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Law Of Tangents
In trigonometry, the law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. In Figure 1, , , and are the lengths of the three sides of the triangle, and , , and are the angles ''opposite'' those three respective sides. The law of trigonometric function, tangents states that : \frac = \frac . The law of tangents, although not as commonly known as the law of sines or the law of cosines, is equivalent to the law of sines, and can be used in any case where two sides and the included angle, or two angles and a side, are known. Proof To prove the law of tangents one can start with the law of sines: : \frac = \frac. Let : d = \frac = \frac so that : a = d \sin\alpha \quad\text \quad b = d \sin\beta. It follows that : \frac = \frac = \frac . Using the List of trigonometric identities#Product-to-sum and sum-to-product identities, trigonometric identity, the factor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle With Notations 2
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-Collinearity, collinear, determine a unique triangle and simultaneously, a unique Plane (mathematics), plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Types of triangle The terminology for categorizing triangles is more than two thousand years old, having been defined on the very first page of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geometry are the natural analog of straight lines in Euclidean space. For any pair of distinct non- antipodal points on the sphere, there is a unique great circle passing through both. (Every great circle through any point also passes through its antipodal point, so there are infinitely many great circles through two antipodal points.) The shorter of the two great-circle arcs between two distinct points on the sphere is called the ''minor arc'', and is the shortest surface-path between them. Its arc length is the great-circle distance between the points (the intrinsic distance on a sphere), and is proportional to the measure of the central angle formed by the two points and the center of the sphere. A great circle is the largest circle that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation. Trigonometry is known for its many identities. These trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation. History Sumerian astronomers studied angle measure, using a division of circles into 360 degrees. They, and later the Babylonians, studied the ratios of the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Half-angle Formula
In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle. The tangent of half an angle is the stereographic projection of the circle onto a line. Among these formulas are the following: : \begin \tan \tfrac12( \eta \pm \theta) &= \frac = \frac = -\frac, \\ 0pt \tan \tfrac12 \theta &= \frac = \frac = \frac, & & (\eta = 0) \\ 0pt \tan \tfrac12 \theta &= \frac = \frac = \csc\theta-\cot\theta, & & (\eta = 0) \\ 0pt \tan \tfrac12 \big(\theta \pm \tfrac12\pi \big) &= \frac = \sec\theta \pm \tan\theta = \frac, & & \big(\eta = \tfrac12\pi \big) \\ 0pt \tan \tfrac12 \big(\theta \pm \tfrac12\pi \big) &= \frac = \frac = \frac, & & \big(\eta = \tfrac12\pi \big) \\ 0pt \frac &= \pm\sqrt \\ 0pt \tan \tfrac12 \theta &= \pm \sqrt \\ 0pt \end From these one can derive identities expressing the sine, cosine, and tangent as functions of tangents of half-angles: : \begin \sin \alpha & = \frac \\ pt\cos \alpha & = \frac \\ pt\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Half-side Formula
In spherical trigonometry, the half side formula relates the angles and lengths of the sides of spherical triangles, which are triangles drawn on the surface of a sphere and so have curved sides and do not obey the formulas for plane triangles. Formulas On a unit sphere, the half-side formulas are. : \begin \tan\left(\frac\right) & = R \cos (S- A) \\ pt\tan \left(\frac\right) & = R \cos (S- B) \\ pt\tan \left(\frac\right) & = R \cos (S - C) \end where * ''a'', ''b'', ''c'' are the lengths of the sides respectively opposite angles ''A'', ''B'', ''C'', * S = \frac(A+B+ C) is half the sum of the angles, and * R=\sqrt. The three formulas are really the same formula, with the names of the variables permuted. To generalize to a sphere of arbitrary radius ''r'', the lengths ''a'',''b'',''c'' must be replaced with * a \rightarrow a/r * b \rightarrow b/r * c \rightarrow c/r so that ''a'',''b'',''c'' all have length scales, instead of angular scales. See also * Spherical law o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mollweide's Formula
In trigonometry, Mollweide's formula is a pair of relationships between sides and angles in a triangle. A variant in more geometrical style was first published by Isaac Newton in 1707 and then by in 1746. Thomas Simpson published the now-standard expression in 1748. Karl Mollweide republished the same result in 1808 without citing those predecessors. It can be used to check the consistency of solutions of triangles.Ernest Julius Wilczynski, ''Plane Trigonometry and Applications'', Allyn and Bacon, 1914, page 105 Let ''a'', ''b'', and ''c'' be the lengths of the three sides of a triangle. Let ''α'', ''β'', and ''γ'' be the measures of the angles opposite those three sides respectively. Mollweide's formulas are : \begin \frac c = \frac , \\ 0mu \frac c = \frac . \end Relation to other trigonometric identities Because in a planar triangle \tfrac12\gamma = \tfrac12\pi - \tfrac12(\alpha + \beta), these identities can alternately be written in a form in which they ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Cotangents
