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Solution Of Triangles
Solution of triangles () is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation. Solving plane triangles A general form triangle has six main characteristics (see picture): three linear (side lengths ) and three angular (). The classical plane trigonometry problem is to specify three of the six characteristics and determine the other three. A triangle can be uniquely determined in this sense when given any of the following: *Three sides (SSS) *Two sides and the included angle (SAS, side-angle-side) *Two sides and an angle not included between them (SSA), if the side length adjacent to the angle is shorter than the other side length. *A side and the two angles adjacent to it (ASA) *A side, the angle opposite to it and an angle adjacent to it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometry
Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. 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 me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Cosines
In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides , , and , opposite respective angles , , and (see Fig. 1), the law of cosines states: \begin c^2 &= a^2 + b^2 - 2ab\cos\gamma, \\[3mu] a^2 &= b^2+c^2-2bc\cos\alpha, \\[3mu] b^2 &= a^2+c^2-2ac\cos\beta. \end The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if is a right angle then , and the law of cosines special case, reduces to . The law of cosines is useful for solution of triangles, solving a triangle when all three sides or two sides and their included angle are given. Use in solving triangles The theorem is used in solution of triangles, i.e., to find (see Figure 3): *the third side of a triangle if two sides and the angle between them is known: c = \sqrt\,; *the angles of a triangle if the three sides are known: \gamma = \arccos\l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Secrets Of Triangles
''The Secrets of Triangles: A Mathematical Journey'' is a popular mathematics book on the geometry of triangles. It was written by Alfred S. Posamentier and , and published in 2012 by Prometheus Books. Topics The book consists of ten chapters, with the first six concentrating on triangle centers while the final four cover more diverse topics including the area of triangles, inequalities involving triangles, straightedge and compass constructions, and fractals. Beyond the classical triangle centers (the circumcenter, incenter, orthocenter, and centroid) the book covers other centers including the Brocard points, Fermat point, Gergonne point, and other geometric objects associated with triangle centers such as the Euler line, Simson line, and nine-point circle. The chapter on areas includes both trigonometric formulas and Heron's formula for computing the area of a triangle from its side lengths, and the chapter on inequalities includes the Erdős–Mordell inequalit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angle Bisector
In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'', a line that passes through the midpoint of a given segment, and the ''angle bisector'', a line that passes through the apex of an angle (that divides it into two equal angles). In three-dimensional space, bisection is usually done by a bisecting plane, also called the ''bisector''. Perpendicular line segment bisector Definition *The perpendicular bisector of a line segment is a line which meets the segment at its midpoint perpendicularly. *The perpendicular bisector of a line segment AB also has the property that each of its points X is equidistant from segment AB's endpoints: (D)\quad , XA, = , XB, . The proof follows from , MA, =, MB, and Pythagoras' theorem: :, XA, ^2=, XM, ^2+, MA, ^2=, XM, ^2+, MB, ^2=, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Altitude (geometry)
In geometry, an altitude of a triangle is a line segment through a given vertex (called '' apex'') and perpendicular to a line containing the side or edge opposite the apex. This (finite) edge and (infinite) line extension are called, respectively, the '' base'' and ''extended base'' of the altitude. The point at the intersection of the extended base and the altitude is called the ''foot'' of the altitude. The length of the altitude, often simply called "the altitude" or "height", symbol , is the distance between the foot and the apex. The process of drawing the altitude from a vertex to the foot is known as ''dropping the altitude'' at that vertex. It is a special case of orthogonal projection. Altitudes can be used in the computation of the area of a triangle: one-half of the product of an altitude's length and its base's length (symbol ) equals the triangle's area: /2. Thus, the longest altitude is perpendicular to the shortest side of the triangle. The altitudes are al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Median (geometry)
In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side. Every triangle has exactly three medians, one from each vertex, and they all intersect at the triangle's centroid. In the case of isosceles and equilateral triangles, a median bisects any angle at a vertex whose two adjacent sides are equal in length. The concept of a median extends to tetrahedra. Relation to center of mass Each median of a triangle passes through the triangle's centroid, which is the center of mass of an infinitely thin object of uniform density coinciding with the triangle. Thus, the object would balance at the intersection point of the medians. The centroid is twice as close along any median to the side that the median intersects as it is to the vertex it emanates from. Equal-area division Each median divides the area of the triangle in half, hence the name, and hence a triangular object of uniform density would balance on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resolve Triangle With C Alpha Beta
Resolve may refer to: Music * ''Resolve'' (Lagwagon album), 2005 * ''Resolve'' (Poppy Ackroyd album), 2018 * "Resolve" (song), by the Foo Fighters * ''Resolve'', an album by Last Tuesday Film and television * '' The Resolve'', a 1915 American silent short drama film * "Resolve" (''One Tree Hill'' episode) * "Resolve" (''Tales of the Jedi'') Other uses * Determination * ''Resolve'', a British tugboat, formerly '' Empire Zona'' * Operation Resolve, an underwater search for the wreckage of South African Airways Flight 295 * Claris Resolve, a spreadsheet program * DaVinci Resolve, video editing software * to ''resolve'' a server address, in the Domain Name System See also * Resolved (other) * Resolution (other) Resolution(s) may refer to: Common meanings * Resolution (debate), the statement which is debated in policy debate * Resolution (law), a written motion adopted by a deliberative body * New Year's resolution, a commitment that an individual m .. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right Triangle
A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle is called the '' hypotenuse'' (side c in the figure). The sides adjacent to the right angle are called ''legs'' (or ''catheti'', singular: '' cathetus''). Side a may be identified as the side ''adjacent'' to angle B and ''opposite'' (or ''opposed to'') angle A, while side b is the side adjacent to angle A and opposite angle B. Every right triangle is half of a rectangle which has been divided along its diagonal. When the rectangle is a square, its right-triangular half is isosceles, with two congruent sides and two congruent angles. When the rectangle is not a square, its right-triangular half is scalene. Every triangle whose base is the diameter of a circle and whose apex lies on the circle is a right triangle, with the right angle at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resolve Triangle With B C Beta 2 Solutions
Resolve may refer to: Music * ''Resolve'' (Lagwagon album), 2005 * ''Resolve'' (Poppy Ackroyd album), 2018 * "Resolve" (song), by the Foo Fighters * ''Resolve'', an album by Last Tuesday Film and television * '' The Resolve'', a 1915 American silent short drama film * "Resolve" (''One Tree Hill'' episode) * "Resolve" (''Tales of the Jedi'') Other uses * Determination * ''Resolve'', a British tugboat, formerly '' Empire Zona'' * Operation Resolve, an underwater search for the wreckage of South African Airways Flight 295 * Claris Resolve, a spreadsheet program * DaVinci Resolve, video editing software * to ''resolve'' a server address, in the Domain Name System See also * Resolved (other) * Resolution (other) Resolution(s) may refer to: Common meanings * Resolution (debate), the statement which is debated in policy debate * Resolution (law), a written motion adopted by a deliberative body * New Year's resolution, a commitment that an individual m .. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heron's Formula
In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths Letting be the semiperimeter of the triangle, s = \tfrac12(a + b + c), the area is A = \sqrt. It is named after first-century engineer Heron of Alexandria (or Hero) who proved it in his work ''Metrica'', though it was probably known centuries earlier. Example Let be the triangle with sides , , and . This triangle's semiperimeter is s = \tfrac12(a+b+c)= \tfrac12(4+13+15) = 16 therefore , , , and the area is \begin A &= \\ mu&= \\ mu&= 24. \end In this example, the triangle's side lengths and area are integers, making it a Heronian triangle. However, Heron's formula works equally well when the side lengths are real numbers. As long as they obey the strict triangle inequality, they define a triangle in the Euclidean plane whose area is a positive real number. Alternate expressions Heron's formula can also be written in terms of just ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
