Blog Entry © Tuesday, March 17, 2026, by James Pate Williams, Jr., Comparison of Power Series Arctangent and Arcsine Functions with the C++ Built-In Functions

#pragma once

//https://scipp-legacy.pbsci.ucsc.edu/~haber/ph116A/taylor11.pdf

class Functions
{
public:
	static void initialize();
	static double factorial(int n);
	static double arccosine(double x, int n);
	static double arccosecant(double x);
	static double arccotangent(double x);
	static double arcsecant(double x);
	static double arcsine(double x, int n);
	static double arctangent(double x, int n);
};

#include "Functions.h"
#include <math.h> 

double pi2 = 0.0;

double Functions::factorial(int n)
{
	double nfactorial = 1.0;

	for (int i = 2; i <= n; i++)
		nfactorial *= i;

	return nfactorial;
}

void Functions::initialize()
{
	pi2 = 2.0 * atan(1.0);
}

double Functions::arccosine(double x, int n)
{
	double sum = pi2 - arcsine(x, n);
	return sum;
}

double Functions::arccosecant(double x)
{
	double sum = 0;
	return sum;
}

double Functions::arccotangent(double x)
{
	double sum = 0;
	return sum;
}

double Functions::arcsecant(double x)
{
	double sum = 0;
	return sum;
}

double Functions::arcsine(double x, int n)
{
	double sum = 0.0;

	if (fabs(x) <= 1.0)
	{
		for (int i = n; i >= 0; i--)
		{
			double ifact = factorial(i);
			double i2 = 2.0 * i, i21 = 2 * i + 1;
			double coeff = factorial(i2) /
				(pow(2, i2) * ifact * ifact * (i21));
			sum += coeff * pow(x, i21);
		}
	}

	return sum;
}

double Functions::arctangent(double x, int n)
{
	double sum = 0.0;

	if (fabs(x) <= 1.0)
	{
		double one = 0.0;

		if (n % 2 == 0)
			one = 1.0;
		else
			one = -1.0;

		for (int i = 2 * n + 1; i >= 0; i--)
		{
			one = -one;
			double i21 = 2 * i + 1.0;
			sum += one * pow(x, i21) / i21;
		}
	}

	return sum;
}

// InvTrigConsoleCPP.cpp (c) Monday, March 16, 2026
// by James Pate Williams, Jr., BA, BS, MSwE, PhD

#include "Functions.h"
#include <math.h>
#include <iomanip>
#include <iostream>

static void CreateTable(
	char title[],
	double a, double b, int n, int nPts,
	double(*fx)(double, int), double(*gx)(double))
{
	double x = a;
	double deltaX = (b - a) / nPts;
	std::cout << title << std::endl;
	std::cout << "# Terms = " << (2 * n + 2) << std::endl;
	std::cout << "x" << '\t' << "fx" << "\t\t";
	std::cout << "sx" << "\t\t" << "error" << std::endl;

	for (int i = 0; i < nPts; i++)
	{
		double f = fx(x, n);
		double s = gx(x);
		double e = 0.0;

		if (fabs(s) != 0)
			e = 100.0 * fabs(f - s) / fabs(s);

		std::cout << std::fixed << std::setprecision(4);
		std::cout << std::setw(4);
		std::cout << x << '\t';
		std::cout << std::fixed << std::setprecision(11);
		std::cout << std::setw(12);
		std::cout << f << '\t';
		std::cout << std::fixed << std::setprecision(11);
		std::cout << std::setw(12);
		std::cout << s << '\t';
		std::cout << e << std::endl;
		x += deltaX;
	}
}

int main()
{
	Functions::initialize();

	while (true)
	{
		char line[128] = "";
		std::cout << "== Menu == " << std::endl;
		std::cout << "1 arcsine" << std::endl;
		std::cout << "2 arctangent" << std::endl;
		std::cout << "3 exit" << std::endl;
		std::cout << "option # : ";
		std::cin.getline(line, 128);
		char option = line[0];

		if (option == '3')
			break;

		if (option < '1' || option > '2')
		{
			std::cout << "invalid option" << std::endl;
			continue;
		}

		std::cout << "# points: ";
		std::cin.getline(line, 128);
		int nPts = atoi(line);

		std::cout << "# terms: ";
		std::cin.getline(line, 128);
		int n = atoi(line);

		if (option == '1')
		{
			char title[] = "Series arcsin Versus C++ asin";
			CreateTable(title, 0.0, 1.0, n, nPts,
				Functions::arcsine, asin);
		}
		else if (option == '2')
		{
			char title[] = "Series arctan Versus C++ atan";
			CreateTable(title, 0.0, 1.0, n, nPts,
				Functions::arctangent, atan);
		}
	}

	return 0;
}