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TRIGONOMETRY
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TRIGONOMETRY
Trigonometry (from Greek trigōnon "triangle"
+ metron "measure") is a branch of mathematics that
studies triangles and the relationships between the
lengths of their sides and the angles between those
sides.
Trigonometry defines the trigonometric
functions, which describe those
relationships and have applicability
to cyclical phenomena, such as waves.
The field evolved during the third
century BC as a branch
of geometry used extensively for
astronomical studies . It is also the
foundation of the practical art
of surveying.
HISTORY Of TRIGONOMETRY
Classical Greek
mathematicians (such
as Euclid and Archimed
es) studied the
properties
of chords and inscribed
angles in circles, and
proved theorems that
are equivalent to
modern trigonometric
formulae, although they
presented them
geometrically rather
than algebraically.
The modern sine
function was first
defined in the Surya
Siddhanta, and its
properties were
further documented
by the 5th
century Indian
mathematician and
astronomer Aryabh
ata.
These Greek and Indian works were translated
and expanded by medieval Islamic
mathematicians. By the 10th century, Islamic
mathematicians were using all six
trigonometric functions, had tabulated their
values, and were applying them to problems
in spherical geometry.
The Father of Trigonometry
The
first trigonometri
c table was
apparently
compiled
by Hipparchus,
who is now
consequently
known as "the
father of
RIGHT TRIANGLE
A right triangle or right-
angled triangle is
a triangle in which
one angle is a right
angle (that is, a 90-
degree angle). The
relation between the
sides and angles of a
right triangle is the
basis for trigonometry.
The side opposite the right
angle is called
the hypotenuse (side c in
the figure above). The
sides adjacent to the right
angle are called legs.
Side a may be identified as
the side adjacent to angle
B and opposed
to (or opposite) angle A,
while side b is the
side adjacent to angle
A and opposed to angle B.
PYTHAGORAS THEOREM
The Pythagorean theorem:
The sum of the areas of the
two squares on the legs (a
and b) equals the area of the
square on the hypotenuse
(c).
If one angle of a triangle is 90 degrees
and one of the other angles is known, the
third is thereby fixed, because the three
angles of any triangle add up to 180
degrees. The two acute angles therefore
add up to 90 degrees: they
are complementary angles. The shape of
a triangle is completely determined,
except for similarity, by the angles.
TRIGONOMETRIC RATIOS
Once the angles are known,
the ratios of the sides are
determined, regardless of the
overall size of the triangle. If the
length of one of the sides is
known, the other two are
determined. These ratios are given
by the following trigonometric
functions of the known angle A,
where a, b and c refer to the
lengths of the sides in the
The hypotenuse is the side opposite to the 90 degree angle in a
right triangle; it is the longest side of the triangle, and one of the
two sides adjacent to angle A. The adjacent leg is the other side
that is adjacent to angle A. The opposite side is the side that is
opposite to angle A. The terms perpendicular and base are
sometimes used for the opposite and adjacent sides respectively.
The reciprocals of these functions are named the cosecant (csc or
cosec), secant (sec), and cotangent (cot), respectively:
TRIGONOMETRIC FUNCTIONS
STANDARD IDENTITIES
Identities are those equations that hold true for any value
REDUCTION FORMULA
Sin (90-A) =Cos A
Tan (90-A)= Cot
A
Cosec (90-A)= Sec
A
Calculator1) This calculates the value of
trigonometric functions of
different angles.
2) First enter whether you
want enter the angle in
radian or in degree.
3) Then enter the required
trigonometric function in
the format given below:
4) Enter 1 for Sin
5) Enter 2 for Cosine
6) Enter 3 for tangent
7) Enter 4 for Cosecant
8) Enter 5 for Secant
9) Enter 6 for cotangent
10)Then enter the magnitude
of angle.
Applications of trigonometry
Fields that use trigonometry or trigonometric functions
include astronomy (especially for locating apparent positions of
celestial objects, in which spherical trigonometry is essential)
and hence navigation (on the oceans, in aircraft, and in
space), music theory, audio synthesis, acoustics, optics,
analysis of financial markets, electronics, probability
theory, statistics, biology, medical imaging (CAT
scans and ultrasound)
Application of trigonometry in
Astronomy
1) Since ancient times trigonometry was used in
astronomy
2) The technique triangulation is used to measure the to
nearby stars .
