In mathematics, one of a set of functions related to the hyperbola in the same way the trigonometric functions relate to the circle. They are the hyperbolic sine, cosine, tangent, secant, cotangent, and cosecant (written “sinh,” “cosh,” etc.). The hyperbolic equivalent of the fundamental trigonometric identity is cosh²math.z − sinh²math.z = 1. The hyperbolic sine and cosine, particularly useful for finding special types of integrals, can be defined in terms of exponential functions: sinhmath.x = (math.e^math.x − math.e^−math.x) ÷ 2 and coshmath.x = (math.e^math.x + math.e^−math.x) ÷ 2

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Non-Euclidean geometry, useful in modeling interstellar space, that rejects the parallel postulate, proposing instead that at least two lines through any point not on a given line are parallel to that line. Though many of its theorems are identical to those of Euclidean geometry, others differ. For example, two parallel lines converge in one direction and diverge in the other, and the angles of a triangle add up to less than 180°.

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