Trigonometry

The trigonometry (of the old Greek τρίγωνος/ trígonos , “triangular”, and μέτρον/ métron , “measurement”) is a branch of the Mathématiques which treats reports/ratios of distances and angles in the Triangle S and of goniometrical functions such as sine, cosine and tangent.

Presentation

History of trigonometry

The origins of trigonometry go up with civilizations of ancient Egypt, of Mésopotamie and of the valley of Indus, there is more than 4000 years. It would seem that the Babylonians based trigonometry on a numerical system with bases 60. Lagadha (-1350; -1200) is the first mathematician to use the geometry and trigonometry for astronomy. The majority of its work are destroyed today.

The first use of sine appears in the Sulba Sutras in India, between 800 and 500 before J.C., where the sine of π /4 (45°) is correctly calculated like 1/√2 in a problem of construction of a of the same circle surface than a given square (opposite of the Quadrature of the circle).

The Greek mathematician Hipparque de Nicée (-190; -120) built the first trigonometrical tables in the shape of tables of cords: they made correspond to each value of the angle in the center, the length of the cord intercepted in the circle, for a given fixed ray. This calculation corresponds to the double of the sine of the angle half, and thus gives, in a certain way, which we call today a table of sine. However, the tables of Hipparque not having arrived to us, they are known for us only by the Egyptian mathematician Ptolémée, which published them, in years 100, with their method of construction in his Almageste . Thus they were redécouvertes at the end of the Middle Ages by Georg von Purbach and its pupil Regiomontanus.

The Indian mathematician Aryabhata, in 499, gives a table of the sines and the cosine. He uses zya for sine, kotizya for cosine and otkram zya for the reverse of the sine. He introduces also the Sinus pours.

Another Indian mathematician, Brahmagupta, use into 628 the numerical Interpolation to calculate the value of the sines until the second order.

Omar Khayyam (1048 - 1131) combines the use of trigonometry and the Théorie of the approximation to provide methods of solutions of equations algebraic by the geometry.

Detailed methods of constructions of tables of sine and cosine for all the angles are written by the mathematician Bhaskara in 1150. It develops also the spherical Trigonométrie.

At the 13th century, Aldine Nasir Tusi, following Bhaskara, is probably one of the first to regard trigonometry as a discipline distinct from mathematics.

Lastly, at the 14th century, Al-Kashi carries out trigonometrical function tables at the time of its studies in astronomy. The mathematician Silesia N Bartholomäus Pitiscus publishes a remarkable work on trigonometry in 1595, whose title ( Trigonometria ) gave its name to the discipline.

Applications

See also: Applications of trigonometry

The applications of trigonometry are immense. In particular, it is used in Astronomie with the technique of Triangulation which makes it possible to measure the distance between stars. Other fields where trigonometry intervenes (nonexhaustive list): Acoustic, Optical, electronic, Statistical, economy, Biology, Chemistry, Medicine, Physical, Meteorology, Geodesy, Geography, Cartography, Cryptography, etc

Trigonometry

A possible definition of the goniometrical functions is to use the right-angled triangles, i.e. the triangles which have a right angle (90° (degree S) or π/2 Radian S).

And because the sum of the Angle S of a Triangle made 180° (or π radians), the largest angle in such a triangle is the right angle. The longest side in a right-angled triangle, i.e. the side opposed to the largest angle (the right angle), is called the Hypoténuse .

In the figure on the right, the angle $\backslash \; widehat\; \{ACB\}$ form the right angle. Side AB the hypotenuse.

The goniometrical functions are defined thus, with the angle $\backslash \; widehat\; \{VAT\}\; =\; A$:

$\backslash \; sin\; has\; =\; \{\backslash \; mbox\; \{opp\}\; \backslash \; over\; \backslash \; mbox\; \{hyp\}\}\; =\; \{has\; \backslash \; over\; C\}$

\ qquad \ cos has = {\ mbox {adj} \ over \ mbox {hyp}} = {B \ over C} \ qquad \ tan has = {\ mbox {opp} \ over \ mbox {adj}} = {has \ over B}

With opp for opposed side, adj for adjacent side and hyp for hypotenuse.

