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The tangent half-angle substitution relates an angle to the slope of a line. Introducing a new variable = , sines and cosines can be expressed as rational functions of , and can be expressed as the product of and a rational function of , as follows: = +, = +, = +.
A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as many of the rules are redundant, and can be proven with the other rules.
In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables. The intent is that when expressed in new variables, the problem may become simpler, or equivalent to a better understood problem.
In Boolean algebra, a formula is in conjunctive normal form (CNF) or clausal normal form if it is a conjunction of one or more clauses, where a clause is a disjunction of literals; otherwise put, it is a product of sums or an AND of ORs.
The formula for an integration by parts is () ′ = [() ()] ′ (). Beside the boundary conditions , we notice that the first integral contains two multiplied functions, one which is integrated in the final integral ( g ′ {\displaystyle g'} becomes g {\displaystyle g} ) and one which is differentiated ( f {\displaystyle f} becomes f ...
In logic, a rule of replacement [1] [2] [3] is a transformation rule that may be applied to only a particular segment of an expression. A logical system may be constructed so that it uses either axioms, rules of inference, or both as transformation rules for logical expressions in the system. Whereas a rule of inference is always applied to a ...
[6] Considering mathematics as a formal language, a variable is a symbol from an alphabet, usually a letter like x, y, and z, which denotes a range of possible values. [7] If a variable is free in a given expression or formula, then it can be replaced with any of the values in its range. [8] Certain kinds of bound variables can be substituted too.
Transformation into negation normal form can increase the size of a formula only linearly: the number of occurrences of atomic formulas remains the same, the total number of occurrences of and is unchanged, and the number of occurrences of in the normal form is bounded by the length of the original formula.