coth( z ) Show The hyperbolic cotangent of z in SageMath. Defined by \[ \coth z = \frac{ \cosh z }{ \sinh z } \]Plot on the real axis: Series expansion about the origin: Special values: Related functions: tanh arccoth acoth Function category: sagemath-docs In mathematics, the inverse functions of hyperbolic functions are referred to as inverse hyperbolic functions or area hyperbolic functions. There are six inverse hyperbolic functions, namely, inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant, inverse hyperbolic secant, and inverse hyperbolic cotangent functions. These functions are depicted as sinh-1 x, cosh-1 x, tanh-1 x, csch-1 x, sech-1 x, and coth-1 x. With the help of an inverse hyperbolic function, we can find the hyperbolic angle of the corresponding hyperbolic function. Function name Function Formula Domain RangeInverse hyperbolic sinesinh-1 x ln[x + √(x2 + 1)] (-∞, ∞) (-∞, ∞) Inverse hyperbolic cosinecosh-1x ln[x + √(x2 – 1) [1, ∞) [0, ∞) Inverse hyperbolic tangent tanh-1 x ½ ln[(1 + x)/(1 – x)] (-1,1) (-∞, ∞) Inverse hyperbolic cosecant csch-1 x ln[(1 + √(x2 + 1)/x] (-∞, ∞) (-∞, ∞) Inverse hyperbolic secant sech-1 x ln[(1 + √(1 – x2)/x] (0, 1] [0, ∞) Inverse hyperbolic cotangent coth-1 x ½ ln[(x + 1)/(x – 1)] (-∞, -1) or (1, ∞) (-∞, ∞) Inverse hyperbolic sine Functionsinh-1 x = ln[x + √(x2 + 1)] Proof:
Inverse hyperbolic cosine Functioncosh-1 x = ln[x + √(x2 – 1)] Proof:
Inverse hyperbolic tangent functiontanh-1 x = ½ ln[(1 + x)/(1 – x)] = ½ [ln(1 + x) – ln(1 – x)] Proof:
Inverse hyperbolic cosecant function csch-1 x = ln[(1 + √(x2 + 1)/x] Proof:
Inverse hyperbolic secant function sech-1 x = ln[(1 + √(1 – x2)/x] Proof:
Inverse hyperbolic cotangent function coth-1 x = ½ ln[(x + 1)/(x – 1)] Proof:
Derivates of inverse hyperbolic functions Inverse hyperbolic function Derivative sinh-1x 1/√(x2 + 1) cosh-1 x 1/√(x2 – 1), x>1 tanh-1x 1/(1 – x2), |x| < 1 csch-1 x 1/{|x|√(1 + x2)}, x ≠ 0 sech-1 x -1/[x√(1 – x2)], 0 < x < 1 coth-1 x 1/(1 – x2), |x| > 1 Sample ProblemsProblem 1: If sinh x = 4, then prove that x = loge(4 + √17). Solution:
Problem 2: Prove that tanh-1 (sin x) = cosh-1 (sec x). Solution:
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