Sine Rule with Derivation, Example and Implementation

Given angles(in degrees) A, C, and the side c, corresponding to the figure below, the task is to find the remaining two sides a and b.

Examples:

Input: A = 45, C = 35, c = 23
Output:
28.35 
39.49
Explanation:
a is 28.35 and b is 39.49

Input: A = 45, C = 45, c = 10
Output:
10
14.14

Approach: The idea is to use Sine rule. It states that the sides of any triangle are proportional to the sine of the angles opposite to them. a / Sin(A) = b / Sin(B) = c / Sin(C). The derivation is described below:

As is evident from the figure above:

A perpendicular of length h has been drawn on BC from A. From General trigonometric rules:

SinB=h/c——–(1)

SinC=h/b——–(2)

From the above two equations, we get:

c x SinB=b x SinC

Or b/SinB=c/SinC—–(3)

Similarly, if a perpendicular is drawn from B to AC, we can get:

a/SinA=c/SinC——-(4)

From Equations (3) and (4), we get:

a/SinA=b/SinB=c/SinC  

Follow the steps below to solve the problem:

• Change the angles A and C from degrees to radians to be able to be used in the inbuilt functions.
• Calculate the angle B using the observation that sums of angles of a triangle sums up to 180 degrees.
• Use the Sine rule to calculate the sides a and b.

Below is the implementation of the above approach:

C++14

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to calculate remaining two sides
void findSides(double A, double C, double c)
{
    // Calculate angle B
    double B = 180 - (A + C);

    // Convert angles to their respective radians for
    // using trigonometric functions
    A = A * (3.14159 / 180);
    C = C * (3.14159 / 180);
    B = B * (3.14159 / 180);

    // Sine rule
    double a = (c / sin(C)) * sin(A);
    double b = (c / sin(C)) * sin(B);

    // Precision of 2 decimal spaces
    cout << fixed << setprecision(2);

    // Print the answer
    cout << a << endl;
    cout << b << endl;
}

// Driver Code
int main()
{
    // Input
    double A = 45.0;
    double C = 35.0;
    double c = 23;

    // Function Call
    findSides(A, C, c);
    return 0;
}

Java

// Java program for the above approach
class GFG{

// Function to calculate remaining two sides
static void findSides(double A, double C,
                      double c)
{

    // Calculate angle B
    double B = 180 - (A + C);

    // Convert angles to their respective
    // radians for using trigonometric functions
    A = A * (3.14159 / 180);
    C = C * (3.14159 / 180);
    B = B * (3.14159 / 180);

    // Sine rule
    double a = (c / Math.sin(C)) * Math.sin(A);
    double b = (c / Math.sin(C)) * Math.sin(B);

    // Print the answer
    System.out.println(String.format("%.2f", a));
    System.out.println(String.format("%.2f", b));
}

// Driver code
public static void main(String[] args)
{

    // Input
    double A = 45.0;
    double C = 35.0;
    double c = 23;

    // Function Call
    findSides(A, C, c);
}
}

// This code is contributed by abhinavjain194

Python3

# Python3 program for the above approach
import math

# Function to calculate remaining two sides
def findSides(A, C, c):

    # Calculate angle B
    B = 180 - (A + C)

    # Convert angles to their respective radians
    # for using trigonometric functions
    A = A * (3.14159 / 180)
    C = C * (3.14159 / 180)
    B = B * (3.14159 / 180)

    # Sine rule
    a = (c / math.sin(C)) * math.sin(A)
    b = (c / math.sin(C)) * math.sin(B)

    # Precision of 2 decimal spaces

    # Print the answer
    print("{0:.2f}".format(a))
    print("{0:.2f}".format(b))

# Driver Code

# Input
A = 45.0
C = 35.0
c = 23

# Function Call
findSides(A, C, c)

# This code is contributed by target_2

C#

// C# program for the above approach
using System;
class GFG{

// Function to calculate remaining two sides
static void findSides(double A, double C,
                      double c)
{

    // Calculate angle B
    double B = 180 - (A + C);

    // Convert angles to their respective
    // radians for using trigonometric functions
    A = A * (3.14159 / 180);
    C = C * (3.14159 / 180);
    B = B * (3.14159 / 180);

    // Sine rule
    double a = (c / Math.Sin(C)) * Math.Sin(A);
    double b = (c / Math.Sin(C)) * Math.Sin(B);

    // Print the answer
    Console.WriteLine("{0:F2}",a);
    Console.WriteLine("{0:F2}",b);
}

// Driver code
public static void Main(String[] args)
{

    // Input
    double A = 45.0;
    double C = 35.0;
    double c = 23;

    // Function Call
    findSides(A, C, c);
}
}

// This code is contributed by shivanisinghss2110

Javascript

<script>
       // JavaScript program for the above approach


       // Function to calculate remaining two sides
       function findSides(A, C, c)
       {

           // Calculate angle B
           let B = 180 - (A + C);

           // Convert angles to their respective radians for
           // using trigonometric functions
           A = A * (3.14159 / 180);
           C = C * (3.14159 / 180);
           B = B * (3.14159 / 180);

           // Sine rule
           let a = (c / Math.sin(C)) * Math.sin(A);
           let b = (c / Math.sin(C)) * Math.sin(B);

           // Precision of 2 decimal spaces

           // Print the answer
           document.write(a.toPrecision(4) + "<br>");
           document.write(b.toPrecision(4) + "<br>");
       }

       // Driver Code

       // Input
       let A = 45.0;
       let C = 35.0;
       let c = 23;

       // Function Call
       findSides(A, C, c);

 // This code is contributed by Potta Lokesh
   </script>

Output

28.35
39.49

Time Complexity: O(1)
Auxiliary Space: O(1)