Trigonometry (from
Greek trigōnon "triangle" +
metron "measure") is a branch of
mathematics that studies
triangle, particularly
right triangles. Trigonometry deals with
relationships between the sides and the angles of triangles and
with the
trigonometric
functions, which describe those relationships, as well as
describing angles in general and the motion of
waves such as sound and light waves.
Trigonometry is usually taught in
secondary schools either as a separate
course or as part of a
precalculus
course. It has applications in both
pure mathematics and in
applied mathematics, where it is
essential in many branches of science and technology. A branch of
trigonometry, called
spherical
trigonometry, studies triangles on
spheres, and is important in
astronomy and
navigation.
History
Pre-Hellenic societies such as the ancient
Egyptians and
Babylonians lacked the concept of an
angle measure, but they studied the ratios of the sides of similar
triangles and discovered some properties of these ratios. Ancient
Greek mathematicians such as
Euclid and
Archimedes studied the properties of the
chord of an angle and proved theorems that
are equivalent to modern trigonometric formulae, although they
presented them geometrically rather than algebraically. The
sine function in its modern
form was first defined in the
Surya
Siddhanta and its properties were further documented by
the 5th century
Indian
mathematician and astronomer
Aryabhata.
These Indian works were translated and expanded by
medieval Islamic scholars. 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. At about the same
time,
Chinese mathematicians
developed trigonometry independently, although it was not a major
field of study for them. Knowledge of trigonometric functions and
methods reached Europe via Latin translations of the works of
Persian and Arabic astronomers such as
Al Battani and
Nasir al-Din
al-Tusi.
One of the earliest works on trigonometry by
a European mathematician is De Triangulis by the 15th
century German
mathematician Regiomontanus.
Trigonometry was still so little known in 16th century Europe that
Nicolaus Copernicus devoted two
chapters of
De
revolutionibus orbium coelestium to explaining its basic
concepts.
Overview
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 right triangle is completely determined, up
to
similarity, by the angles.
This means that once one of the other angles is known, the
ratios of the various sides are always the same
regardless of the overall size of the triangle. 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 accompanying figure:
- The sine function (sin), defined as the ratio
of the side opposite the angle to the hypotenuse.
- : \sin
A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{\,c\,}\,.
- The cosine function (cos), defined as the
ratio of the adjacent leg to the hypotenuse.
- : \cos
A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{\,c\,}\,.
- The tangent function (tan), defined as the
ratio of the opposite leg to the adjacent leg.
- : \tan
A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{\,b\,}=\frac{\sin
A}{\cos A}\,.
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.
With these functions one can answer virtually all questions about
arbitrary triangles by using the
law of
sines and the
law of cosines.
These laws can be used to compute the remaining angles and sides of
any triangle as soon as two sides and an angle or two angles and a
side or three sides are known. These laws are useful in all
branches of geometry, since every
polygon
may be described as a finite combination of triangles.
Image:sin drawing process.gif|Graphing process of
y =
sin(
x) using a unit circle.Image:tan drawing
process.gif|Graphing process of
y = tan(
x) using
a unit circle.Image:csc drawing process.gif|Graphing process of
y = csc(
x) using a unit circle.
Extending the definitions
Graphs of the functions sin(
x) and cos(
x), where
the angle
x is measured in radians.
The above definitions apply to angles between 0 and 90 degrees (0
and π/2
radians) only. Using the
unit circle, one can extend them to all positive
and negative arguments (see
trigonometric function). The
trigonometric functions are
periodic, with a period of 360 degrees or
2π radians. That means their values repeat at those
intervals.
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 function
cis is particularly useful
- \operatorname{cis}\,x = \cos x + i\sin x \! = e^{ix}.
See
Euler's and
De Moivre's formulas.
Mnemonics
A common use of
mnemonics is to remember
facts and relationships in trigonometry. For example, the
sine,
cosine, and
tangent ratios in a
right triangle can be remembered by representing them as strings of
letters, as in SOH-CAH-TOA.
- Sine = Opposite ÷
Hypotenuse
- Cosine = Adjacent ÷
Hypotenuse
- Tangent = Opposite ÷
Adjacent
The memorization of this mnemonic can be aided by expanding it into
a phrase, such as "
Some
Officers
Have
Curly
Auburn
Hair
Till
Old
Age". Any memorable phrase
constructed of words beginning with the letters S-O-H-C-A-H-T-O-A
will serve.
Calculating trigonometric 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 cis) and their inverses. Most allow a choice
of angle measurement methods: degrees, radians and, sometimes,
grad. Most computer
programming languages provide function
libraries that include the trigonometric functions. The
floating point unit hardware
incorporated into the microprocessor chips used in most personal
computers have built-in instructions for calculating trigonometric
functions.
Applications of trigonometry
There are an enormous number of uses of trigonometry and
trigonometric functions. For instance, the technique of
triangulation is used in
astronomy to measure the distance to nearby stars,
in
geography to measure distances between
landmarks, and in
satellite
navigation systems. The sine and cosine functions are
fundamental to the theory of
periodic
functions such as those that describe sound and
light waves.
Fields which make use of trigonometry or trigonometric functions
include
astronomy (especially, for
locating the apparent positions of celestial objects, in which
spherical trigonometry is essential) and hence
navigation (on the oceans, in aircraft, and in
space),
music theory,
acoustics,
optics, analysis
of financial markets,
electronics,
probability theory,
statistics,
biology,
medical imaging (
CAT scans and
ultrasound),
pharmacy,
chemistry,
number
theory (and hence
cryptology),
seismology,
meteorology,
oceanography, many
physical sciences, land
surveying and
geodesy,
architecture,
phonetics,
economics,
electrical engineering,
mechanical engineering,
civil engineering,
computer graphics,
cartography,
crystallography and
game development.
Common formulae
Certain equations involving trigonometric functions are true for
all angles and are known as
trigonometric identities.
There are some identities which equate an expression to a different
expression involving the same angles and these are listed in
List of trigonometric
identities, and then there are the triangle identities which
relate the sides and angles of a given triangle and these are
listed below.
In the following identities,
A,
B and
C
are the angles of a triangle and
a,
b and
c are the lengths of sides of the triangle opposite the
respective angles.
Law of sines
The
law of sines (also
known as the "sine rule") for an arbitrary triangle states:
- \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} =
2R,
where
R is the radius of the
circumcircle of the triangle:
- R = \frac{abc}{\sqrt{(a+b+c)(a-b+c)(a+b-c)(b+c-a)}}.
Another law involving sines can be used to calculate the area of a
triangle. If you know two sides and the angle between the sides,
the area of the triangle becomes:
- \mbox{Area} = \frac{1}{2}a b\sin C.
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:
- c^2=a^2+b^2-2ab\cos C ,\,
or equivalently:
- \cos C=\frac{a^2+b^2-c^2}{2ab}.\,
Law of tangents
The
law of
tangents:
-
\frac{a-b}{a+b}=\frac{\tan\left[\tfrac{1}{2}(A-B)\right]}{\tan\left[\tfrac{1}{2}(A+B)\right]}
See also
Notes
- Boyer p215
- Boyer p237, p274
References
- Christopher M. Linton (2004). From Eudoxus to Einstein: A
History of Mathematical Astronomy . Cambridge University
Press.
- Weisstein, Eric W. "Trigonometric Addition Formulas". Wolfram
MathWorld.
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