Difference between revisions of "Trig Functions"

Latest revision as of 16:19, 18 July 2018

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.

Important

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) ) )

@@ Line 2: / Line 2: @@
 [[Category:Doc Status C]] <!-- For Lumina use, do not change -->
-== Trigonometric 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.
 <tip title="Important">
-Analytica's trigonometric functions operate using ''degrees'', not radians.
+Analytica's trigonometric functions operate using ''degrees'' as parameters (or value returned), not radians.
 </tip>
+== Trigonometric Functions ==
-== Function Radians ==
+== Functions Radians and Degrees ==
 :[[Radians]](Degrees: atomic numeric)
+Converts an angle measure in degrees to the equivalent in radians.
-Converts an angle measure expressed in degrees to the equivalent measure expressed in radians.
-== Function Degrees ==
 :[[Degrees]](Radians: numeric atomic)
-Converts an angle measure expressed in radians to an equivalent degree measure.
+Converts an angle expressed in radians to an equivalent in degrees.
 == Functions Cos, Sin and Tan ==
@@ Line 24: / Line 24: @@
 == Functions ArcCos, ArcSin, ArcTan, ArcTan2 ==
+Inverse trig functions.  Results are in Degrees.
 :[[ArcCos]](X: Numeric atomic)
 :[[ArcSin]](X: Numeric atomic)
@@ Line 29: / Line 31: @@
 :[[ArcTan2]](Y, X: Numeric atomic)
-Inverse trig functions.  Results are in Degrees.
 The range of the results are as follows:
@@ Line 59: / Line 61: @@
 :<math>SinH(x) = {{e^{Radians(x)} - e^{-Radians(x)}}\over 2}</math>
 :<math>TanH(x) = {{e^{Radians(x)} - e^{-Radians(x)}}\over {e^{Radians(x)} + e^{-Radians(x)}}}</math>
-:<math>e^{Radians(x)} = CosH(x) + SinH(x)</code>
+:<math>e^{Radians(x)} = CosH(x) + SinH(x)</math>
-:<math>e^{Radians(x)} = CosH(x) - SinH(x)</code>
+:<math>e^{Radians(x)} = CosH(x) - SinH(x)</math>
-The hyperbolic functions are also defined for complex numbers «x».
+The hyperbolic functions are also defined for [[Complex Numbers|complex numbers]] «x».
 == Functions ArcCosH, ArcSinH, ArcTanH ==
 Inverse hyperbolic trigonometric functions.  The returned value is in degrees.
+<code>ArcCosH(x)</code> returns the value such that <code>CosH(ArcCosH(x))=x</code> and <code>ArcCosH(CosH(x))=x</code>.
+<code>ArcSinH(x)</code> returns the value such that <code>SinH(ArcSinH(x))=x</code> and <code>ArcSinH(SinH(x))=x</code>.
+<code>ArcTanH(x)</code> returns the value such that <code>TanH(ArcTanH(x))=x</code> and <code>ArcTanH(TanH(x))=x</code>.
+These are equivalent to the following expressions.
+:<code>ArcCosH(x) = [[Degrees]]( [[Ln]]( x + [[Sqrt]]( x^2-1 ) ) )</code>
+:<code>ArcSinH(x) = [[Degrees]]( [[Ln]]( x + [[Sqrt]]( x^2+1 ) ) )</code>
+:<code>ArcTanH(x) = [[Degrees]]( 0.5 * [[Ln]]( (1+x) / (1-x) ) )</code>
 ==See Also==