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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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CameAHoppin’
Old Hi iOld Hi i
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
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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.
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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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