# 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 \$