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Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β.
It is also quite possible for (S, ∗) to have no identity element, [15] such as the case of even integers under the multiplication operation. [3] Another common example is the cross product of vectors, where the absence of an identity element is related to the fact that the direction of any nonzero cross product is always orthogonal to any ...
The plus–minus sign or plus-or-minus sign (±) and the complementary minus-or-plus sign (∓) are symbols with broadly similar multiple meanings. In mathematics , the ± sign generally indicates a choice of exactly two possible values, one of which is obtained through addition and the other through subtraction .
Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric. [1] More generally, these terms may apply to an inequation or inequality; the right-hand side is everything on the right side of a test operator in an expression, with LHS defined similarly.
2. Equivalence class: given an equivalence relation, [] often denotes the equivalence class of the element x. 3. Integral part: if x is a real number, [] often denotes the integral part or truncation of x, that is, the integer obtained by removing all digits after the decimal mark.
If R is a ring other than the zero ring, then 0 is a (two-sided) zero divisor, because any nonzero element x satisfies 0x = 0 = x 0. If R is the zero ring, in which 0 = 1, then 0 is not a zero divisor, because there is no nonzero element that when multiplied by 0 yields 0. Some references include or exclude 0 as a zero divisor in all rings by ...
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The standard form of Schur's is the case of this inequality where x = a, y = b, z = c, k = 1, ƒ(m) = m r. [ 1 ] Another possible extension states that if the non-negative real numbers x ≥ y ≥ z ≥ v {\displaystyle x\geq y\geq z\geq v} with and the positive real number t are such that x + v ≥ y + z then [ 2 ]