3) In 240 B.C, a mathematician named Eratosthenes
discovered the radius of the earth using trigonometry
and geometry
4) In 2001 , a group of European astronomers did an
experiment that started in 1997 about the distance of
Venus from the sun.
conclusion
TRIGONOMETRY IS A BRANCH OF
MATHEMATICS WITH SEVERAL
IMPORTANT AND USEFUL
APPLICATIONS.HENCE IT
ATTRACTSMORE AND MORE
RESEARCHWITH SEVERAL THEORIES
PUBLISHED YEAR AFTER YEAR.
THANK YOU

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Trigonometry

  • 2. TRIGONOMETRY Trigonometry (from Greek trigōnon "triangle" + metron "measure") is a branch of mathematics that studies triangles and the relationships between the lengths of their sides and the angles between those sides.
  • 3. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. The field evolved during the third century BC as a branch of geometry used extensively for astronomical studies . It is also the foundation of the practical art of surveying.
  • 4. HISTORY Of TRIGONOMETRY Classical Greek mathematicians (such as Euclid and Archimed es) studied the properties of chords and inscribed angles in circles, and proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically.
  • 5. The modern sine function was first defined in the Surya Siddhanta, and its properties were further documented by the 5th century Indian mathematician and astronomer Aryabh ata.
  • 6. These Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, and were applying them to problems in spherical geometry.
  • 7. The Father of Trigonometry The first trigonometri c table was apparently compiled by Hipparchus, who is now consequently known as "the father of
  • 8. RIGHT TRIANGLE A right triangle or right- angled triangle is a triangle in which one angle is a right angle (that is, a 90- degree angle). The relation between the sides and angles of a right triangle is the basis for trigonometry.
  • 9. The side opposite the right angle is called the hypotenuse (side c in the figure above). The sides adjacent to the right angle are called legs. Side a may be identified as the side adjacent to angle B and opposed to (or opposite) angle A, while side b is the side adjacent to angle A and opposed to angle B.
  • 10. PYTHAGORAS THEOREM The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).
  • 11. If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a triangle is completely determined, except for similarity, by the angles. TRIGONOMETRIC RATIOS
  • 12. Once the angles are known, the ratios of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the
  • 13.
  • 14. The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle, and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec), and cotangent (cot), respectively:
  • 16. STANDARD IDENTITIES Identities are those equations that hold true for any value
  • 17. REDUCTION FORMULA Sin (90-A) =Cos A Tan (90-A)= Cot A Cosec (90-A)= Sec A
  • 18. Calculator1) This calculates the value of trigonometric functions of different angles. 2) First enter whether you want enter the angle in radian or in degree. 3) Then enter the required trigonometric function in the format given below: 4) Enter 1 for Sin 5) Enter 2 for Cosine 6) Enter 3 for tangent 7) Enter 4 for Cosecant 8) Enter 5 for Secant 9) Enter 6 for cotangent 10)Then enter the magnitude of angle.
  • 19. Applications of trigonometry Fields that use trigonometry or trigonometric functions include astronomy (especially for locating apparent positions of celestial objects, in which spherical trigonometry is essential) and hence navigation (on the oceans, in aircraft, and in space), music theory, audio synthesis, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound)
  • 20. Application of trigonometry in Astronomy 1) Since ancient times trigonometry was used in astronomy 2) The technique triangulation is used to measure the to nearby stars . 3) In 240 B.C, a mathematician named Eratosthenes discovered the radius of the earth using trigonometry and geometry 4) In 2001 , a group of European astronomers did an experiment that started in 1997 about the distance of Venus from the sun.
  • 21. conclusion TRIGONOMETRY IS A BRANCH OF MATHEMATICS WITH SEVERAL IMPORTANT AND USEFUL APPLICATIONS.HENCE IT ATTRACTSMORE AND MORE RESEARCHWITH SEVERAL THEORIES PUBLISHED YEAR AFTER YEAR. THANK YOU