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Inverse trigonometric functions

In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. They are the inverses of the sine, tangent, cotangent and cosecant functions, are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are used in engineering, navigation and geometry. There are several notations used for the inverse trigonometric functions; the most common convention is to name inverse trigonometric functions using an arc- prefix: arcsin, arctan, etc. This notation arises from the following geometric relationships: When measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ, where r is the radius of the circle. Thus, in the unit circle, "the arc whose cosine is x" is the same as "the angle whose cosine is x", because the length of the arc of the circle in radii is the same as the measurement of the angle in radians. In computer programming languages the inverse trigonometric functions are called by the abbreviated forms asin, atan.

The notations sin−1, cos−1, tan−1, etc. as introduced by John Herschel in 1813, are used as well in English-language sources, this convention complies with the notation of an inverse function. This might appear to conflict logically with the common semantics for expressions like sin2, which refer to numeric power rather than function composition, therefore may result in confusion between multiplicative inverse and compositional inverse; the confusion is somewhat ameliorated by the fact that each of the reciprocal trigonometric functions has its own name—for example, −1 = sec. Certain authors advise against using it for its ambiguity. Another convention used by a few authors is to use a majuscule first letter along with a −1 superscript: Sin−1, Cos−1, Tan−1, etc; this avoids confusion with the multiplicative inverse, which should be represented by sin−1, cos−1, etc. Since 2009, the ISO 80000-2 standard has specified the "arc" prefix for the inverse functions. Since none of the six trigonometric functions are one-to-one, they are restricted in order to have inverse functions.

Therefore, the ranges of the inverse functions are proper subsets of the domains of the original functions. For example, using function in the sense of multivalued functions, just as the square root function y = √x could be defined from y2 = x, the function y = arcsin is defined so that sin = x. For a given real number x, with − 1 ≤ x ≤ 1, there are multiple numbers; when only one value is desired, the function may be restricted to its principal branch. With this restriction, for each x in the domain the expression arcsin will evaluate only to a single value, called its principal value; these properties apply to all the inverse trigonometric functions. The principal inverses are listed in the following table. If x is allowed to be a complex number the range of y applies only to its real part. Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is by considering the geometry of a right-angled triangle, with one side of length 1, another side of length x applying the Pythagorean theorem and definitions of the trigonometric ratios.

Purely algebraic derivations are longer. Complementary angles: arccos ⁡ = π 2 − arcsin ⁡ arccot ⁡ = π 2 − arctan ⁡ arccsc ⁡ = π 2 − arcsec ⁡ Negative arguments: arcsin ⁡ = − arcsin ⁡ arccos ⁡ = π − arccos ⁡ arctan

1st Infantry Regiment (France)

The 1st Infantry Regiment is an infantry regiment of the French Army, founded in 1479 as one of the oldest regiments in active service in the world. It is an offspring of the bande de Picardie under the Ancien regime, one of the five oldest regiments in France, it distinguished itself, as the 1ère Demi-Brigade d'Infanterie de Ligne, during the French Revolutionary Wars at the Battle of Fleurus, the Battle of Messkirch and the Battle of Biberach. The regiment has been patroned by the city of Saint-Amand-Montrond since 12 April 2003; the 1er RI is the only French infantry regiment to feature a squadron specialised in urban warfare, the Groupe Commando d'Investigation. The CGI is, with the Compagnie de Combat en zone Urbaine of the 2nd Foreign Parachute Regiment, the benchmark for urban combat in the French military; the flag bears the names of the battle in which the 1st Infantry Regiment took part for the French Republic: Valmy 1792, Fleurus 1794, Moeskirch 1800, Biberach 1800, Miliana 1842, Guise 1914, Verdun-l'Yser 1916-1917, La Somme 1916, L'Ourcq 1918, Résistance Berry 1944, AFN 1952-1962.

During the Napoleonic Wars the regiment fought at the battles of Caldiero, Salamanca, Lützen, Dresden, Montmirail, Qatre Bras and Waterloo. The Fourragère of the Médaille militaire was attributed on 10 September 1918, presented by General Castelneau on 19 October 1918 in Alsace, after the regiment was Mentioned in Despatches at the order of the Army for the fourth time, it is adorned of the olive of the Croix de Guerre 1914-1918, of the olive of the Croix de Guerre 1939-1945 for two mentions in despatches awarded during the Phony War and in the French Resistance


Redstockings known as Redstockings of the Women's Liberation Movement, is a radical feminist group, founded in January 1969 in New York City. The group's name is derived from bluestocking, a term used to disparage feminist intellectuals of earlier centuries, red, for its association with the revolutionary left; the group was started by Ellen Willis and Shulamith Firestone in February 1969, after the breakup of New York Radical Women. Other early members included Kathie Sarachild, Patricia Mainardi, Barbara Leon, Lucinda Cisler, Irene Peslikis, Alix Kates Shulman. Firestone soon split with the group to form New York Radical Feminists, along with Anne Koedt. Rita Mae Brown was briefly a member during 1970; the group was active in New York City, where most of the group's members resided, also in Gainesville, Florida. A group called Redstockings West was started in San Francisco in 1969, but was independent of the East Coast group. Redstockings went through several phases of inactivity. One of the group's earliest actions was on February 13, 1969, when members stormed a hearing of the New York State Joint Legislative Committee on Public Health, considering abortion law reform.

