|
Ailles Rectangle
The Ailles rectangle is a rectangle constructed from four right-angled triangles which is commonly used in geometry classes to find the values of trigonometric functions of 15° and 75°. It is named after Douglas S. Ailles who was a high school teacher at Kipling Collegiate Institute in Toronto. Construction A Special right triangle#30°–60°–90° triangle, 30°–60°–90° triangle has sides of length 1, 2, and \sqrt. When two such triangles are placed in the positions shown in the illustration, the smallest rectangle that can enclose them has width 1+\sqrt and height \sqrt. Drawing a line connecting the original triangles' top corners creates a Special right triangle#45°–45°–90° triangle, 45°–45°–90° triangle between the two, with sides of lengths 2, 2, and (by the Pythagorean theorem) 2\sqrt. The remaining space at the top of the rectangle is a right triangle with acute angles of 15° and 75° and sides of \sqrt-1, \sqrt+1, and 2\sqrt. Derived trig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectangle
In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containing a right angle. A rectangle with four sides of equal length is a ''square''. The term "wikt:oblong, oblong" is used to refer to a non-square rectangle. A rectangle with Vertex (geometry), vertices ''ABCD'' would be denoted as . The word rectangle comes from the Latin ''rectangulus'', which is a combination of ''rectus'' (as an adjective, right, proper) and ''angulus'' (angle). A #Crossed rectangles, crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals (therefore only two sides are parallel). It is a special case of an antiparallelogram, and its angles are not right angles an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right-angled 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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometric Functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis. The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent functions. Their multiplicative inverse, reciprocals are respectively the cosecant, the secant, and the cotangent functions, which are less used. Each of these six trigonometric functions has a corresponding Inverse trigonometric functions, inverse function, and an analog among the hyperbolic functions. The oldest definitions of trigonometric functions, related to right-an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kipling Collegiate Institute
Kipling Collegiate Institute (Kipling CI, KCI, or Kipling) is a public high school in Toronto, Ontario, Canada. It is located in the former suburb of Etobicoke under the management of the Toronto District School Board, operating since 1960. History Kipling Collegiate was constructed in 1959 and opened its doors in September 1960. The school was built in a modernist design by architect Gordon Adamson. The school merged with Scarlett Heights Entrepreneurial Academy in 2018 and attempts to rename the school failed. See also * Education in Ontario * List of secondary schools in Ontario The following is a list of secondary schools in Ontario. Secondary education policy in the Canadian province of Ontario is governed by the Ministry of Education. Secondary education in Ontario includes Grades 9 to 12. The following list includes ... References External links Kipling Collegiate InstituteTDSB Profile {{Toronto High Schools High schools in Toronto Education in Etobicoke Schoo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toronto
Toronto ( , locally pronounced or ) is the List of the largest municipalities in Canada by population, most populous city in Canada. It is the capital city of the Provinces and territories of Canada, Canadian province of Ontario. With a population of 2,794,356 in 2021, it is the List of North American cities by population, fourth-most populous city in North America. The city is the anchor of the Golden Horseshoe, an urban agglomeration of 9,765,188 people (as of 2021) surrounding the western end of Lake Ontario, while the Greater Toronto Area proper had a 2021 population of 6,712,341. As of 2024, the census metropolitan area had an estimated population of 7,106,379. Toronto is an international centre of business, finance, arts, sports, and culture, and is recognized as one of the most multiculturalism, multicultural and cosmopolitanism, cosmopolitan cities in the world. Indigenous peoples in Canada, Indigenous peoples have travelled through and inhabited the Toronto area, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Right Triangle
A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio. Knowing the relationships of the angles or ratios of sides of these special right triangles allows one to quickly calculate various lengths in geometric problems without resorting to more advanced methods. Angle-based ''Angle-based'' special right triangles are specified by the relationships of the angles of which the triangle is composed. The angles of these triangles are such that the larger (right) angle, which is 90 degrees or radians, is equal to the sum of the other two angles ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagorean Theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. The theorem can be written as an equation relating the lengths of the sides , and the hypotenuse , sometimes called the Pythagorean equation: :a^2 + b^2 = c^2 . The theorem is named for the Ancient Greece, Greek philosopher Pythagoras, born around 570 BC. The theorem has been Mathematical proof, proved numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both Geometry, geometric proofs and Algebra, algebraic proofs, with some dating back thousands of years. When Euclidean space is represented by a Cartesian coordinate system in analytic geometry, Euclidean distance satisfies th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stack Exchange
Stack Exchange is a network of question-and-answer (Q&A) websites on topics in diverse fields, each site covering a specific topic, where questions, answers, and users are subject to a reputation award process. The reputation system allows the sites to be self-moderating. Stack Exchange is composed of 173 communities bringing in over 100 million unique visitors each month. the three most active sites in the network are Stack Overflow (which focuses on computer programming), Mathematics, and Ask Ubuntu (focusing on the Linux distribution Ubuntu). All sites in the network are modeled after the initial site Stack Overflow which was created by Jeff Atwood and Joel Spolsky in 2008. Further Q&A sites in the network are established, defined, and eventually if found relevant brought to creation by registered users through a special site named Area 51. User contributions since May 2, 2018 are licensed under Creative Commons Attribution-ShareAlike 4.0 International. Older c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Trigonometric Values
In mathematics, the values of the trigonometric functions can be expressed approximately, as in \cos (\pi/4) \approx 0.707, or exactly, as in \cos (\pi/ 4)= \sqrt 2 /2. While trigonometric tables contain many approximate values, the exact values for certain angles can be expressed by a combination of arithmetic operations and square roots. The angles with trigonometric values that are expressible in this way are exactly those that can be constructed with a compass and straight edge, and the values are called constructible numbers. Common angles The trigonometric functions of angles that are multiples of 15°, 18°, or 22.5° have simple algebraic values. These values are listed in the following table for angles from 0° to 45° (see below for proofs). In the table below, the label "Undefined" represents a ratio 1:0. If the codomain of the trigonometric functions is taken to be the real numbers these entries are undefined, whereas if the codomain is taken to be the projectiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle Geometry
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals a straight angle (180 degrees or π radians). The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the ''base'', in which case the opposite vertex is called the ''apex''; the shortest segment between the base and apex is the ''height''. The area of a triangle equals one-half the product of height and base length. In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, and any three points that do not all lie on the same straight line determine a unique triangle situated within a u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
