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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.
The natural product is associative and commutative and distributes over the natural sum. The natural product is always greater or equal to the usual product, but it may be strictly greater. For example, the natural product of ω and 2 is ω · 2 (the usual product), but this is also the natural product of 2 and ω. Under natural addition, the ...
The nimber square of a Fermat 2-power x is equal to 3x/2 as evaluated under the ordinary multiplication of natural numbers. The smallest algebraically closed field of nimbers is the set of nimbers less than the ordinal ω ω ω , where ω is the smallest infinite ordinal.
The sign of the square root needs to be chosen properly—note that if 2 π is added to θ, the quantities inside the square roots are unchanged, but the left-hand-sides of the equations change sign. Therefore, the correct sign to use depends on the value of θ.
In trigonometry, Mollweide's formula is a pair of relationships between sides and angles in a triangle. [1] [2]A variant in more geometrical style was first published by Isaac Newton in 1707 and then by Friedrich Wilhelm von Oppel [] in 1746.
the set of natural numbers, irrespective of including or excluding zero, the set of all integers, any infinite subset of the integers, such as the set of all square numbers or the set of all prime numbers, the set of all rational numbers, the set of all constructible numbers (in the geometric sense), the set of all algebraic numbers,
A set that has the same cardinality as the set of natural numbers, meaning its elements can be listed in a sequence without end. cov(I) covering number The covering number cov(I) of an ideal I of subsets of X is the smallest number of sets in I whose union is X. critical 1.
For the avoidance of ambiguity, zero will always be a valid possible constituent of "sums of two squares", so for example every square of an integer is trivially expressible as the sum of two squares by setting one of them to be zero. 1. The product of two numbers, each of which is a sum of two squares, is itself a sum of two squares.