99guru -- santosoku: Trigonometry

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"Trig" redirects here. For other uses, see Trig (disambiguation).

The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. Calculating the final position of the astronaut at the end of the arm requires repeated use of the trigonometric functions of those angles.

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

Trigonometry (from Ancient Sanskrit "Trikona" or triangle and "Mati" or measure. Also from Greek trigōnon "triangle" + metron "measure")^[1] is a branch of mathematics that studies triangles, particularly right triangles. Trigonometry deals with relationships between the sides and the angles of triangles, and with trigonometric functions, which describe those relationships and 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 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

Ancient Egyptian and Babylonian mathematicians 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. The ancient Greeks transformed trigonometry into an ordered science.^[3]

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. Claudius PtolemyHipparchus' Chords in a Circle in his Almagest.^[4] 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 Aryabhata.^[5] 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. At about the same time, ChineseLatin translations of the works of Persian and Arabic astronomers such as Al Battani and Nasir al-Din al-Tusi.^[6] 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. expanded upon mathematicians developed trigonometry independently, although it was not a major field of study for them. Knowledge of trigonometric functions and methods reached Europe via

Driven by the demands of navigation and the growing need for accurate maps of large areas, trigonometry grew to be a major branch of mathematics.^[7] Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595.^[8] Gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry. The works of James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series.^[9] Also in the 18th century, Brook Taylor defined the general Taylor series.^[10]

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.

Graphing process of yx) using a unit circle. = sin(

Graphing process of yx) using a unit circle. = tan(

Graphing process of yx) using a unit circle. = csc(

Common formulas

Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. Some identities equate an expression to a different expression involving the same angles. These are listed in List of trigonometric identities. Triangle identities that relate the sides and angles of a given triangle 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 circumscribed circle 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:

$\mbox{Area} = \frac{1}{2}a b\sin C.$