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Trigonometry

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   Trigonometry (from the Greek trigonon = three angles and metron =
   measure ) is a branch of mathematics which deals with triangles,
   particularly triangles in a plane where one angle of the triangle is 90
   degrees ( right triangles). 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,
   the trigonometric functions, and with calculations based on these
   functions.

   Trigonometry has important applications in many branches of pure
   mathematics as well as of applied mathematics and, consequently, much
   of science.

Overview

Basic definitions

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

   The shape of a right triangle is completely determined, up to
   similarity, by the value of either of the other two 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 size of the triangle. These
   ratios are traditionally described by the following trigonometric
   functions of the known angle:
     * The sine function (sin), defined as the ratio of the leg opposite
       the angle to the hypotenuse.
     * The cosine function (cos), defined as the ratio of the adjacent leg
       to the hypotenuse.
     * The tangent function (tan), defined as the ratio of the opposite
       leg to the adjacent leg.

   The adjacent leg is the side of the angle that is not the hypotenuse.
   The hypotenuse is the side opposite to the 90 degree angle in a right
   triangle; it is the longest side of the triangle.

          \sin A = \frac{\mbox{opposite}}{\mbox{hypotenuse}} \qquad \cos A
          = \frac{\mbox{adjacent}}{\mbox{hypotenuse}} \qquad \tan A =
          \frac{\mbox{opposite}}{\mbox{adjacent}} = \frac{\sin A}{\cos A}

   The reciprocals of these functions are named the cosecant (csc), 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.

Extending the definitions

   Graphs of the functions sin(x) and cos(x), where the angle x is
   measured in radians.
   Enlarge
   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 may 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 formula.

Mnemonics

   The sine, cosine and tangent ratios in right triangles can be
   remembered by SOH CAH TOA (sine-opposite-hypotenuse
   cosine-adjacent-hypotenuse tangent-opposite-adjacent). It is commonly
   referred to as "Sohcahtoa" by some American mathematics teachers, who
   liken it to a (nonexistent) Native American girl's name. See
   trigonometry mnemonics for other memory aids. Also another way to
   remember is "Some Old Hags, Cant Always Hide, Their Old Age"

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.

Early history of trigonometry

   Table of Trigonometry, 1728 Cyclopaedia
   Enlarge
   Table of Trigonometry, 1728 Cyclopaedia

   Trigonometry was probably invented for the purposes of astronomy. The
   origins of trigonometry can be traced to the civilizations of ancient
   Egypt, Mesopotamia and the Indus Valley, more than 4000 years ago. The
   common practice of measuring angles in degrees, minutes and seconds
   comes from the Babylonian's base sixty system of numeration.

   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. Ptolemy further developed
   trigonometric calculations circa 100 AD.

   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).

   The ancient Sinhalese, when constructing reservoirs in the Anuradhapura
   kingdom, used trigonometry to calculate the gradient of the water flow.

   The Indian mathematician Aryabhata in 499, gave tables of half chords
   which are now known as sine tables, along with cosine tables. He used
   zya for sine, kotizya for cosine, and otkram zya for inverse sine, and
   also introduced the versine.

   Another Indian mathematician, Brahmagupta in 628, used an interpolation
   formula to compute values of sines, up to the second order of the
   Newton- Stirling interpolation formula.

   In the 10th century, the Persian mathematician and astronomer Abul Wáfa
   introduced the tangent function and improved methods of calculating
   trigonometry tables. He established the angle addition identities, e.g.
   sin (a + b), and discovered the sine formula for spherical geometry:

          \frac{\sin(A)}{\sin(a)} = \frac{\sin(B)}{\sin(b)} =
          \frac{\sin(C)}{\sin(c)}

   Also in the late 10th and early 11th centuries, the Egyptian astronomer
   Ibn Yunus performed many careful trigonometric calculations and
   demonstrated the formula cos(a)cos(b) = (1 / 2)[cos(a + b) + cos(a −
   b)].

   Indian mathematicians were the pioneers of variable computations
   algebra for use in astronomical calculations along with trigonometry.
   Lagadha (circa 1350- 1200 BC) is the first person thought to have used
   geometry and trigonometry for astronomy, in his Vedanga Jyotisha.

