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Using the usual notations for a triangle (see the figure at the upper right), where a, b, c are the lengths of the three sides, A, B, C are the vertices opposite those three respective sides, α, β, γ are the corresponding angles at those vertices, s is the semiperimeter, that is, s = a + b + c / 2 , and r is the radius of the inscribed circle, the law of cotangents states that
The torsion-free spin connection is defined by d σ a + ω a b ∧ σ c η b c = 0 {\displaystyle d\sigma ^{a}+\omega ^{ab}\wedge \sigma ^{c}\eta _{bc}=0} The contorsion tensor gives the difference between a connection with torsion, and a corresponding connection without torsion.
More formulas of this nature can be given, as explained by Ramanujan's theory of elliptic functions to alternative bases. Perhaps the most notable hypergeometric inversions are the following two examples, involving the Ramanujan tau function τ {\displaystyle \tau } and the Fourier coefficients j {\displaystyle \mathrm {j} } of the J-invariant ...
In mathematics, the inverse trigonometric functions (occasionally also called antitrigonometric, [1] cyclometric, [2] or arcus functions [3]) are the inverse functions of the trigonometric functions, under suitably restricted domains.
The above formula is known as the shoelace formula or the surveyor's formula. If we locate the vertices in the complex plane and denote them in counterclockwise sequence as a = x A + y A i , b = x B + y B i , and c = x C + y C i , and denote their complex conjugates as a ¯ {\displaystyle {\bar {a}}} , b ¯ {\displaystyle {\bar {b}}} , and c ...
PHOTO: One person was injured when a holiday drone show in Orlando on Dec. 21, 2024, went haywire and several of the unmanned aerial devices crashed into each other and plummeted to the ground ...
A top Federal Reserve official said Monday that he is leaning toward supporting an interest rate cut when the Fed meets in two weeks but that evidence of persistent inflation before then could ...
The optimal speed if both energy and distance traveled in a given time are taken into account (with some "price" for each) may be faster or slower than the speed giving the lowest COT. Previously it was thought that a running person (unlike running animals) uses the same energy whether they run a distance slowly or fast.