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MOOC# Trigonometric Functions, Logarithms and Exponentials

Description

Ce cours donne les connaissances fondamentales liées aux fonctions trigonométriques, logarithmiques et exponentielles. La présentation des concepts et des propositions est soutenue par une grande gamme de figures et d'animations, ainsi que par des exemples qui illustrent la mise en oeuvre des connaissances acquises.

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Instructor

Related courses (243)

Related concepts (188)

Related publications (4)

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Trigonometry () is a branch of mathematics concerned with relationships between angles and ratios of 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.

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.

In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

Lectures in this MOOC (36)

In this thesis, we study the stochastic heat equation (SHE) on bounded domains and on the whole Euclidean space $\R^d.$ We confirm the intuition that as the bounded domain increases to the whole space, both solutions become arbitrarily close to one another. Both vanishing Dirichlet and Neumann boundary conditions are considered.We first study the nonlinear SHE in any space dimension with multiplicative correlated noise and bounded initial data. We prove that the solutions to SHE on an increasing sequence of domains converge exponentially fast to the solution to SHE on $\R^d.$ Uniform convergence on compact set is obtained for all $p$-moments. The conditions that need to be imposed on the noise are the same as those required to ensure existence of a random field solution. A Gronwall-type iteration argument is used together with uniform bounds on the solutions, which are surprisingly valid for the entire sequence of increasing domains.We then study SHE in space dimension $d\ge 2$ with additive white noise and bounded initial data. Even though both solutions need to be considered as distributions, their difference is proved to be smooth. If fact, the order of smoothness depends only on the regularity of the boundary of the increasing sequence of domains. We prove that the Fourier transform, in the sense of distributions, of the solution to SHE on $\R^d$ do not have any locally mean-square integrable representative. Therefore, convergence is studied in local versions of Sobolev spaces. Again, exponential rate is obtained.Finally, we study the Anderson model for SHE with correlated noise and initial data given by a measure. We obtain a special expression for the second moment of the difference of the solution on $\R^d$ with that on a bounded domain. The contribution of the initial condition is made explicit. For example, exponentially fast convergence on compact sets is obtained for any initial condition with polynomial growth. More interestingly, from a given convergence rate, we can decide whether some initial data is admissible.

Angle and FunctionsMOOC: Trigonometric Functions, Logarithms and Exponentials

Delves into fundamental special functions like trigonometric, logarithmic, and exponential functions, emphasizing the concept of angles in various applications.

Angle Measurement: Trigonometric ConceptsMOOC: Trigonometric Functions, Logarithms and Exponentials

Explores angle measurement using a unit circle, radians, and trigonometric concepts, including clock hand movements and infinite angles.

Trigonometric Circle: Interpretation and ApplicationsMOOC: Trigonometric Functions, Logarithms and Exponentials

Explores the interpretation of angles on the trigonometric circle and its applications in circular motion.

Angle Measurement: Radians and DegreesMOOC: Trigonometric Functions, Logarithms and Exponentials

Covers angle measurement using radians and degrees, including unit conversion and trigonometric circle symmetries.

Trigonometric Functions: Sin and CosMOOC: Trigonometric Functions, Logarithms and Exponentials

Covers the definition of sine and cosine functions, their graphs, and notable points.

We study the system of linear partial differential equations given by dw + a Lambda w = f, on open subsets of R-n, together with the algebraic equation da Lambda u = beta, where a is a given 1-form, f is a given (k + 1)-form, beta is a given k + 2-form, w and u are unknown k-forms. We show that if rank[da] >= 2(k+1) those equations have at most one solution, if rank[da] equivalent to 2m >= 2(k + 2) they are equivalent with beta = df + a Lambda f and if rank[da] equivalent to 2m >= 2(n - k) the first equation always admits a solution. Moreover, the differential equation is closely linked to the Poincare lemma. Nevertheless, as soon as a is nonexact, the addition of the term a Lambda w drastically changes the problem.

2018Mika Tapani Göös, Siddhartha Jain

We exhibit an unambiguous k-DNF formula that requires CNF width (Omega) over tilde (k(2)), which is optimal up to logarithmic factors. As a consequence, we get a near-optimal solution to the Alon-Saks-Seymour problem in graph theory (posed in 1991), which asks: How large a gap can there be between the chromatic number of a graph and its biclique partition number? Our result is also known to imply several other improved separations in query and communication complexity.