Given the value of sin(?) and a variable n<=15. The task is to find the value of sin(n?) using property of trigonometric functions.
Examples:
Input: sin(?)=0.5, n=1 Output: 0.5 Input: sin(?)=0.5, n=10 Output: -0.866025
Approach: This problem can be solved using De moivre’s theoremand Binomial theorem
Now, we have both sin(?) and cos(?). Put the value in equation to get your answer.
Below is the implementation of above approach:
C++
// C++ Program to find the value of sin(n?)#include <bits/stdc++.h>#define ll long long int#define MAX 16using namespace std;ll nCr[MAX][MAX] = { 0 };// This function use to calculate the// binomial coefficient upto 15void 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 ofdouble findCosNTheta(double sinTheta, ll n){ // find cosTheta from sinTheta double cosTheta = sqrt(1 - sinTheta * sinTheta); // store required answer double ans = 0; // use to toggle sign in sequence. ll toggle = 1; for (int i = 1; i <= n; i += 2) { ans = ans + nCr[n][i] * pow(cosTheta, n - i) * pow(sinTheta, i) * toggle; toggle = toggle * -1; } return ans;}// Driver code.int main(){ binomial(); double sinTheta = 0.5; ll n = 10; cout << findCosNTheta(sinTheta, n) << endl; return 0;} |
Java
// Java Program to find the value of sin(n?)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 findCosNTheta(double sinTheta, int n) { // find cosTheta from sinTheta double cosTheta = Math.sqrt(1 - sinTheta * sinTheta); // store required answer double ans = 0; // use to toggle sign in sequence. long toggle = 1; for (int i = 1; i <= n; i += 2) { ans = ans + nCr[n][i] * Math.pow(cosTheta, n - i) * Math.pow(sinTheta, i) * toggle; toggle = toggle * -1; } return ans; } // Driver code public static void main (String args[]){ binomial(); double sinTheta = 0.5; int n = 10; System.out.println(findCosNTheta(sinTheta, n)); }// This code is contributed by ANKITRAI1} |
Python3
# Python3 program to find the # value of sin(n-theta) import math MAX=16nCr=[[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 findCosNTheta(sinTheta,n): # find sinTheta from sinTheta cosTheta = math.sqrt(1 - sinTheta * sinTheta) # to store required answer ans = 0 # use to toggle sign in sequence. toggle = 1 for i in range(1,n+1,2): ans = (ans + nCr[n][i]*(cosTheta**(n - i)) *(sinTheta**i) * toggle) toggle = toggle * -1 return ans # Driver code if __name__=='__main__': binomial() sinTheta = 0.5 n = 10 print(findCosNTheta(sinTheta, n)) # this code is contributed by sahilshelangia |
C#
// C# Program to find the value of sin(n?)using System;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 cos(n-theta) static double findCosNTheta(double sinTheta, int n) { // find cosTheta from sinTheta double cosTheta = Math.Sqrt(1 - sinTheta * sinTheta); // store required answer double ans = 0; // use to toggle sign in sequence. long toggle = 1; for (int i = 1; i <= n; i += 2) { ans = ans + nCr[n, i] * Math.Pow(cosTheta, n - i) * Math.Pow(sinTheta, i) * toggle; toggle = toggle * -1; } return ans; } // Driver code public static void Main (){ binomial(); double sinTheta = 0.5; int n = 10; Console.Write(findCosNTheta(sinTheta, n));}}// This code is contributed by ChitraNayal |
PHP
<?php// PHP Program to find the value of sin(n?)$MAX=16;$nCr=array_fill(0,$MAX,array_fill(0,$MAX,0));// This function use to calculate the// binomial coefficient upto 15function 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][$j - 1]; } }}// Function to find the value offunction findCosNTheta($sinTheta,$n){ global $MAX,$nCr; // find cosTheta from sinTheta $cosTheta = sqrt(1 - $sinTheta * $sinTheta); // store required answer $ans = 0; // use to toggle sign in sequence. $toggle = 1; for ($i = 1; $i <= $n; $i += 2) { $ans = $ans + $nCr[$n][$i] * pow($cosTheta, $n - $i) * pow($sinTheta, $i) * $toggle; $toggle = $toggle * -1; } return $ans;}// Driver code. binomial(); $sinTheta = 0.5; $n = 10; echo findCosNTheta($sinTheta, $n);// this code is contributed by mits?> |
Javascript
<script>// Javascript Program to find the value of sin(n?)MAX = 16var nCr = Array.from(Array(MAX), () => new Array(MAX));// This function use to calculate the// binomial coefficient upto 15function 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 offunction findCosNTheta(sinTheta, n){ // find cosTheta from sinTheta var cosTheta = Math.sqrt(1 - sinTheta * sinTheta); // store required answer var ans = 0; // use to toggle sign in sequence. var toggle = 1; for (var i = 1; i <= n; i += 2) { ans = ans + nCr[n][i] * Math.pow(cosTheta, n - i) * Math.pow(sinTheta, i) * toggle; toggle = toggle * -1; } return ans.toFixed(6);}// Driver code.binomial();var sinTheta = 0.5;var n = 10;document.write( findCosNTheta(sinTheta, n));</script> |
Output:
-0.866025
Time Complexity: O(MAX2)
Auxiliary Space: O(MAX2)
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