In trigonometry, the law of cotangentsThe Universal Encyclopaedia of Mathematics, Pan Reference Books, 1976, page 530. English version George Allen and Unwin, 1964. Translated from the German version Meyers Rechenduden, 1960. is a relationship among the lengths of the sides of a triangle and the cotangents of the halves of the three angles. This is also known as the Cot Theorem. Just as three quantities whose equality is expressed by the law of sines are equal to the diameter of the circumscribed circle of the triangle (or to its reciprocal, depending on how the law is expressed), so also the law of cotangents relates the radius of the inscribed circle of a triangle (the inradius) to its sides and angles. Statement Using the usual notations for a triangle (see the figure at the upper right), where , , are the lengths of the three sides, , , are the vertices opposite those three respective sides, , , are the corresponding angles at those vertices, is the semi-perimeter, that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Cosines
In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states :c^2 = a^2 + b^2 - 2ab\cos\gamma, where denotes the angle contained between sides of lengths and and opposite the side of length . For the same figure, the other two relations are analogous: :a^2=b^2+c^2-2bc\cos\alpha, :b^2=a^2+c^2-2ac\cos\beta. The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if the angle is a right angle (of measure 90 degrees, or radians), then , and thus the law of cosines reduces to the Pythagorean theorem: :c^2 = a^2 + b^2. The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known. History Though the notion of the cosine was not yet developed in his time, Euclid's '' Elements'', dating back to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Sines
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\, 2R, where , and are the lengths of the sides of a triangle, and , and are the opposite angles (see figure 2), while is the radius of the triangle's circumcircle. When the last part of the equation is not used, the law is sometimes stated using the reciprocals; \frac \,=\, \frac \,=\, \frac. The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. It can also be used when two sides and one of the non-enclosed angles are known. In some such cases, the triangle is not uniquely determined by this data (called the ''ambiguous case'') and the technique gives two possible values for the enclosed angle. The law of sines is one of two trigonometric equations commonly app ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nasir Al-Din Al-Tusi
Muhammad ibn Muhammad ibn al-Hasan al-Tūsī ( fa, محمد ابن محمد ابن حسن طوسی 18 February 1201 – 26 June 1274), better known as Nasir al-Din al-Tusi ( fa, نصیر الدین طوسی, links=no; or simply Tusi in the West), was a Persian polymath, architect, philosopher, physician, scientist, and theologian. Nasir al-Din al-Tusi was a well published author, writing on subjects of math, engineering, prose, and mysticism. Additionally, al-Tusi made several scientific advancements. In astronomy, al-Tusi created very accurate tables of planetary motion, an updated planetary model, and critiques of Ptolemaic astronomy. He also made strides in logic, mathematics but especially trigonometry, biology, and chemistry. Nasir al-Din al-Tusi left behind a great legacy as well. Tusi is widely regarded as one of the greatest scientists of medieval Islam, since he is often considered the creator of trigonometry as a mathematical discipline in its own right. The Muslim sc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics In Medieval Islam
Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics ( Aryabhata, Brahmagupta). Important progress was made, such as full development of the decimal place-value system to include decimal fractions, the first systematised study of algebra, and advances in geometry and trigonometry. Arabic works played an important role in the transmission of mathematics to Europe during the 10th—12th centuries. Concepts Algebra The study of algebra, the name of which is derived from the Arabic word meaning completion or "reunion of broken parts", flourished during the Islamic golden age. Muhammad ibn Musa al-Khwarizmi, a Persian scholar in the House of Wisdom in Baghdad was the founder of algebra, is along with the Greek mathematician Diophantus, known as the father of algebra. In his book '' The Compendious Book on Calculation by Completion and Balancin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ibn Mu'adh Al-Jayyani
Abū ʿAbd Allāh Muḥammad ibn Muʿādh al-Jayyānī ( ar, أبو عبد الله محمد بن معاذ الجياني; 989, Cordova, Al-Andalus – 1079, Jaén, Al-Andalus) was an Arab, mathematician, Islamic scholar, and Qadi from Al-Andalus (in present-day Spain). Al-Jayyānī wrote important commentaries on Euclid's '' Elements'' and he wrote the first known treatise on spherical trigonometry. Life Little is known about his life. Confusion exists over the identity of ''al-Jayyānī'' of the same name mentioned by ibn Bashkuwal (died 1183), Qur'anic scholar, Arabic Philologist, and expert in inheritance laws (farāʾiḍī). It is unknown whether they are the same person. ''The book of unknown arcs of a sphere'' Al-Jayyānī wrote ''The book of unknown arcs of a sphere'', which is considered "the first treatise on spherical trigonometry", although spherical trigonometry in its ancient Hellenistic form was dealt with by earlier mathematicians such as Menelaus of Alexa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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