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This equation has the same two solutions as the original one: = and = We can also modify the solution set by squaring both sides, because this will make any negative values in the ranges of the equation positive, causing extraneous solutions.
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.
An element y is called (simply) an inverse of x if xyx = x and y = yxy. Every regular element has at least one inverse: if x = xzx then it is easy to verify that y = zxz is an inverse of x as defined in this section. Another easy to prove fact: if y is an inverse of x then e = xy and f = yx are idempotents, that is ee = e and ff = f.
In a vector space, the additive inverse −v (often called the opposite vector of v) has the same magnitude as v and but the opposite direction. [11] In modular arithmetic, the modular additive inverse of x is the number a such that a + x ≡ 0 (mod n) and always exists. For example, the inverse of 3 modulo 11 is 8, as 3 + 8 ≡ 0 (mod 11). [12]
The positive part and negative part of a function are used to define the Lebesgue integral for a real-valued function. Analogously to this decomposition of a function, one may decompose a signed measure into positive and negative parts — see the Hahn decomposition theorem.
The proofs given in this article use these definitions, and thus apply to non-negative angles not greater than a right angle. For greater and negative angles , see Trigonometric functions . Other definitions, and therefore other proofs are based on the Taylor series of sine and cosine , or on the differential equation f ″ + f = 0 ...
The ones' complement of a binary number is the value obtained by inverting (flipping) all the bits in the binary representation of the number. The name "ones' complement" [1] refers to the fact that such an inverted value, if added to the original, would always produce an "all ones" number (the term "complement" refers to such pairs of mutually additive inverse numbers, here in respect to a ...
The resultant sign from multiplication when both are positive or one is positive and the other is negative can be illustrated so long as one uses the positive factor to give the cardinal value to the implied repeated addition or subtraction operation, or in other words, -5 x 2 = -5 + -5 = -10, or 10 ÷ -2 = 10 - 2 - 2 - 2 - 2 - 2 = 0 (the ...