# Trig Functions

Analytica includes a full range of trigonometric functions, including Sin, Cos, Tan, their inverses, ASin, ACos, ATan, and ATan2, and hyperbolic functions, SinH, CosH, TanH, and their inverses, ASinH, ACosH, and ATanH.

Analytica's trigonometric functions operate using *degrees* as parameters (or value returned), not radians.

## Trigonometric Functions

## Functions Radians and Degrees

- Radians(Degrees: atomic numeric)

Converts an angle measure in degrees to the equivalent in radians.

- Degrees(Radians: numeric atomic)

Converts an angle expressed in radians to an equivalent in degrees.

## Functions Cos, Sin and Tan

Basic trigonometric functions. Parameter is in degrees.

`Sin(90) → 1`

`Cos(Degrees(Pi)) → -1`

## Functions ArcCos, ArcSin, ArcTan, ArcTan2

Inverse trig functions. Results are in Degrees.

- ArcCos(X: Numeric atomic)
- ArcSin(X: Numeric atomic)
- ArcTan(X: Numeric atomic)
- ArcTan2(Y, X: Numeric atomic)

The range of the results are as follows:

Function Range (in degrees) ArcCos 0 to 180 ArcSin -90 to 90 ArcTan -90 to 90 ArcTan2 -180 to 180

Note: `ArcTan2(0, 0)`

returns 0.

## Functions CosH, SinH, TanH

Hyperbolic trig functions. The parameter is in degrees.

An xy-graph of `Sin(x)`

vs. `Cos(x)`

plots a circle.
Analoguously, an xy-graph of `SinH(x)`

vs. `CosH(x)`

plots a hyperbola (on the right side of the y-axis):

Although the parameter is specified in degrees, it does not denote an angle to the point on the hyperbola. «x» is referred to as the *hyperbolic angle* and is defined to be the area of the *hyperbolic sector* times [math]\displaystyle{ 360 / \pi }[/math]. Conversely, the area of the hyperbolic sector is [math]\displaystyle{ x \pi / 360 }[/math].

- [math]\displaystyle{ CosH(x) = {{e^{Radians(x)} + e^{-Radians(x)}}\over 2} }[/math]
- [math]\displaystyle{ SinH(x) = {{e^{Radians(x)} - e^{-Radians(x)}}\over 2} }[/math]
- [math]\displaystyle{ TanH(x) = {{e^{Radians(x)} - e^{-Radians(x)}}\over {e^{Radians(x)} + e^{-Radians(x)}}} }[/math]
- [math]\displaystyle{ e^{Radians(x)} = CosH(x) + SinH(x) }[/math]
- [math]\displaystyle{ e^{Radians(x)} = CosH(x) - SinH(x) }[/math]

The hyperbolic functions are also defined for complex numbers «x».

## Functions ArcCosH, ArcSinH, ArcTanH

Inverse hyperbolic trigonometric functions. The returned value is in degrees.

`ArcCosH(x)`

returns the value such that `CosH(ArcCosH(x))=x`

and `ArcCosH(CosH(x))=x`

.

`ArcSinH(x)`

returns the value such that `SinH(ArcSinH(x))=x`

and `ArcSinH(SinH(x))=x`

.

`ArcTanH(x)`

returns the value such that `TanH(ArcTanH(x))=x`

and `ArcTanH(TanH(x))=x`

.

These are equivalent to the following expressions.

`ArcCosH(x) = Degrees( Ln( x + Sqrt( x^2-1 ) ) )`

`ArcSinH(x) = Degrees( Ln( x + Sqrt( x^2+1 ) ) )`

`ArcTanH(x) = Degrees( 0.5 * Ln( (1+x) / (1-x) ) )`

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