Compruebe si existe un punto en el sector del círculo o no.

Tenemos un círculo centrado en el origen (0, 0). Como entrada, se nos proporciona el ángulo inicial del sector circular y el tamaño del sector circular en porcentaje. 

Ejemplos: 

Input :  Radius = 8 
         StartAngle = 0 
         Percentage = 12 
         x = 3 y = 4 
Output : Point (3, 4) exists in the circle 
         sector

Input : Radius = 12 
        Startangle = 45
        Percentage = 25  
        x = 3 y = 4 
Output : Point (3, 4) does not exist in 
         the circle sector

Fuente: wikibooks.org

En esta imagen, el ángulo inicial es de 0 grados, el radio r y suponga que el porcentaje del área coloreada es del 12 %, luego calculamos el ángulo final como 360/porcentaje + ángulo inicial .

Para encontrar si un punto (x, y) existe en un sector circular (centrado en el origen) o no, encontramos las coordenadas polares de ese punto y luego seguimos los siguientes pasos:

1. Convierta x, y en coordenadas polares usando este 
   Ángulo = atan(y/x); Radio = sqrt(x * x + y * y);
2. Entonces el Ángulo debe estar entre el Ángulo Inicial y el Ángulo Final, y el Radio entre 0 y su Radio.

// C++ program to check if a point lies inside a circle
// sector.
#include<bits/stdc++.h>
using namespace std;
 
void checkPoint(int radius, int x, int y, float percent,
                                         float startAngle)
{
    // calculate endAngle
    float endAngle = 360/percent + startAngle;
 
    // Calculate polar co-ordinates
    float polarradius = sqrt(x*x+y*y);
    float Angle = atan(y/x);
 
    // Check whether polarradius is less then radius of circle
    // or not and Angle is between startAngle and endAngle
    // or not
    if (Angle>=startAngle && Angle<=endAngle && polarradius<radius)
        printf("Point (%d, %d) exist in the circle sector\n", x, y);
    else
        printf("Point (%d, %d) does not exist in the circle sector\n",
                                                              x, y);
}
 
// Driver code
int main()
{
    int radius = 8, x = 3, y = 4;
    float percent  = 12, startAngle = 0;
    checkPoint(radius, x, y, percent, startAngle);
    return 0;
}

// Java program to check if
// a point lies inside a circle
// sector.
 
class GFG
{
static void checkPoint(int radius, int x, int y, float percent,
                                         float startAngle)
{
 
    // calculate endAngle
    float endAngle = 360/percent + startAngle;
  
    // Calculate polar co-ordinates
    double polarradius = Math.sqrt(x*x+y*y);
    double Angle = Math.atan(y/x);
  
    // Check whether polarradius is
    // less then radius of circle
    // or not and Angle is between
    // startAngle and endAngle
    // or not
    if (Angle>=startAngle && Angle<=endAngle && polarradius<radius)
        System.out.print("Point"+"("+x+","+y+")"+
        " exist in the circle sector\n");
    else
        System.out.print("Point"+"("+x+","+y+")"+
        " exist in the circle sector\n");
}
 
// Driver Program to test above function
public static void main(String arg[])
{
    int radius = 8, x = 3, y = 4;
    float percent  = 12, startAngle = 0;
    checkPoint(radius, x, y, percent, startAngle);
}
}
 
// This code is contributed
// by Anant Agarwal.

# Python3 program to check if a point
# lies inside a circle sector.
import math
 
def checkPoint(radius, x, y, percent, startAngle):
 
    # calculate endAngle
    endAngle = 360 / percent + startAngle
 
    # Calculate polar co-ordinates
    polarradius = math.sqrt(x * x + y * y)
    Angle = math.atan(y / x)
 
    # Check whether polarradius is less
    # then radius of circle or not and
    # Angle is between startAngle and
    # endAngle or not
    if (Angle >= startAngle and Angle <= endAngle
                        and polarradius < radius):
        print("Point (", x, ",", y, ") "
              "exist in the circle sector")
    else:
        print("Point (", x, ",", y, ") "
              "does not exist in the circle sector")
 
# Driver code
radius, x, y = 8, 3, 4
percent, startAngle = 12, 0
 
checkPoint(radius, x, y, percent, startAngle)
 
# This code is contributed by
# Smitha Dinesh Semwal

// C# program to check if a point lies
// inside a circle sector.
using System.IO;
using System;
 
class GFG {
     
    static void checkPoint(int radius, int x, int y,
                    float percent, float startAngle)
    {
         
        // calculate endAngle
        float endAngle = 360 / percent + startAngle;
     
        // Calculate polar co-ordinates
        float polarradius =
                    (float)Math.Sqrt(x * x + y * y);
                     
        float Angle = (float)Math.Atan(y / x);
     
        // Check whether polarradius is less then
        // radius of circle or not and Angle is
        // between startAngle and endAngle or not
        if (Angle >= startAngle && Angle <= endAngle
                            && polarradius < radius)
            Console.Write("Point ({0}, {1}) exist in "
                         + "the circle sector", x, y);
        else
            Console.Write("Point ({0}, {1}) does not "
                + "exist in the circle sector", x, y);
    }
     
    // Driver code
    public static void Main()
    {
        int radius = 8, x = 3, y = 4;
        float percent = 12, startAngle = 0;
        checkPoint(radius, x, y, percent, startAngle);
    }
}
 
// This code is contributed by Smitha Dinesh Semwal

<script>
 
// Javascript program to check if
// a point lies inside a circle
// sector.
function checkPoint(radius, x, y, percent, startAngle)
{
     
    // Calculate endAngle
    let endAngle = 360 / percent + startAngle;
    
    // Calculate polar co-ordinates
    let polarradius = Math.sqrt(x * x + y * y);
    let Angle = Math.atan(y / x);
    
    // Check whether polarradius is
    // less then radius of circle
    // or not and Angle is between
    // startAngle and endAngle
    // or not
    if (Angle >= startAngle &&
        Angle <= endAngle &&
        polarradius < radius)
        document.write("Point" + "(" + x + 
                       "," + y + ")" +
        " exist in the circle sector\n");
    else
        document.write("Point" + "(" + x +
                       "," + y + ")" +
        " exist in the circle sector\n");
}
      
// Driver code   
let radius = 8, x = 3, y = 4;
let percent  = 12, startAngle = 0;
 
checkPoint(radius, x, y, percent, startAngle);
 
// This code is contributed by splevel62
           
</script>

Producción : 

Point(3, 4) exists in the circle sector

Complejidad temporal:  O(1)
Espacio auxiliar: O(1)

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