The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric...

In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for...

are several equivalent ways for defining trigonometric functions, and the proofs of the trigonometric identities between them depend on the chosen definition...

between sine and cosine is sometimes called the fundamental Pythagorean trigonometric identity. In similar triangles, the ratios of the sides are the same...

before the nineteenth century Pythagorean theorem Pythagorean triple Pythagorean prime Pythagorean trigonometric identity Table of Pythagoras, another...

sides, expanding the squared binomial, and then applying the Pythagorean trigonometric identity ⁠ cos 2 ⁡ γ + sin 2 ⁡ γ = 1 {\displaystyle \cos ^{2}\gamma...

}-45^{\circ })=\sin(45^{\circ })} . Substituting into the Pythagorean trigonometric identity sin ⁡ ( 45 ∘ ) 2 + cos ⁡ ( 45 ∘ ) 2 = 1 {\displaystyle \sin(45^{\circ...

{\displaystyle \cosh ^{2}x-\sinh ^{2}x=1} which is analogous to the Pythagorean trigonometric identity. One also has sech 2 ⁡ x = 1 − tanh 2 ⁡ x csch 2 ⁡ x = coth...

formula on both sides with squared hypotenuse resulting in the Pythagorean trigonometric identity, the sum of a squared sine and a squared cosine equals 1:...

cotangent identity Lagrange's trigonometric identities Morrie's law Proofs of trigonometric identities Pythagorean trigonometric identity Tangent half-angle...

widely used in the study of dynamical systems. Angle measure Pythagorean trigonometric identity Riemannian circle Radian Unit disk Unit sphere Unit hyperbola...

_{1})=\operatorname {sinc} (\theta )~I.} The analog of the Pythagorean trigonometric identity holds: sin 2 ⁡ X + cos 2 ⁡ X = I {\displaystyle \sin ^{2}X+\cos...

Papyrus) and Babylonian mathematics. Trigonometry was also prevalent in Kushite mathematics. Systematic study of trigonometric functions began in Hellenistic...

is a generalized Pythagorean theorem for inner-product spaces (which can have an uncountable infinity of basis vectors). The identity asserts that the...

tables of values for trigonometric ratios (also called trigonometric functions) such as sine. Throughout history, trigonometry has been applied in areas...

trigonometric functions has a corresponding inverse function, and an analog among the hyperbolic functions. The oldest definitions of trigonometric functions...

^{2}\theta }}\\[4pt]={}&{-1}\end{aligned}}} Note the use of the Pythagorean trigonometric identity sin 2 ⁡ θ + cos 2 ⁡ θ = 1. {\displaystyle \sin ^{2}\theta...

\right\|^{2}\left(1-\cos ^{2}\theta \right).} Invoking the Pythagorean trigonometric identity one obtains: ‖ a × b ‖ = ‖ a ‖ ‖ b ‖ | sin ⁡ θ | , {\displaystyle...

four-square identity, related to quaternions in the same way Lagrange's identity Other Parseval's identity Pythagorean trigonometric identity acceleration...

Chords were used extensively in the early development of trigonometry. The first known trigonometric table, compiled by Hipparchus in the 2nd century BC,...

The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change...

Nonhypotenuse number Plimpton 322 Pythagorean prime Pythagorean quadruple Quadric Tangent half-angle formula Trigonometric identity Long (1972, p. 48) Robson...

mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential...

substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity. One of the most prominent...

In trigonometry, it is common to use mnemonics to help remember trigonometric identities and the relationships between the various trigonometric functions...

\right\|\left\|\mathbf {B} \right\|}}\quad (-\pi <\theta \leq \pi )} The Pythagorean trigonometric identity then provides: ‖ A × B ‖ 2 + ( A ⋅ B ) 2 = ‖ A ‖ 2 ‖ B ‖ 2...

The following proof is very similar to one given by Raifaizen. By the Pythagorean theorem we have b 2 = h 2 + d 2 {\displaystyle b^{2}=h^{2}+d^{2}} and...

quickly. It used the identities for the trigonometric functions of sums and differences of angles in terms of the products of trigonometric functions of those...

squaring the sine and cosine outputs, adding them together, (Pythagorean trigonometric identity) subtracting a constant, and applying the difference to a...

In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle. The tangent of half...