Program to find the value of tan(nΘ)

Given a value of tan(?) and a variable n <=15. The task is to find the value of tan(n?) using property of trigonometric functions.
Examples:

Input: tan(?) = 0.3, n = 10
Output: -2.15283

Input: tan(?) = 0.3, n = 5
Output: 0.37293

The problem can be solved using De moivre’s theorem and Binomial theorem as described below:
Using De-Moivre’s theorem, we have:

Below is the implementation of above approach:

C++

// C++ program to find the value 
// of cos(n-theta) 
 
#include <bits/stdc++.h>
#define ll long long int
#define MAX 16

using namespace std;
ll nCr[MAX][MAX] = { 0 };
 
// This function use to calculate the
// binomial coefficient upto 15

void binomial()
{
 
    // use simple DP to find coefficient

    for (int i = 0; i < MAX; i++) {

        for (int j = 0; j <= i; j++) {

            if (j == 0 || j == i)

                nCr[i][j] = 1;

            else

                nCr[i][j] = nCr[i - 1][j] + nCr[i - 1][j - 1];

        }

    }
}
 
// Function to find the value of

double findTanNTheta(double tanTheta, ll n)
{

    // store required answer

    double ans = 0, numerator = 0, denominator = 0;
 
    // use to toggle sign in sequence.

    ll toggle = 1;
 
    // calculate numerator

    for (int i = 1; i <= n; i += 2) {

        numerator = numerator + nCr[n][i] * pow(tanTheta, i) * toggle;

        toggle = toggle * -1;

    }
 
    // calculate denominator

    denominator = 1;

    toggle = -1;
 
    for (int i = 2; i <= n; i += 2) {

        numerator = numerator + nCr[n][i] * pow(tanTheta, i) * toggle;

        toggle = toggle * -1;

    }
 
    ans = numerator / denominator;
 
    return ans;
}
 
// Driver code.

int main()
{

    binomial();

    double tanTheta = 0.3;
 
    ll n = 10;
 
    cout << findTanNTheta(tanTheta, n) << endl;

    return 0;
}

Java

// Java program to find the value 
// of cos(n-theta) 

public class GFG {

    private static final int MAX = 16 ;

    static long nCr[][] = new long [MAX][MAX] ;
 
    // This function use to calculate the 

    // binomial coefficient upto 15 

    static void binomial() 

    { 
 
        // use simple DP to find coefficient 

        for (int i = 0; i < MAX; i++) { 

            for (int j = 0; j <= i; j++) { 

                if (j == 0 || j == i) 

                    nCr[i][j] = 1; 

                else

                    nCr[i][j] = nCr[i - 1][j] + nCr[i - 1][j - 1]; 

            } 

        } 

    } 
 
    // Function to find the value of 

    static double findTanNTheta(double tanTheta, int n) 

    { 

        // store required answer 

        double ans = 0, numerator = 0, denominator = 0; 

        // use to toggle sign in sequence. 

        long toggle = 1; 

        // calculate numerator 

        for (int i = 1; i <= n; i += 2) { 

            numerator = numerator + nCr[n][i] * Math.pow(tanTheta, i) * toggle; 

            toggle = toggle * -1; 

        } 

        // calculate denominator 

        denominator = 1; 

        toggle = -1; 

        for (int i = 2; i <= n; i += 2) { 

            numerator = numerator + nCr[n][i] * Math.pow(tanTheta, i) * toggle; 

            toggle = toggle * -1; 

        } 

        ans = numerator / denominator; 

        return ans; 

    }
 
    // Driver code

    public static void main(String args[])

    {

        binomial(); 

        double tanTheta = 0.3; 

        int n = 10; 

        System.out.println(findTanNTheta(tanTheta, n));

    }
// This code is contributed by ANKITRAI1 
}

Python3

# Python3 program to find the value 
# of cos(n-theta) 
 
import math 

MAX=16

nCr=[[0 for i in range(MAX)] for i in range(MAX)] 
 
# Function to calculate the binomial 
# coefficient upto 15 

def binomial(): 

    # use simple DP to find coefficient 

    for i in range(MAX): 

        for j in range(0,i+1): 

            if j == 0 or j == i: 

                nCr[i][j] = 1

            else: 

                nCr[i][j] = nCr[i - 1][j] + nCr[i - 1][j - 1] 
 
# Function to find the value of cos(n-theta) 

def findTanNTheta(tanTheta,n): 

    # store required answer

    numerator=0

    denominator=1

    # to store required answer 

    ans = 0

    # use to toggle sign in sequence. 

    toggle = 1
 
    # calculate numerator

    for i in range(1,n+1,2): 

        numerator = (numerator + nCr[n][i]*

                    (tanTheta**(i)) * toggle) 

        toggle = toggle * -1

    # calculate denominator 

    toggle=-1

    for i in range(2,n+1,2): 

        numerator = (numerator + nCr[n][i]*

                    (tanTheta**i) * toggle) 

        toggle = toggle * -1

    ans=numerator/denominator

    return ans
 
# Driver code 

if __name__=='__main__': 

    binomial() 

    tanTheta = 0.3

    n = 10

    print(findTanNTheta(tanTheta, n)) 