   Persian mathematician Omar Khayyám ( 1048- 1131) combined trigonometry
   and approximation theory to provide methods of solving algebraic
   equations by geometrical means. Khayyam solved the cubic equation x^3 +
   200x = 20x^2 + 2000 and found a positive root of this cubic by
   considering the intersection of a rectangular hyperbola and a circle.
   An approximate numerical solution was then found by interpolation in
   trigonometric tables.

   Detailed methods for constructing a table of sines for any angle were
   given by the Indian mathematician Bhaskara in 1150, along with some
   sine and cosine formulae. Bhaskara also developed spherical
   trigonometry.

   The 13th century Persian mathematician Nasir al-Din Tusi, along with
   Bhaskara, was probably the first to treat trigonometry as a distinct
   mathematical discipline. Nasir al-Din Tusi in his Treatise on the
   Quadrilateral was the first to list the six distinct cases of a right
   angled triangle in spherical trigonometry.

   In the 14th century, Persian mathematician al-Kashi and Timurid
   mathematician Ulugh Beg (grandson of Timur) produced tables of
   trigonometric functions as part of their studies of astronomy.

   The mathematician Bartholemaeus Pitiscus published an influential work
   on trigonometry in 1595 which may have coined the word "trigonometry".

Applications of trigonometry

   Marine sextants like this are used to measure the angle of the sun or
   stars with respect to the horizon. Using trigonometry and an accurate
   clock, the position of the ship can then be determined from several
   such measurements.
   Enlarge
   Marine sextants like this are used to measure the angle of the sun or
   stars with respect to the horizon. Using trigonometry and an accurate
   clock, the position of the ship can then be determined from several
   such measurements.

   There are an enormous number of applications 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. Many express
   important geometric relationships. For example, the Pythagorean
   identities are an expression of the Pythagorean Theorem. Here are some
   of the more commonly used identities, as well as the most important
   formulae connecting angles and sides of an arbitrary triangle. For more
   identities see trigonometric identity.

Trigonometric identities

Pythagorean identities

   \sin^2 A+\cos^2 A = 1 ,\,

                            1+\tan^2 A=\sec^2 A ,\,

                                                   1+\cot^2 A=\csc^2 A \,

Sum and difference identities

          \sin (A + B) = \sin A \cos B + \cos A \sin B \,
          \cos (A + B) = \cos A \cos B - \sin A \sin B \,
          \tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \,

          \sin (A - B) = \sin A \cos B - \cos A \sin B \,
          \cos (A - B) = \cos A \cos B + \sin A \sin B \,
          \tan (A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} =
          \frac{\cot B - \cot A}{1 + \cot A \cot B} \,

Double-angle identities

          \sin 2A = 2 \sin A \cos A \,

          \cos 2A = \cos^2 A - \sin^2 A = 2 \cos^2 A -1 = 1-2 \sin^2 A =
          {1 - \tan^2 A \over 1 + \tan^2 A} \,

          \tan 2A = {2 \tan A \over 1 - \tan^2 A} = {2 \cot A \over \cot^2
          A - 1} = {2 \over \cot A - \tan A} \,

Half-angle identities

   Note that \pm in these formulae does not mean both are correct, it
   means it may be either one, depending on the value of A/2.

          \sin \frac{A}{2} = \pm \sqrt{\frac{1-\cos A}{2}} \,

          \cos \frac{A}{2} = \pm \sqrt{\frac{1+\cos A}{2}} \,

          \tan \frac{A}{2} = \pm \sqrt{\frac{1-\cos A}{1+\cos A}} = \frac
          {\sin A}{1+\cos A} = \frac {1-\cos A}{\sin A} \,

Triangle identities

Law of sines

   The law of sines for an arbitrary triangle states:

          \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c},

   or equivalently:

          \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

Law of cosines

   The law of cosines (also known as the cosine formula) is an extension
   of the Pythagorean theorem to arbitrary triangles:

          c^2 = a^2 + b^2 − 2abcosC,

   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[\frac{1}{2}(A+B)]}{\tan[\frac{1}{2}(A-B)]}

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