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Product-to-sum and sum-to-product identities Proof of the sum-and-difference-to-product cosine identity for prosthaphaeresis calculations using an isosceles triangle. The product-to-sum identities or prosthaphaeresis formulae can be proven by expanding their right-hand sides using the angle addition theorems.
Proofs of trigonometric identities. There are several equivalent ways for defining trigonometric functions, and the proofs of the trigonometric identities between them depend on the chosen definition. The oldest and most elementary definitions are based on the geometry of right triangles. The proofs given in this article use these definitions ...
Sum and difference formulas. The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to Ptolemy. One can also produce them algebraically using ...
In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus. For example, many asymptotic expansions are derived from the ...
t. e. In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
The derivative of the function at a point is the slope of the line tangent to the curve at the point. Slope of the constant function is zero, because the tangent line to the constant function is horizontal and it's angle is zero. In other words, the value of the constant function, y, will not change as the value of x increases or decreases.
A finite difference is a mathematical expression of the form f (x + b) − f (x + a).If a finite difference is divided by b − a, one gets a difference quotient.The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems.
The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic functions that does not involve complex numbers. The graph of the function a cosh(x/a) is the catenary, the curve formed by a uniform flexible chain, hanging freely between two fixed points under uniform gravity.
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