# this code is contributed by sahilshelangia

// C# program to find the value 
// of cos(n-theta) 
 
using System;

public class GFG {

    private static int MAX = 16 ;

    static long[,] nCr = new long [MAX,MAX] ;

    // This function use to calculate the 

    // binomial coefficient upto 15 

    static void binomial() 

    { 

        // use simple DP to find coefficient 

        for (int i = 0; i < MAX; i++) { 

            for (int j = 0; j <= i; j++) { 

                if (j == 0 || j == i) 

                    nCr[i,j] = 1; 

                else

                    nCr[i,j] = nCr[i - 1,j] + nCr[i - 1,j - 1]; 

            } 

        } 

    } 

    // Function to find the value of 

    static double findTanNTheta(double tanTheta, int n) 

    { 

        // store required answer 

        double ans = 0, numerator = 0, denominator = 0; 

        // use to toggle sign in sequence. 

        long toggle = 1; 

        // calculate numerator 

        for (int i = 1; i <= n; i += 2) { 

            numerator = numerator + nCr[n,i] * Math.Pow(tanTheta, i) * toggle; 

            toggle = toggle * -1; 

        } 

        // calculate denominator 

        denominator = 1; 

        toggle = -1; 

        for (int i = 2; i <= n; i += 2) { 

            numerator = numerator + nCr[n,i] * Math.Pow(tanTheta, i) * toggle; 

            toggle = toggle * -1; 

        } 

        ans = numerator / denominator; 

        return ans; 

    }

    // Driver code

    public static void Main()

    {

        binomial(); 

        double tanTheta = 0.3; 

        int n = 10; 

        Console.Write(findTanNTheta(tanTheta, n));

    }
 
}

PHP

<?php
// PHP program to find the value 
// of cos(n-theta) 
 
$MAX=16;

$nCr=array_fill(0, $MAX, array_fill(0, $MAX, 0));
 
// This function use to calculate the
// binomial coefficient upto 15

function binomial()
{

global $MAX,$nCr;

    // use simple DP to find coefficient

    for ($i = 0; $i < $MAX; $i++) {

        for ($j = 0; $j <= $i; $j++) {

            if ($j == 0 || $j == $i)

                $nCr[$i][$j] = 1;

            else

                $nCr[$i][$j] = $nCr[$i - 1][$j] + $nCr[$i - 1][$j - 1];

        }

    }
}
 
// Function to find the value of

function findTanNTheta($tanTheta, $n)
{

    global $MAX,$nCr;

    // store required answer

    $ans = 0;

    $numerator = 0;

    $denominator = 0;
 
    // use to toggle sign in sequence.

    $toggle = 1;
 
    // calculate numerator

    for ($i = 1; $i <= $n; $i += 2) {

        $numerator = $numerator + $nCr[$n][$i] * pow($tanTheta, $i) * $toggle;

        $toggle = $toggle * -1;

    }
 
    // calculate denominator

    $denominator = 1;

    $toggle = -1;
 
    for ($i = 2; $i <= $n; $i += 2) {

        $numerator = $numerator + $nCr[$n][$i] * pow($tanTheta, $i) * $toggle;

        $toggle = $toggle * -1;

    }
 
    $ans = $numerator / $denominator;
 
    return $ans;
}
 
// Driver code.
 
    binomial();

    $tanTheta = 0.3;
 
    $n = 10;
 
    echo findTanNTheta($tanTheta, $n);

// This code is contributed by mits
?>

Javascript

<script>
 
// Javascript program to find the value
// of cos(n-theta)
 
MAX = 16

var nCr = Array.from(Array(MAX), ()=> new Array(MAX));
 
// This function use to calculate the
// binomial coefficient upto 15

function binomial()
{
 
    // use simple DP to find coefficient

    for (var i = 0; i < MAX; i++) {

        for (var j = 0; j <= i; j++) {

            if (j == 0 || j == i)

                nCr[i][j] = 1;

            else

                nCr[i][j] = nCr[i - 1][j] + 

                nCr[i - 1][j - 1];

        }

    }
}
 
// Function to find the value of

function findTanNTheta(tanTheta, n)
{

    // store required answer

    ans = 0, numerator = 0, denominator = 0;
 
    // use to toggle sign in sequence.

    toggle = 1;
 
    // calculate numerator

    for (var i = 1; i <= n; i += 2) {

        numerator = numerator + nCr[n][i] * 

        Math.pow(tanTheta, i) * toggle;

        toggle = toggle * -1;

    }
 
    // calculate denominator

    denominator = 1;

    toggle = -1;
 
    for (var i = 2; i <= n; i += 2) {

        numerator = numerator + nCr[n][i] * 

        Math.pow(tanTheta, i) * toggle;

        toggle = toggle * -1;

    }
 
    ans = numerator / denominator;
 
    return ans.toFixed(5);
}
 
// Driver code.
binomial();

var tanTheta = 0.3;

var n = 10;
document.write( findTanNTheta(tanTheta, n));
 
</script>

Output:

-2.15283

Time Complexity: O(n + MAX²)

Auxiliary Space: O(MAX²)

Article Tags :

C++ Programs

DSA

Mathematical

binomial coefficient