Average mnemotechnics to retain the goniometrical functions: SOHCAHTOA (S: Sine, C: Cosine, T: Tangent, O: Opposed, H: Hypotenuse, Adjacent a: ) or then " break " (CAHSOHTOA) vulgar thus easier

They are the goniometrical functions most important and there exists about it much of different. They were defined for the angles between 0° and 90° (either between 0 and π/2 radians). By using the Circle unit, one can extend this definition.

Formulas of trigonometry

See also: Amorce=Pour to find all the other formulas of trigonometry, to see the article, trigonometrical Identity

Fundamental identity

Some With one is the angle a:

$\backslash \; cos^\; \{2\}\; has\; +\; \backslash \; sin^\; \{2\}\; has\; =\; 1\; \backslash ,$

Formulas of addition and difference

$\backslash \; sin\; (has\; -\; B)\; =\; \backslash \; sin\; has\; \backslash \; cos\; B\; -\; \backslash \; cos\; has\; \backslash \; sin\; B\; \backslash ,$

$\backslash \; sin\; (has\; +\; B)\; =\; \backslash \; sin\; has\; \backslash \; cos\; B\; +\; \backslash \; cos\; has\; \backslash \; sin\; B\; \backslash ,$

$\backslash \; cos\; (has\; -\; B)\; =\; \backslash \; cos\; has\; \backslash \; cos\; B\; +\; \backslash \; sin\; has\; \backslash \; sin\; B\; \backslash ,$

$\backslash \; cos\; (has\; +\; B)\; =\; \backslash \; cos\; has\; \backslash \; cos\; B\; -\; \backslash \; sin\; has\; \backslash \; sin\; B\; \backslash ,$

$\backslash \; tan\; (has\; -\; B)\; =\; \backslash \; frac\; \{\backslash \; tan\; has\; -\; \backslash \; tan\; B\}\; \{1\; +\; \backslash \; tan\; has\; \backslash \; tan\; B\}\; \backslash ,$

$\backslash \; tan\; (has\; +\; B)\; =\; \backslash \; frac\; \{\backslash \; tan\; has\; +\; \backslash \; tan\; B\}\; \{1\; -\; \backslash \; tan\; has\; \backslash \; tan\; B\}\; \backslash ,$

Formulas of the arc half

These formulas are important to retain because they intervene in very many problems. While posing: $t\; =\; \backslash \; tan\; \backslash \; frac\; \{has\}\; \{2\}$, one has then:

$\backslash \; sin\; \backslash \; has\; =$

$\backslash \; cos\; \backslash \; has\; =$

$\backslash \; tan\; \backslash \; has\; =$

Formulas of development and factorization

Développement

$\backslash \; cos\; has\; \backslash \; times\; \backslash \; cos\; B\; =$

$\backslash \; sin\; has\; \backslash \; times\; \backslash \; sin\; B\; =$

$\backslash \; cos\; has\; \backslash \; times\; \backslash \; sin\; B\; =$

Factorization

$\backslash \; cos\; p\; +\; \backslash \; cos\; Q\; =\; 2\; \backslash \; cos\; \backslash \; left\; (\backslash \; right)\; \backslash \; cos\; \backslash \; left\; (\backslash \; right)$

$\backslash \; cos\; p\; -\; \backslash \; cos\; Q\; =\; -2\; \backslash \; sin\; \backslash \; left\; (\backslash \; right)\; \backslash \; sin\; \backslash \; left\; (\backslash \; right)$

$\backslash \; sin\; p\; +\; \backslash \; sin\; Q\; =\; 2\; \backslash \; sin\; \backslash \; left\; (\backslash \; right)\; \backslash \; cos\; \backslash \; left\; (\backslash \; right)$

$\backslash \; sin\; p\; -\; \backslash \; sin\; Q\; =\; 2\; \backslash \; sin\; \backslash \; left\; (\backslash \; right)\; \backslash \; cos\; \backslash \; left\; (\backslash \; right)$

Al-Kashi theorem or Law of the cosine

$a^2\; =\; b^2\; +\; c^2\; -\; 2\; \backslash ,\; B\; \backslash ,\; C\; \backslash \; cdot\; cos\; \backslash \; A$

This formula has an particular importance in triangulation and was used for the origin in astronomy (see Théorème of Al-Kashi for more details). One must to the mathematician Ghiyath Al-Kashi, of the school of Samarkand, to put the theorem in a form usable for the Triangulation during the 15th century.

