public void TestArc3()
{
var func = MathS.Arccos(2 * x);
AssertEqEntity(func.Derive(x).Simplify(), (-2) / MathS.Sqrt(1 + (-4) * MathS.Sqr(x)));
}
public void Setup()
{
exprEasy = x + MathS.Sqr(x) - 3;
exprMedium = MathS.Sin(x + MathS.Cos(x)) + MathS.Sqrt(x + MathS.Sqr(x));
exprHard = MathS.Sin(x + MathS.Arcsin(x)) / (MathS.Sqr(x) + MathS.Cos(x)) * MathS.Arccos(x / 1200 + 0.00032 / MathS.Cotan(x + 43));
exprSolvable = MathS.FromString("3arccos(2x + a)3 + 6arccos(2x + a)2 - a3 + 3");
}
[TestMethod] public void ArcTrig() => TestSimplify(@"\arcsin\left(\arccos\left(\arctan\left(\arccot\left(x\right)\right)\right)\right)", MathS.Arcsin(MathS.Arccos(MathS.Arctan(MathS.Arccotan(x)))));
/// <summary>
/// Returns a set of possible roots of a function, e. g.
/// sin(x) = a =>
/// x = arcsin(a) + 2 pi n
/// x = pi - arcsin(a) + 2 pi n
/// </summary>
/// <param name="func"></param>
/// <param name="value"></param>
/// <param name="x"></param>
/// <returns></returns>
public static Set InvertFunctionEntity(FunctionEntity func, Entity value, Entity x)
{
Entity a = func.Children[0];
Entity b = func.Children.Count == 2 ? func.Children[1] : null;
int arg = func.Children.Count == 2 && func.Children[1].FindSubtree(x) != null ? 1 : 0;
var n = Utils.FindNextIndex(func + value, "n");
var res = new Set();
var pi = MathS.pi;
Set GetNotNullEntites(Set set)
{
return(set.FiniteWhere(el => el.entType != Entity.EntType.NUMBER || el.GetValue().IsDefinite()));
}
switch (func.Name)
{
// Consider case when sin(sin(x)) where double-mention of n occures
case "sinf":
{
// sin(x) = value => x = arcsin(value) + 2pi * n
res.AddRange(GetNotNullEntites(FindInvertExpression(a, MathS.Arcsin(value) + 2 * pi * n, x)));
// sin(x) = value => x = pi - arcsin(value) + 2pi * n
res.AddRange(GetNotNullEntites(FindInvertExpression(a, pi - MathS.Arcsin(value) + 2 * pi * n, x)));
return(res);
}
case "cosf":
{
// cos(x) = value => x = arccos(value) + 2pi * n
res.AddRange(GetNotNullEntites(FindInvertExpression(a, MathS.Arccos(value) + 2 * pi * n, x)));
// cos(x) = value => x = -arccos(value) + 2pi * n
res.AddRange(GetNotNullEntites(FindInvertExpression(a, -MathS.Arccos(value) - 2 * pi * n, x)));
return(res);
}
case "tanf":
{
var inverted = FindInvertExpression(a, MathS.Arctan(value) + pi * n, x);
// tan(x) = value => x = arctan(value) + pi * n
res.AddRange(GetNotNullEntites(inverted));
return(res);
}
case "cotanf":
{
var inverted = FindInvertExpression(a, MathS.Arccotan(value) + pi * n, x);
// cotan(x) = value => x = arccotan(value)
res.AddRange(GetNotNullEntites(inverted));
return(res);
}
case "arcsinf":
// arcsin(x) = value => x = sin(value)
if (EntityInBounds(value, ArcsinFrom, ArcsinTo))
{
return(GetNotNullEntites(FindInvertExpression(a, MathS.Sin(value), x)));
}
else
{
return(Empty);
}
case "arccosf":
// arccos(x) = value => x = cos(value)
if (EntityInBounds(value, ArccosFrom, ArccosTo))
{
return(GetNotNullEntites(FindInvertExpression(a, MathS.Cos(value), x)));
}
else
{
return(Empty);
}
case "arctanf":
// arctan(x) = value => x = tan(value)
return(GetNotNullEntites(FindInvertExpression(a, MathS.Tan(value), x)));
case "arccotanf":
// arccotan(x) = value => x = cotan(value)
return(GetNotNullEntites(FindInvertExpression(a, MathS.Cotan(value), x)));
case "logf":
if (arg != 0)
{
// log(x, a) = value => x = a ^ value
return(GetNotNullEntites(FindInvertExpression(b, MathS.Pow(a, value), x)));
}
else
{
// log(a, x) = value => a = x ^ value => x = a ^ (1 / value)
return(GetNotNullEntites(FindInvertExpression(a, MathS.Pow(b, 1 / value), x)));
}
default:
throw new SysException("Unknown function");
}
}
public void TestPatt10()
{
var expr = MathS.Arcsin(x * 3) + MathS.Arccos(x * 1);
Assert.IsTrue(expr.Simplify() == MathS.Arccos(x) + MathS.Arcsin(3 * x));
}
public void TestPatt7()
{
var expr = MathS.Arcsin(x * 3) + MathS.Arccos(x * 3);
Assert.IsTrue(expr.Simplify() == 0.5 * MathS.pi);
}
public void TestArc3()
{
var func = MathS.Arccos(2 * x);
Assert.Equal((-1) / MathS.Sqrt(1 - MathS.Sqr(2 * x)) * 2, func.Differentiate(x).Simplify());
}
public void TestArc3()
{
var func = MathS.Arccos(2 * x);
Assert.IsTrue(func.Derive(x).Simplify() == (-2) * MathS.Pow(1 + (-4) * MathS.Sqr(x), -0.5));
}