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Trigonometry
Hypotenuse
Adjacent
Op
posi
te
T- 1-855-694-8886Email- info@iTutor.com
By iTutor.com
TRIGONOMETRY
The word ‘trigonometry’ is derived from the Greek words ‘tri’(meaning three), ‘gon’ (meaning sides) and ‘metron’ (meaning measure).
Trigonometry is the study of relationships between the sides and angles of a triangle.
Early astronomers used it to find out the distances of the stars and planets from the Earth.
Even today, most of the technologically advanced methods used in Engineering and Physical Sciences are based on trigonometrical concepts.
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RIGHT TRIANGLE
A triangle in which one angle is equal to 90 is called right triangle.
The side opposite to the right angle is known as hypotenuse.
AC is the hypotenuse The other two sides are
known as legs.
AB and BC are the legs
Trigonometry deals with Right Triangles
A
CB
Hypotenuse
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PYTHAGORAS THEOREM
In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of areas of the squares whose sides are the two legs.
A
CB
Hypotenuse
(Hypotenuse)2 = (Perpendicular)2 + (Base)2
AC2 = BC2 + AB2
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PYTHAGORAS THEOREMPythagoras Theorem Proof: Given: Δ ABC is a right angled triangle where
B = 900 And AB = P, BC= b and AC = h.
To Prove: h2 = p2 + b2
Construction : Draw a BD from B to AC , where AD = x and CB = h-x ,
Proof : In Δ ABC and Δ ABD,
Δ ABC Δ ABD --------(AA)In Δ ABC and Δ BDC both are similar
So by these similarity,
p
b
h
A
B C
D
x
(h-x)
PYTHAGORAS THEOREM
Or P2 = x × h And b2 = h (h – x)
Adding both L.H.S. and R.H. S. Then
p2 + b2 = (x × h) + h (h – x)
Or p2 + b2 = xh + h2 – hx
Hence the Pythagoras theorem
p2 + b2 = h2
bxh
hb
px
hp And
p
b
h
A
B C
D
x
(h-x)
TRIGONOMETRIC RATIOS
Let us take a right triangle ABC
Here, ∠ ACB () is an acute angle.
The position of the side AB with respect to angle . We call it the side opposite to angle .
AC is the hypotenuse of the right triangle and the side BC is a part of . So, we call it the side adjacent to angle .
A
CB
Hypotenuse
Sid
e op
posi
te t
o a
ng
le
Side adjacent to angle
‘’
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TRIGONOMETRIC RATIOS
The trigonometric ratios of the angle C in right ABC as follows :
Sine of C =
=
Cosine of C=
=
A
CB
Hypotenuse
Sid
e op
posi
te t
o a
ng
le
Side adjacent to angle
‘’
Side opposite to C Hypotenuse
ABAC
Side adjacent to C Hypotenuse
BCAC
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TRIGONOMETRIC RATIOS
Tangent of C =
=
Cosecant of C=
=
Secant of C =
A
CB
Hypotenuse
Sid
e op
posi
te t
o a
ng
le
Side adjacent to angle
‘’
Side opposite to CSide adjacent to CAB
BC
Side adjacent to C
Hypotenuse
Side opposite to C
Hypotenuse
ACAB
ACAB
=
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TRIGONOMETRIC RATIOS
Cotangent of C
Above Trigonometric Ratio arbitrates as sin C, cos C, tan C , cosec C , sec C, Cot C .