To solve a triangle

It is, being given a side and two adjacent angles, or an angle and two sides adjacent, or with the rigor two sides B and C and the angle B, to find the triangle corresponding, i.e., has, B, C, has, C (and to check one of the rules not applied in the process). One solves this kind of problem using the preceding formulas (more the formula of obvious projection has = b.cosC +c.cosB).

Example: On axis OX, OB = 1 and OC = 1.5. OBM = 60° and OCM = 30° To find M: One solves as follows: to make the diagram; M is in (x= 0.75; there = 0.45) approximately. To reason: triangle BMC: B = 120°, C = 30° thus M = 30°; thus isosceles triangle in b: BM = 0.5; then CM = 2. (0.5) .cos C = sqrt (3) /2. That is to say H the projection of M on the axis: HM = there and angle HMB = 30°. It results from it that there = sqrt (3) /4 = 0,433 and X = 1 (0,5) /2 = 0,75. The distance OM = sqrt (3) /2 = MC, and azimuth of M = 30°, angle OMB = 90°.

It is rare from the cadastral point of view that the cases are also simple.

In general one requests 4 from 5 ChS (significant figures): the calculators considerably reduced the rather tiresome work of " reduction of the triangles". Let us point out what the measurement of the degree of the terrestrial meridian line of Paris was carried out kind between Malvoisine and Montlhéry by the Picard abbot, about 1660? .

Surface S of the triangle is calculated by the formula of the sines or the formula of Héron (of Alexandria) which from of deduced: S ² = p (Pa) (Pb) (PC).

detailed Article: Resolution of a triangle

Some famous problems

• the Quadrature of the circle: to build a of the same square surface than a circle given using a rule and of a compass, this problem is insoluble.

• the approximation sin has = has - K has ³ with K = 1/6 when has is small (in radian!) can result from the formula sin (3A) = 3.sinA -4 sin ³ a: the terms has ³ of it give: - k27 = -3k -4, CQFD.

• the arrow of a cord AB under tightening the arc AOB = 2 α: that is to say I medium of AB and CD the diameter passing by I: ID = arrow such as F (2R- F) = (R sin α) ²

• Surface of the miter: S = R ² - sin (2.$\backslash \; alpha$) /2 when alpha is very small, one compares this surface with that of the osculatory parabola 1/3 f.AB (theorem of Archimedes): the difference is of a nature higher than 3.

• formula of Thing (1706): maybe With the arc whose tangent is 1/5 and B that whose arc is 1/239: then 4A - B = $\backslash \; pi$/4, which gives a good approximation of pi, " enough rapidement". This formula spreads.

• constructible regular polygons: the heptagon and the nonagon are impossible, but the polygon at 17 sides (Heptadécagone) is constructible (theorem of Gauss at 19 years: 1796); on the other hand one can build by folding (cf Origami) the heptagon and the nonagon. One finds nevertheless easily has = 360°/7, then sin has .sin 2A .sin 3A = sqrt (7) /8 and for the cosine 1/8 cf article. Similar formulas exist for the nonagon.

• algorithm CORDIC of Briggs and redécouvert by Volker: or how your calculator go does also quickly?

See too

• trigonometrical Identité
• Cercle goniometrical unit
• Function
• hyperbolic Fonction
• Trigonométrie of Wildberger
• Trigonométrie complexes
• trigonometrical Sens

Simple: Trigonometry Zh-min-nan: Saⁿ-kak-hoat Zh-yue: 三角學

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