Step by Step solution about How to resolve the algorithm Law of cosines - triples step by step in the C# programming language

Overview: This C# code demonstrates how to find triples of integers (a, b, c) that represent the sides of a triangle given an angle (degrees) and a maximum side length. The code uses the Law of Cosines to determine if the triples satisfy the triangle inequality theorem.

Key Functions:

• Main2(): The entry point of the program. It invokes other functions to print examples of triangle triples.
• PrintTriples(): Prints the triples that satisfy the given criteria. It includes optional parameters for specifying the range of side lengths and whether the sides should be distinct.
• FindTriples(): Generates triples of integers that meet the Law of Cosines and have a maximum side length. It uses a formula derived from the Law of Cosines to calculate the third side (c) and filters out triples where c is not an integer.
• NotAllTheSameLength(): A lambda function that checks if all sides of a triple are not equal. This is used for filtering triples when the notAllTheSameLength parameter is set to true.

Key Concepts:

• Law of Cosines: A mathematical formula that relates the side lengths and angles of a triangle.
• Enumerables (from, Range): Language features used to generate sequences of integers.
• Linq (Where): Language feature for filtering sequences of values.
• Tuple: A data type that represents a small collection of values.

Usage:

1. Set the angle and maximum side length in the PrintTriples() calls.
2. Specify any additional criteria, such as not allowing sides of equal length, by setting the corresponding parameters.
3. Run the program to generate and display the triangle triples.

Example Output:

60 degree triangles in range 1..13
6 13 15
7 13 14
8 13 13
9 13 12
10 13 11
11 13 10
5 solutions

90 degree triangles in range 1..13
5 12 13
6 8 10
6 9 11
7 7 10
7 8 9
8 6 10
8 7 9
9 6 11
9 7 10
10 5 11
10 6 9
10 7 8
12 5 13
12 6 8
13 5 12
15 solutions

120 degree triangles in range 1..13
7 9 12
8 8 12
9 7 12
10 6 12
11 5 12
12 4 12
13 3 12
7 solutions

60 degree triangles in range 1..10000 where not all sides are the same
6 13 15
7 13 14
8 13 13
9 13 12
10 13 11
4 456 456
5 456 457
6 456 458
7 456 459
8 456 460
9 456 461
29 1980 1980
30 1980 1981
31 1980 1982
32 1980 1983
33 1980 1984
34 1980 1985
40 1631 1900
41 1631 1901
42 1631 1902
43 1631 1903
44 1631 1904
45 1631 1905
192 688 959
...
1087 solutions

using System;
using System.Collections.Generic;
using static System.Linq.Enumerable;

public static class LawOfCosinesTriples
{
    public static void Main2() {
        PrintTriples(60, 13);
        PrintTriples(90, 13);
        PrintTriples(120, 13);
        PrintTriples(60, 10_000, true, false);
    }

    private static void PrintTriples(int degrees, int maxSideLength, bool notAllTheSameLength = false, bool print = true) {
        string s = $"{degrees} degree triangles in range 1..{maxSideLength}";
        if (notAllTheSameLength) s += " where not all sides are the same";
        Console.WriteLine(s);
        int count = 0;
        var triples = FindTriples(degrees, maxSideLength);
        if (notAllTheSameLength) triples = triples.Where(NotAllTheSameLength);
        foreach (var triple in triples) {
            count++;
            if (print) Console.WriteLine(triple);
        }
        Console.WriteLine($"{count} solutions");
    }

    private static IEnumerable<(int a, int b, int c)> FindTriples(int degrees, int maxSideLength) {
        double radians = degrees * Math.PI / 180;
        int coefficient = (int)Math.Round(Math.Cos(radians) * -2, MidpointRounding.AwayFromZero);
        int maxSideLengthSquared = maxSideLength * maxSideLength;
        return
            from a in Range(1, maxSideLength)
            from b in Range(1, a)
            let cc = a * a + b * b + a * b * coefficient
            where cc <= maxSideLengthSquared
            let c = (int)Math.Sqrt(cc)
            where c * c == cc
            select (a, b, c);
    }

    private static bool NotAllTheSameLength((int a, int b, int c) triple) => triple.a != triple.b || triple.a != triple.c;
}