Dadas las coordenadas de los tres vértices del triángulo en el plano 2D, la tarea es encontrar los tres ángulos.
Ejemplo:
Input : A = (0, 0),
B = (0, 1),
C = (1, 0)
Output : 90, 45, 45
Para resolver este problema usamos a continuación la Ley de los cosenos .
c^2 = a^2 + b^2 - 2(a)(b)(cos beta)
Después de reorganizar
beta = acos( ( a^2 + b^2 - c^2 ) / (2ab) )
En trigonometría, la ley de los cosenos (también conocida como fórmula del coseno o regla del coseno) relaciona las longitudes de los lados de un triángulo con el coseno de uno de sus ángulos.
First, calculate the length of all the sides. Then apply above formula to get all angles in radian. Then convert angles from radian into degrees.
A continuación se muestra la implementación de los pasos anteriores.
C++
// Code to find all three angles
// of a triangle given coordinate
// of all three vertices
#include <iostream>
#include <utility> // for pair
#include <cmath> // for math functions
using namespace std;
#define PI 3.1415926535
// returns square of distance b/w two points
int lengthSquare(pair<int,int> X, pair<int,int> Y)
{
int xDiff = X.first - Y.first;
int yDiff = X.second - Y.second;
return xDiff*xDiff + yDiff*yDiff;
}
void printAngle(pair<int,int> A, pair<int,int> B,
pair<int,int> C)
{
// Square of lengths be a2, b2, c2
int a2 = lengthSquare(B,C);
int b2 = lengthSquare(A,C);
int c2 = lengthSquare(A,B);
// length of sides be a, b, c
float a = sqrt(a2);
float b = sqrt(b2);
float c = sqrt(c2);
// From Cosine law
float alpha = acos((b2 + c2 - a2)/(2*b*c));
float beta = acos((a2 + c2 - b2)/(2*a*c));
float gamma = acos((a2 + b2 - c2)/(2*a*b));
// Converting to degree
alpha = alpha * 180 / PI;
beta = beta * 180 / PI;
gamma = gamma * 180 / PI;
// printing all the angles
cout << "alpha : " << alpha << endl;
cout << "beta : " << beta << endl;
cout << "gamma : " << gamma << endl;
}
// Driver code
int main()
{
pair<int,int> A = make_pair(0,0);
pair<int,int> B = make_pair(0,1);
pair<int,int> C = make_pair(1,0);
printAngle(A,B,C);
return 0;
}
Java
// Java Code to find all three angles
// of a triangle given coordinate
// of all three vertices
import java.awt.Point;
import static java.lang.Math.PI;
import static java.lang.Math.sqrt;
import static java.lang.Math.acos;
class Test
{
// returns square of distance b/w two points
static int lengthSquare(Point p1, Point p2)
{
int xDiff = p1.x- p2.x;
int yDiff = p1.y- p2.y;
return xDiff*xDiff + yDiff*yDiff;
}
static void printAngle(Point A, Point B,
Point C)
{
// Square of lengths be a2, b2, c2
int a2 = lengthSquare(B,C);
int b2 = lengthSquare(A,C);
int c2 = lengthSquare(A,B);
// length of sides be a, b, c
float a = (float)sqrt(a2);
float b = (float)sqrt(b2);
float c = (float)sqrt(c2);
// From Cosine law
float alpha = (float) acos((b2 + c2 - a2)/(2*b*c));
float betta = (float) acos((a2 + c2 - b2)/(2*a*c));
float gamma = (float) acos((a2 + b2 - c2)/(2*a*b));
// Converting to degree
alpha = (float) (alpha * 180 / PI);
betta = (float) (betta * 180 / PI);
gamma = (float) (gamma * 180 / PI);
// printing all the angles
System.out.println("alpha : " + alpha);
System.out.println("betta : " + betta);
System.out.println("gamma : " + gamma);
}
// Driver method
public static void main(String[] args)
{
Point A = new Point(0,0);
Point B = new Point(0,1);
Point C = new Point(1,0);
printAngle(A,B,C);
}
}
Python3
# Python3 code to find all three angles
# of a triangle given coordinate
# of all three vertices
import math
# returns square of distance b/w two points
def lengthSquare(X, Y):
xDiff = X[0] - Y[0]
yDiff = X[1] - Y[1]