If the measure of angle C is ‘’ then the ratios are :
sin , cos , tan , cosec , sec and cot
A
CB
Hypotenuse
Sid
e op
posi
te t
o a
ng
le
Side adjacent to angle
‘’
Side opposite to C
Side adjacent to C AB
BC= =
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RELATION AMONG T-RATIO
Tan = Cosec = 1 / Sin
Sec = 1 / Cos
Cot = Cos / Sin
= 1 / Tan
A
CB
p
b
h
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cossin
TABLE
1. Sin = p / h
2. Cos = b / h
3. Tan = p / b
4. Cosec = h / p
5. Sec = h / b
6. Cot = b / p
A
CB
p
b
h
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TRIGONOMETRIC RATIOS OF SOME SPECIFIC ANGLES
Trigonometric Ratios of 45°In Δ ABC, right-angled at B, if one angle is 45°, thenthe other angle is also 45°, i.e., ∠ A = ∠ C = 45°So, BC = ABNow, Suppose BC = AB = a.Then by Pythagoras Theorem,
AC2 = BC2 + AB2 = a2 + a2
AC2 = 2a2 , or AC = a2
A
CB
450
a
a
450
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TRIGONOMETRIC RATIOS OF 45°
Sin 450 = = = = 1/2
Cos 450 = = = = 1/2
Tan 450 = = = = 1
Cosec 450 = 1 / sin 450 = 1 / 1/2 = 2
Sec 450 = 1 / cos 450 = 1 / 1/2 = 2
Cot 450 = 1 / tan 450 = 1 / 1 = 1
Side opposite to 450Hypotenuse
ABAC
aa2
Side adjacent to 450Hypotenuse
BCAC
a
Side opposite to 450Side adjacent to 450
ABBC
aa
a2
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TRIGONOMETRIC RATIOS OF 30° AND 60°
Consider an equilateral triangle ABC. Since each angle in an equilateral triangle is 60°, therefore,
∠ A = ∠ B = ∠ C = 60°.Draw the perpendicular AD from A to the side BC,
Now Δ ABD Δ ACD ≅ --------- (S. A. S)Therefore, BD = DCand ∠ BAD = ∠ CAD -----------(CPCT)Now observe that:Δ ABD is a right triangle, right-angled at D with ∠
BAD = 30° and ∠ ABD = 60°
600
300
A
B D C
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TRIGONOMETRIC RATIOS OF 30° AND 60°
As you know, for finding the trigonometric ratios, we need to know the lengths of the sides of the triangle.
So, let us suppose that AB = 2a.BD = ½ BC = a
AD2 = AB2 – BD2 = (2a)2 - (a)2 = 3a2
AD = a3
Now Trigonometric ratios
Sin 300 = =
= = ½
600
300
A
B D C
2a
2a 2a
a aSide opposite to 300Hypotenuse
BDAB
a2a
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TRIGONOMETRIC RATIOS OF 30° AND 60°
Cos 300 = = = 3 / 2
Tan 300 = = = 1 / 3
Cosec 300 = 1 / sin 300 = 1 / ½ = 2
Sec 300 = 1 / cos 300 = 1 / 3/2 = 2 / 3
Cot 300 = 1 / tan 300 = 1 / 1/3 = 3
Now trigonometric ratios of 600
ADAB
a32a
BDAD
aa3
300
A
B D C
2a
2a 2a
a a
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TRIGONOMETRIC RATIOS OF 30° AND 60°
Sin 600 = = = 3 / 2
Cos 600 = = = ½
Tan 600 = = = 3
Cosec 600 = 1 / Sin 600 = 1 /3 / 2 = 2 / 3
Sec 600 = 1 / cos 600 = 1 / ½ = 2
Cot 600 = 1 / tan 600 = 1 /3
ADAB
a32a
BDAB
a2a
ADBD
a3a
600
A
B D C
2a
2a 2a
a a
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VALUES OF TRIGONOMETRIC FUNCTION
T. Ratios 0 30 45 60 90
Sine 0 ½ 1/2 3/2 1
Cosine 1 3/2 1/2 ½ 0
Tangent 0 1/ 3 1 3 Not defined
Cosecant Not defined 2 2 2/ 3 1
Secant 1 2/ 3 2 2 Not defined
Cotangent Not defined
3 1 1/ 3 0
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RELATION OF ANGLES WITH T. RATIOS
Relation of with Sin when 00 900 The greater the value of ‘’, the greater is the
value of Sin.Smallest value of Sin = 0Greatest value of Sin = 1
Relation of with Cos when 00 900 The greater the value of ‘’, the smaller is the
value of Cos.Smallest value of Cos = 0Greatest value of Cos = 1
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RELATION OF ANGLES WITH T. RATIOS
Relation of with tan when 00 900 Tan increases as ‘’ increases But ,tan is not defined at ‘’ = 900 Smallest value of tan = 0
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SOME MORE FORMULAE If 00 900
1. Sin(900- ) = Cos 2. Cos(900- ) = Sin
If 00< 900
1. Tan(900- ) = Cot 2. Sec(900- ) = Cosec
If 00 < 900
1. Cot(900- )= Tan 2. Cosec(900- ) = Sec
A
CB
p
b
h
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SOME MORE FORMULAE
Sin2 +Cos2 = 1
Sec2 -Tan2 = 1
Cosec2 - Cot2 = 1
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The End
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