Introduction to trigonometry

INTRODUCTION TO TRIGONOMETRY
DONE BY
HANIYA HEDAYTH
GRADE: 10TH A

What is trigonometry?
 Trigonometry (from Greek trigōnon,
"triangle" and metron, "measure"[1]) is a
branch of mathematics that studies
relationships involving lengths
and angles of triangles. The field
emerged in the Hellenistic world during
the 3rd century BC from applications
of geometry to astronomical studies.[2]
 As for the word "trigonometry," it first
appeared as the title of a
book Trigonometria (literally, the
measuring of triangles), published
by Bartholomeo Pitiscus in 1595.

Discovery of trigonometry.
 Sumerian astronomers studied angle
measure, using a division of circles
into 360 degrees.[4] They, and later
the Babylonians, studied the ratios of
the sides of similar triangles and
discovered some properties of these
ratios but did not turn that into a
systematic method for finding sides
and angles of triangles. The ancient
Nubians used a similar method.[5]

 Babylonian and Greek influences
mingled with rich native mathematical
developments in India around AD 500 to
produce a trigonometry closer to its
modern form. Hindu mathematical works
such as that of Aryabhata give tables of
half chords, known by the termjya-
ardha or simply jya, which bears the
following relationship to our modern
concept of sine: jya x = r sinx, as
illustrated below.

Jya here represents the half
chord AM.

 From India the sine function was
introduced to the Arab world in the 8th
century, where the term jya was
transliterated into jiba or jyb.
 Early Latin translations of Arabic
mathematical treatises mistook jiba for
the Arabic word jaib, which can mean the
opening of a woman's garment at the
neck. Accordingly, jaib was translated
into the Latin sinus, which can mean
"fold" (in a garment), "bosom," "bay," or
even "curve." Hence our word "sine."

Hipparchus, credited with compiling the
first trigonometric table, is known as "the
father of trigonometry".[3]

Trigonometric ratios:
 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. 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, band c refer to the lengths of
the sides in the accompanying figure:

In this right triangle: sin A = a/c;
cos A = b/c; tan A = a/b.

Sine function (sin), defined as the ratio of the side opposite the
angle to the hypotenuse.
Cosine function (cos), defined as the ratio of
the adjacent leg to the hypotenuse.
Tangent function (tan), defined as the ratio of the opposite
leg to the adjacent leg.

 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.
 Many people find it easy to remember what sides
of the right triangle are equal to sine, cosine, or
tangent, by memorizing the word SOH-CAH-TOA
(see below under Mnemonics).

The reciprocals of these functions are named
the cosecant (csc or cosec), secant (sec),
and cotangent (cot), respectively:

 The inverse functions are called
the arcsine, arccosine,
and arctangent, respectively. There
are arithmetic relations between these
functions, which are known
as trigonometric identities. The cosine,
cotangent, and cosecant are so
named because they are respectively
the sine, tangent, and secant of the
complementary angle abbreviated to
"co-".

EXTENDING THE DEFINITION:
 The trigonometric functions can be defined in other
ways besides the geometrical definitions above,
using tools from calculus and infinite series. With
these definitions the trigonometric functions can be
defined for complex numbers. The complex
exponential function is particularly useful.

Euler's and De
Moivre's formulas.
 In mathematics, de Moivre's formula (a.k.a. De Moivre's
theorem and De Moivre's identity), named after Abraham
de Moivre, states that for any complex number (and, in
particular, for any real number) x and integer n it holds that
 where i is the imaginary unit (i2 = −1). While the formula was
named after de Moivre, he never stated it in his works.[1]
 The formula is important because it connects complex
numbers and trigonometry. The expression cos x + i sin x is
sometimes abbreviated to cis x.
 By expanding the left hand side and then comparing the real
and imaginary parts under the assumption that x is real, it is
possible to derive useful expressions for cos(n x) and sin(n x)
in terms of cos x and sin x. Furthermore, one can use a
generalization of this formula to find explicit expressions for
the nth roots of unity, that is, complex numbers z such that z
n = 1.

MNEMONICS:
 A common use of mnemonics is to remember
facts and relationships in trigonometry. For
example, the sine, cosine, andtangent ratios in a
right triangle can be remembered by
representing them and their corresponding sides
as strings of letters. For instance, a mnemonic is
SOH-CAH-TOA:[17]
 Sine = Opposite ÷ Hypotenuse Cosine
= Adjacent ÷ Hypotenuse Tangent = Opposite
÷ Adjacent One way to remember the letters is to
sound them out phonetically (i.e., SOH-CAH-
TOA, which is pronounced 'so-k ə-toe-
uh' /soʊkəˈtoʊə/). Another method is to expand
the letters into a sentence, such as
"Some Old Hippy Caught Another Hippy
Trippin' On Acid".[18]

CALCULATING
TRIGNOMETRIC FUNCTIONS:
 Trigonometric functions were among the earliest uses
for mathematical tables. Such tables were incorporated
into mathematics textbooks and students were taught to
look up values and how to interpolate between the
values listed to get higher accuracy. Slide rules had
special scales for trigonometric functions.
 Today scientific calculators have buttons for calculating
the main trigonometric functions (sin, cos, tan, and
sometimes cisand their inverses). Most allow a choice
of angle measurement methods: degrees, radians, and
sometimes gradians. Most computer programming
languages provide function libraries that include the
trigonometric functions. The floating point unithardware
incorporated into the microprocessor chips used in most
personal computers has built-in instructions for
calculating trigonometric functions.[19]

PYTHAGOREAN
IDENTITIES:
Identities are those equations that hold true for any value.
(The following two can be derived from the first.)

All of the trigonometric functions of an angle θ can be
constructed geometrically in terms of a unit circle
centered at O.

Law of sin
 The law of sins (also known as the
"sine rule") for an arbitrary triangle
states:
 Where is the area of the triangle and R is the radius of
the circumscribed circle of the triangle:

 Another law involving sins can be
used to calculate the area of a
triangle.
 Given two sides a and b and the
angle between the sides C, the area of
the triangle is given by half the product
of the lengths of two sides and the
sine of the angle between the two
sides:

Law of cosines:
 The law of cosines (known as the
cosine formula, or the "cos rule") is an
extension of the Pythagorean
theorem to arbitrary triangles:
 or equivalently:

 The law of cosines may be used to
prove Heron's formula, which is
another method that may be used to
calculate the area of a triangle. This
formula states that if a triangle has
sides of lengths a, b, and c, and if the
semi perimeter is:
then the area of the triangle is:
where R is the radius of the circumcircle of
the triangle.

Introduction to trigonometry

More Related Content

What's hot (20)

Viewers also liked (20)

Similar to Introduction to trigonometry (20)

More from haniya hedayth (7)

Recently uploaded (20)

Introduction to trigonometry