# What is sinh x in C.

## Hyperbolic functions - sinh (x), cosh (x) and tanh (x)

### What are hyperbolic functions?

The **Hyperbolic functions** (also *hyperbolic functions* called) **Hyperbolic sine**, **Hyperbolic cosine** and **Hyperbolic tangent** are functions whose name suggests a close relationship with the **Trigonometric functions** Let sine, cosine and tangent close. As we already know, you can use trigonometric functions **(Sine, cosine and tangent) on the unit circle**, which can be described geometrically with $ x ^ 2 + y ^ 2 = 1 $ for $ x, y ~ \ in ~ \ mathbb {R} $.

For the hyperbolic functions we now start with the **Unit hyperbole**, so $ x ^ 2 - y ^ 2 = 1 $ with $ x, y ~ \ in ~ \ mathbb {R} $, from. Here, too, an angle $ \ alpha $ is plotted from the $ x $ axis so that a straight line through the origin is created and we can construct a right-angled triangle. The hyperbolic sine is then the $ x $ coordinate of the intersection of this straight line through the origin with the unit hyperbolic and the hyperbolic cosine of $ \ alpha $ is the corresponding $ y $ coordinate.

The hyperbolic functions are via the **$ e $ function** Are defined. Thus for any $ x ~ \ in ~ \ mathbb {R} $:

**Hyperbolic sine:** $ \ sinh (x) = \ dfrac {e ^ {x} - e ^ {- x}} {2} $

**Hyperbolic cosine:** $ \ cosh (x) = \ dfrac {e ^ {x} + e ^ {- x}} {2} $

**Hyperbolic tangent:** $ \ tanh (x) = \ dfrac {\ sinh (x)} {\ cosh (x)} = \ dfrac {e ^ {x} - e ^ {- x}} {e ^ {x} + e ^ { -x}} $

We can also transfer the relationship between sine and cosine on the unit circle $ (\ text {sine}) ^ 2 + (\ text {cosine}) ^ 2 = 1 $ to the unit hyperbole and obtain it:

$ (\ text {Hyperbolic sine}) ^ 2 - (\ text {Hyperbolic cosine}) ^ 2 = 1 $

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