They objected to the hearing, asking: "Why are 14 men and only one woman on your list of speakers—and she's a nun?" The committee chairman countered that these were the experts on the subject, which further enraged the Redstockings women, whose position was that there were no better experts on abortion than women, that abortion law needed to be repealed rather than reformed. About a month Redstockings soon held its own "hearing", an open meeting in the Washington Square Methodist Church, where twelve women testified about their experiences with illegal abortion; the March "speakout" was Redstockings' opportunity to hear testimony of those they felt were the experts: "We are the true experts, the only experts, we who have had abortions," one of the twelve said. One of the women in attendance was Gloria Steinem, who would years identify the meeting as a milestone in her feminist activism. In the early 1970s, Redstockings were noted for their "speakouts" and Zap and street theater on the issue of abortion rights.

On March 3, 1989, Redstockings met again at the Washington Square Methodist Church to commemorate the 20th anniversary of their 1969 meeting, at a speakout called "Abortion: Women Tell it Like it Is, Ought to Be...1969-1989." More the group leads a project to make available, through the Women's Liberation Archives for Action, radical feminist papers and original source organizing material building on their concept "History for Activist Use". In 2001, they released a book called Confronting the Myth of America: Women's Liberation and National Health Care; as of 2006, the group is active and operates a website, though Sarachild is the only original member still active with the group. The group is a strong advocate of consciousness raising and what they refer to as "The Pro-Woman Line" – the idea that women's submission to male supremacy was a conscious adaptation to their lack of power under patriarchy, rather than internalized "brainwashing" on the part of women, as was held by some other radical feminist groups.

Consciousness raising was the act by which the theory of "the personal is the political" met practice, was more essential to Redstockings' feminism than organizational membership. Redstockings holds the view that all men oppress all women as a class and that it is the responsibility of individual men to give up male supremacy, rather than the responsibility of women to change themselves. Redstockings' relationship to other strands of feminism of the 1970s was complex. Like many other radical feminists, they were critical of liberal feminist groups like the National Organization for Women, whom they viewed as advancing women's liberation only as a type of institutional reform while ignoring the interpersonal power of men over women; the Redstockings were more influenced by Marxism than other radical feminist groups. However, they rejected socialist feminism as subordinating the issue of women's liberation to class struggle. On the other hand, Redstockings were against cultural feminism, which in their view substituted the building of a separatist women's culture for political engagement.

Brooke Williams was a member of the group. Redstockings were opposed to lesbian separatism, seeing interpersonal relationships with men as an important arena of feminist struggle, hence seeing separatism as escapist. Redstockings were opposed to male homosexuality, which they saw as a misogynist rejection of women. Redstockings' line on gay men and lesbians is criticized as homophobic. Notable essays associated with the group include "The Redstockings Manifesto" and "Program for Consciousness-Raising", as well as "The Politics of Housework" by Pat Mainardi. "The Redstockings Manifesto" and "The Politics of Housework" were inclu

Wisconsin Senate, District 7

The 7th District of the Wisconsin Senate is located in Southeastern Wisconsin, is composed of parts of Milwaukee County. Chris Larson is the senator serving the 7th district, he was elected in 2010. The area of the 7th Senate District contains three State Assembly Districts: The 19th The 20th The 21st The district is located within Wisconsin's 4th congressional district, represented by U. S. Representative Gwen Moore. A list of all previous senators from this district: Note: the boundaries of districts have changed over history. Previous politicians of a specific numbered district have represented a different geographic area, due to redistricting. Political subdivisions of Wisconsin Chris Larson official campaign site

2008 Russian Professional Rugby League season

The 2008 Russian Professional Rugby League Season was the fourth season of the new Russian Professional Rugby League. The season saw a huge expansion of the competition with the entrance of 14 teams into the preliminary group stage of the championship. VVA Monino were the eventual champions; this season saw the introduction of an initial group stage whereby the championship was divided into two Divisions, the Eastern Division consisting of six teams, the Western Division consisting of eight, with the West Division further subdivided into two groups of four. At the end of the group stage the top six sides formed a super-group that featured a home and away stage between these six sides; the play-offs again featured a best of three semi final stage with the top side playing against the four side, whilst two versed three. The final itself was a best of three if required, though the surprise finalists Slava Moscow were defeated by VVA Monino 2-0. Round One Round Two Round 3 Round One Round Two Round One Round Two

Louis Edemann

Louis Edemann was an American sound editor. He won the Academy Award for Best Sound Editing during the 61st Academy Awards, he won for the film Who Framed Roger Rabbit, he shared his Oscar with Charles L. Campbell; the Hunchback of Notre Dame II The Tigger Movie Doug's First Movie Amistad The Rock The Client The River Wild Falling Down In the Name of the Father Schindler's List The Three Musketeers A Few Good Men Freejack Hoffa The Mighty Ducks An American Tail: Fievel Goes West City Slickers Hook The Rocketeer Back to the Future Part III DuckTales the Movie: Treasure of the Lost Lamp Flatliners Misery The Rescuers Down Under Young Guns II Back to the Future Part II The Little Mermaid When Harry Met Sally... Who Framed Roger Rabbit Empire of the Sun The Lost Boys The Running Man An American Tail The Money Pit Three Amigos Romancing the Stone Flashdance E. T. the Extra-Terrestrial Louis Edemann on IMDb