Hyperbolic law of cosines information

In hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar...

trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles...

trigonometry. Pair of pants (mathematics) Triangle group For hyperbolic trigonometry: Angle of parallelism Hyperbolic law of cosines Hyperbolic law of sines Lambert...

spherical trigonometry, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles, analogous...

x} and y {\displaystyle y} , sine and cosine can be expressed in terms of real sines, cosines, and hyperbolic functions as sin ⁡ z = sin ⁡ x cosh ⁡ y...

angle is called the angular coordinate, or polar angle. From the hyperbolic law of cosines, we get that the distance between two points given in polar coordinates...

his "Geometriae prima elementa" on p. 66, Taurinus defined the hyperbolic law of cosines A = arccos ⁡ cos ⁡ ( α − 1 ) − cos ⁡ ( β − 1 ) cos ⁡ ( γ − 1 )...

where cosh is the hyperbolic cosine. This formula is a special form of the hyperbolic law of cosines that applies to all hyperbolic triangles: cosh ⁡...

The sine and cosine of a complex number z = x + i y {\displaystyle z=x+iy} can be expressed in terms of real sines, cosines, and hyperbolic functions as...

CORDIC (Jack E. Volder), Linear CORDIC, Hyperbolic CORDIC (John Stephen Walther), and Generalized Hyperbolic CORDIC (GH CORDIC) (Yuanyong Luo et al.)...

and 0 ≤ θ i < 2 π {\displaystyle 0\leq \theta _{i}<2\pi } . The hyperbolic law of cosines allows to measure the distance d i j {\displaystyle d_{ij}} between...

and angles in scalene triangles, with the other being the law of cosines. The law of sines can be generalized to higher dimensions on surfaces with constant...

(u)=\cos \Pi (u)} He also related the velocity addition to the hyperbolic law of cosines: ch ⁡ u = ch ⁡ u 1 c ⁡ h u 2 + sh ⁡ u 1 sh ⁡ u 2 cos ⁡ α ch ⁡...

equilateral hyperboloids of one and two sheets, where he provides the hyperbolic law of cosines. In 1900 Alexander published "Hyperbolic Quaternions" with the...

function Laws for solution of triangles: Law of cosines Spherical law of cosines Law of sines Law of tangents Law of cotangents Mollweide's formula List of integrals...

(/haɪˈpɜːrbələ/ ; pl. hyperbolas or hyperbolae /-liː/ ; adj. hyperbolic /ˌhaɪpərˈbɒlɪk/ ) is a type of smooth curve lying in a plane, defined by its geometric...

obtained the hyperbolic law of cosines through use of his Algebra of Physics. H. Jansen made the hyperboloid model the explicit focus of his 1909 paper...

z=\operatorname {cd} (z;i)} . Similarly, the hyperbolic lemniscate sine slh and hyperbolic lemniscate cosine clh have a square period lattice with fundamental...

proportional to the hyperbolic secant function. The hyperbolic secant function is equivalent to the reciprocal hyperbolic cosine, and thus this distribution...

– inverse hyperbolic cosecant function. (Also written as arcsch.) arcosh – inverse hyperbolic cosine function. arcoth – inverse hyperbolic cotangent function...

two proofs of the cosine rule (Articles 37 and 60) and two proofs of the sine rule (Articles 40 and 42). The page on Spherical law of cosines gives four...

by the law of cosines and law of sines (also called the cosine rule and sine rule). The triangle inequality states that the sum of the lengths of any two...

hyperbolic orbits by Johann Lambert in 1761–1777. Another milestone in orbit determination was Carl Friedrich Gauss's assistance in the "recovery" of...

midpoint ( 1 / 2 {\displaystyle 1/2} ) of the logistic function. Parametrically, hyperbolic cosine and hyperbolic sine give coordinates on the unit hyperbola:...

Trigonometric function Asymptotes Circular functions Periodic functions Law of cosines Law of sines Polar sine Amplitude Dot product Norm (mathematics) (also...

half-angle formula Solution of triangles Law of sines Law of cosines Law of tangents Law of cotangents Mollweide's formula Chebyshev polynomials Conway...

inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent...