Hyperbolic Functions Calculator – sinh, cosh

Hyperbolic functions are built from exponentials rather than circles. At x = 1: eˣ ≈ 2.71828 and e⁻ˣ ≈ 0.36788, so sinh(1) = (2.71828−0.36788)/2 ≈ 1.17520, cosh(1) = (2.71828+0.36788)/2 ≈ 1.54308, and tanh(1) = 1.17520/1.54308 ≈ 0.76159. The calculator evaluates all three (plus reciprocals csch, sech, coth) from the exponential pair, which it prints so the arithmetic is checkable. The governing identity is cosh²x − sinh²x = 1 — 2.381 − 1.381 = 1 ✓ — the hyperbolic cousin of Pythagoras, reflecting the geometry of hyperbolas x²−y² = 1 instead of circles. Key values anchor the family: cosh 0 = 1, sinh 0 = 0, tanh 0 = 0; tanh saturates toward ±1 (tanh 3 ≈ 0.995), which is why neural networks use it as a squashing activation. Catenary cables hang along cosh curves.

1. 1

   Enter x

   Any real number

2. 2

   Compute the exponential pair

   eˣ and e⁻ˣ are displayed first

3. 3

   Form the three functions

   Half the difference, half the sum, and their quotient

4. 4

   Verify the identity

   cosh² − sinh² recomputes to 1 for your input

• 1

  cosh² − sinh² = 1, not the circular sin²+cos² = 1

• 2

  tanh never leaves (−1, 1); tanh 3 ≈ 0.995

• 3

  cosh is even, sinh is odd

• Using circular sine/cosine values for hyperbolic arguments

• Sign slip in sinh: it is half the difference of exponentials

• Expecting tanh to reach exactly 1 — it only approaches

• Forgetting cosh x ≥ 1 for all x, including negatives