return xDiff * xDiff + yDiff * yDiff
def printAngle(A, B, C):
# Square of lengths be a2, b2, c2
a2 = lengthSquare(B, C)
b2 = lengthSquare(A, C)
c2 = lengthSquare(A, B)
# length of sides be a, b, c
a = math.sqrt(a2);
b = math.sqrt(b2);
c = math.sqrt(c2);
# From Cosine law
alpha = math.acos((b2 + c2 - a2) /
(2 * b * c));
betta = math.acos((a2 + c2 - b2) /
(2 * a * c));
gamma = math.acos((a2 + b2 - c2) /
(2 * a * b));
# Converting to degree
alpha = alpha * 180 / math.pi;
betta = betta * 180 / math.pi;
gamma = gamma * 180 / math.pi;
# printing all the angles
print("alpha : %f" %(alpha))
print("betta : %f" %(betta))
print("gamma : %f" %(gamma))
# Driver code
A = (0, 0)
B = (0, 1)
C = (1, 0)
printAngle(A, B, C);
# This code is contributed
# by ApurvaRaj
C#
// C# Code to find all three angles
// of a triangle given coordinate
// of all three vertices
using System;
class GFG
{
class Point
{
public int x, y;
public Point(int x, int y)
{
this.x = x;
this.y = y;
}
}
// returns square of distance b/w two points
static int lengthSquare(Point p1, Point p2)
{
int xDiff = p1.x - p2.x;
int yDiff = p1.y - p2.y;
return xDiff * xDiff + yDiff * yDiff;
}
static void printAngle(Point A, Point B, Point C)
{
// Square of lengths be a2, b2, c2
int a2 = lengthSquare(B, C);
int b2 = lengthSquare(A, C);
int c2 = lengthSquare(A, B);
// length of sides be a, b, c
float a = (float)Math.Sqrt(a2);
float b = (float)Math.Sqrt(b2);
float c = (float)Math.Sqrt(c2);
// From Cosine law
float alpha = (float) Math.Acos((b2 + c2 - a2) /
(2 * b * c));
float betta = (float) Math.Acos((a2 + c2 - b2) /
(2 * a * c));
float gamma = (float) Math.Acos((a2 + b2 - c2) /
(2 * a * b));
// Converting to degree
alpha = (float) (alpha * 180 / Math.PI);
betta = (float) (betta * 180 / Math.PI);
gamma = (float) (gamma * 180 / Math.PI);
// printing all the angles
Console.WriteLine("alpha : " + alpha);
Console.WriteLine("betta : " + betta);
Console.WriteLine("gamma : " + gamma);
}
// Driver Code
public static void Main(String[] args)
{
Point A = new Point(0, 0);
Point B = new Point(0, 1);
Point C = new Point(1, 0);
printAngle(A, B, C);
}
}
// This code is contributed by Rajput-Ji
Javascript
// JavaScript program
// Code to find all three angles
// of a triangle given coordinate
// of all three vertices
// returns square of distance b/w two points
function lengthSquare(X, Y){
let xDiff = X[0] - Y[0];
let yDiff = X[1] - Y[1];
return xDiff*xDiff + yDiff*yDiff;
}
function printAngle(A, B, C){
// Square of lengths be a2, b2, c2
let a2 = lengthSquare(B,C);
let b2 = lengthSquare(A,C);
let c2 = lengthSquare(A,B);
// length of sides be a, b, c
let a = Math.sqrt(a2);
let b = Math.sqrt(b2);
let c = Math.sqrt(c2);
// From Cosine law
let alpha = Math.acos((b2 + c2 - a2)/(2*b*c));
let beta = Math.acos((a2 + c2 - b2)/(2*a*c));
let gamma = Math.acos((a2 + b2 - c2)/(2*a*b));
// Converting to degree
alpha = alpha * 180 / Math.PI;
beta = beta * 180 / Math.PI;
gamma = gamma * 180 / Math.PI;
// printing all the angles
console.log("alpha : ", alpha);
console.log("beta : ", beta);
console.log("gamma : ", gamma);
}
// Driver code
let A = [0, 0];
let B = [0, 1];
let C = [1, 0];
printAngle(A,B,C);
// The code is contributed by Gautam goel (guatamgoel962)
Producción:
alpha : 90 beta : 45 gamma : 45
Complejidad de tiempo: O(log(n)) desde el uso de funciones sqrt incorporadas
Espacio Auxiliar: O(1)
Referencia :
https://en.wikipedia.org/wiki/Law_of_cosines
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Artículo escrito por GeeksforGeeks-1 y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA