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INTRODUCTION TO TRIGNOMETRY OF CLASS 10. IT ALSO INCLUDES ALL TOPIC OF TRIGNOMETRY OF CLASS 10 WITH PHOTOS AND DERIVATIOM
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INTRODUCTION TO TRIGNOMETRY
MADE BY : KRISHNARAJ MISHRA
SUBJECT :MATHS
SUBMITTED TO : NM GIRI S IR
1) Introduction to trignometry
2) History
3) Trignometric ratios
4) Values of trignometric function
5) Trignometric ratios of some specific angles
6) Trignometric ratios of some complementry angles
7) Trignometric identities
8) Conclusion
INTRODUCTION
• The distances or heights can be
found by using some
mathematical techniques , which
come under a branch of
mathematics called
‘trignometry’.
• The word ‘trignometry’ is
derived from Greek words ‘tri’
(meaning three),’gon’(meaning
sides) and metron(meaning
measure).
INTRODUCTION
Trigonometry is the branch of mathematics which
deals with triangles, particularly triangles in a plane
where one angle of the triangle is 90 degrees
Triangles on a sphere are also studied, in spherical
trigonometry.
Trigonometry specifically deals with the relationships
between the sides and the angles of triangles, that is, on
the trigonometric functions, and with calculations
based on these functions.
History
The origins of trigonometry can be traced to the civilizations of ancient Egypt, Mesopotamia and the Indus Valley, more than 4000 years ago.
Some experts believe that trigonometry was originally invented to calculate sundials, a traditional exercise in the oldest books
The first recorded use of trigonometry came from the Hellenistic mathematician Hipparchus circa 150 BC, who compiled a trigonometric table using the sine for solving triangles.
The Sulba Sutras written in India, between 800 BC and 500 BC, correctly compute the sine of π/4 (45°) as 1/√2 in a procedure for circling the square (the opposite of squaring the circle).
Many ancient mathematicians like Aryabhata, Brahmagupta,Ibn Yunus and Al- Kashi made significant contributions in this field(trigonometry).
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.
AB is the hypotenuse
The other two sides are known as legs.
AC and BC are the legs
Trigonometry deals with Right Triangles
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.
In the figure
AB2 = BC2 + AC2
TRIGONOMETRIC RATIOS
Sine(sin) opposite side/hypotenuse
Cosine(cos) adjacent side/hypotenuse
Tangent(tan) opposite side/adjacent side
Cosecant(cosec) hypotenuse/opposite side
Secant(sec) hypotenuse/adjacent side
Cotangent(cot) adjacent side/opposite side
sin = a/c
cos = b/c
tan = a/b
cosec = c/a
sec = c/b
cot = b/a
• In Δ ABC, right-angled at B, if one angle is 45°, then the other angle is also 45°, i.e., ∠ A = ∠ C = 45° .
• Suppose BC = AB = a.
• Then by Pythagoras Theorem, AC2 = AB2 + BC2
= a2 + a2 = 2a2,
Therefore, AC = 2 a
Trigonometric Ratios of 45° sin 45° = side opposite to angle 45° / hypotenuse
=BC/AC = a/a√2 = 1/ √2
cos 45° = side adjacent to angle 45°/ hypotenuse=AB/AC = a/a √2 = 1/ √2
tan 45° =side opposite to angle 45°/ side adjacent to angle 45°
=BC/AB = a/a = 1
cosec 45°=1/sin 45°= √2
sec 45°=1/cos 45°= √2
cot 45°=1/tan 45°= 1
cosec 30°=1/sin 30° = 2
sec 30°=1/cos 30° = 2/√3
cot 30°=1/tan 30° = √3
sin 60°= a√3/2a = √3/2
cos 60°= ½
tan 60°= √3
cos 60°= 2/√3
sec 60°= 2
cot 60°= 1/√3
Trigonometric Ratios of 0°And 90°
Sin 0⁰ = 0
cos 0⁰ = 1
Sin 90⁰ = 1
Cos 90⁰ =0
VALUES OF TRIGONOMETRIC
FUNCTION
0 30 45 60 90
Sine 0 0.5 1/2 3/2 1
Cosine 1 3/2 1/2 0.5 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
RELATION BETWEEN DIFFERENT
TRIGONOMETRIC IDENTITIES
Sine
Cosine
Tangent
Cosecant
Secant
Cotangent
sin (90⁰-A) = cos A
tan (90⁰-A) = cot A
sec (90⁰-A) = cosec A
cos (90⁰-A) = sin A
cot (90⁰-A) = tan A
cosec (90⁰-A) = sec A
Trigonometric identities
Osin2A + cos2A = 1
O1 + tan2A = sec2A
O1 + cot2A = cosec2A
Osin(A+B) = sinAcosB + cosAsin B
Ocos(A+B) = cosAcosB – sinAsinB
O tan(A+B) = (tanA+tanB)/(1 – tanAtan B)
Osin(A-B) = sinAcosB – cosAsinB
Ocos(A-B)=cosAcosB+sinAsinB
O tan(A-B)=(tanA-tanB)(1+tanAtanB)
sin2A =2sinAcosA
cos2A=cos2A - sin2A
tan2A=2tanA/(1-tan2A)
sin(A/2) = ±{(1-cosA)/2}
Cos(A/2)= ±{(1+cosA)/2}
Tan(A/2)= ±{(1-cosA)/(1+cosA)}
Trigonometric identities
Conclusion
Trigonometry is a branch of Mathematics with
several important and useful applications.
Hence it attracts more and more research with
several theories published year after year.
Thank You……..
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