# 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 = 23Output:

28.35

39.49Explanation:

a is 28.35 and b is 39.49

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

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

BCfromA. From General trigonometric rules:

SinB=h/c——–(1)

SinC=h/b——–(2)From the above two equations, we get:

c x SinB=b x SinCOr

b/SinB=c/SinC—–(3)Similarly, if a perpendicular is drawn from

BtoAC, 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)