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Visual proof of the Pythagorean identity: for any angle , the point (,) = (, ) lies on the unit circle, which satisfies the equation + =.Thus, + =. In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables ...
The triangle shaded blue illustrates the identity + = , and the red triangle shows that + = . The basic relationship between the sine and cosine is given by the Pythagorean identity: sin 2 θ + cos 2 θ = 1 , {\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1,}
The dotted vector, in this case B, is differentiated, while the (undotted) A is held constant. The utility of the Feynman subscript notation lies in its use in the derivation of vector and tensor derivative identities, as in the following example which uses the algebraic identity C⋅(A×B) = (C×A)⋅B:
This geometric argument relies on definitions of arc length and area, which act as assumptions, so it is rather a condition imposed in construction of trigonometric functions than a provable property. [2] For the sine function, we can handle other values. If θ > π /2, then θ > 1. But sin θ ≤ 1 (because of the Pythagorean identity), so sin ...
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
In a more compact vector notation, Lagrange's identity is expressed as: [3] ‖ ‖ ‖ ‖ = < (), where a and b are n-dimensional vectors with components that are real numbers. The extension to complex numbers requires the interpretation of the dot product as an inner product or Hermitian dot product.
The resulting identity is one of the most commonly used in mathematics. Among many uses, it gives a simple proof of the AM–GM inequality in two variables. The proof holds in any commutative ring. Conversely, if this identity holds in a ring R for all pairs of elements a and b, then R is commutative. To see this, apply the distributive law to ...
Heine's identity; Hermite's identity; Lagrange's identity; Lagrange's trigonometric identities; List of logarithmic identities; MacWilliams identity; Matrix determinant lemma; Newton's identity; Parseval's identity; Pfister's sixteen-square identity; Sherman–Morrison formula; Sophie Germain identity; Sun's curious identity; Sylvester's ...