse is side BC because it is The hypotenuse is side BC because it is across from the right angle.

B (ref angle)B (ref. angle)

hypotenuse

A CVikasana – Bridge Course 2012

A C

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Side AC is opposite to reference Side AC is opposite to reference angle B. So it is labeled: opposite.

B (ref. angle)

hypotenuse

A Copposite

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The only side unnamed is side AB. yThis must be the adjacent side.

B (ref. angle)

adjacent h tadjacent hypotenuse

A Copposite

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TRIGONOMETRIC RATIOS:

There are six trigonometric ratios:Sine abbreviation sinSine abbreviation sinCosine abbreviation cosTangent abbreviation tanTangent abbreviation tanSecant abbreviation secCosecant abbreviation cosecCosecant abbreviation cosecCotangent abbreviation cot

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Let us define these trigonometric gratios for right angled triangle:

C

site

AB

op

po

s

AB adjacent

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We need a way We need a way to remember all of these ratiosratios…

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ppThroughOurOld HippieOld Hippie OurApartment

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O A OA Sin

Opp

SOH CAH TOAOpp

Hyp

Cos

Adjj

Hyp

T

Old Hi iOld Hi iTanOpp

Old HippieOld Hippie Adj18

Consider the right angled triangle g g gwith sides 3, 4 and 5 units

3 5CHere opp = 3

adj = 4Here the angle A is

denoted as θ. It’ G k l tt

4 ABhyp = 5It’s a Greek letter.

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Keen Eye:As hypotenuse is always yp ygreater than the other sides of the right angled triangle. the right angled triangle. There fore sin and cos are < 1Sec and cosec > 1 for acute Sec and cosec > 1 for acute angle.

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In the ∆ABC, , AB=12cm and BC=5cm then find the values of all trigonometric ratios of the angle C and A.g g

A

By Pythagoras theorem12 13

By Pythagoras theorem

B C5Vikasana – Bridge Course 2012

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In the ∆ABC, , AB=12cm and BC=5cm then find the values of all trigonometric ratios of the angle C and A.g g

AThen AB is opposite side;

12 13C is the ref. angleBC is adjacent side and AC is hypotenuse.

B C5Vikasana – Bridge Course 2012

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AA

12 13

B C5

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ALet us consider A as ALet us consider A as reference angle.Then

12 13Then opposite side=BC=5;adjacent side=AB=12;

B C5

adjacent side AB 12;hypotenuse = AC = 13.

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AA

12 13

B C5

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If tanA=4/3, then find all other trigonometric ratios of A.

Given tanA=4/3. There fore opp=4

4 5and adj = 3

A3

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If 15cotA=8,find sinA and secAIf 15cotA 8,find sinA and secA

Given cotA=8/15

A

15 17Opp=15; adj=8h 17 A8hyp=17

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If then find the value of If , then find the value of

Clearly, opp=8, adj=7 and hyp= 8yp

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KEEN EYEKEEN EYE

sinA is one symbol and sinAT i t i l ti d d l sinA is one symbol and sinAdoes not mean sin X A. Similar, is In short, trigonometric ratios are

written as t-ratiosEvery t-ratio is a real number.

Trigonometrical ratio depends only upon the magnitude of the angle

means (sinA)(sinA). It is

l itt the case for cosA and tanA. written as t-ratios.and upon the length of sides.also written as

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Alternative definition of T-ratios

P(x y)P(x,y)yr

MxO

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It is used in satellite systems and astronomy, aviation, engineering, astronomy, aviation, engineering, land surveying, geography and many other fieldsmany other fields.

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