john212/infinite_series_for_sine.html

## infinite_series_for_sine.html
<!doctype html>
<html>
<head>
    <meta charset="UTF-8">
    <title>Infinite Series for the Sine Function</title>
</head>
<body>

<h2>Infinite Series for the Sine Function</h2>

<P>This program will use one of the many infinite series formulas to calculate
an approximate value for the sine function. This infinite series for the sine
function is the Taylor series and for this case is the same as the Maclaurin series
for the sine function.</P>

<P>This particular infinite series converges rapidly, the first 8 tems
are within about 0.000000000000001 of the actual value of the sine value. Reminder,
the factorial symbol in math "!" means, for n!, to multiply by n(n-1)(n-2) until n(1),
Note: 0! = 1 by definition. For example: 4! = 4*3*2*1 = 24.</P>

<P>Javascript numbers are always double-precision 64 bit binary (base 2) numbers. There is
no "double precision" type.<br>
Reference: <a href="https://indepth.dev/here-is-what-you-need-to-know-about-javascripts-number-type/">https://indepth.dev/here-is-what-you-need-to-know-about-javascripts-number-type/</a></P>

<P>The infinite series for sin(x) where x is in radians and x is a Real number is:</P>

<P>sin(x) = x<sup>1</sup>/1! - x<sup>3</sup>/3! + x<sup>5</sup>/5! - x<sup>7</sup>/7! + x<sup>9</sup>/9!...the pattern continues to infinity.</P>

<P>This series is an alternating series, that is, the terms alternate between
plus and minus.</P>

<P>This program is written in JavaScript.</P>

<P>By Mr. C. March, 2020</P>

<P>Enter angle in radians, default is pi/3 ~ 1.047197551</P>

<P>Enter number of terms to sum, max is 10, default is 8.</P>

<P>*********************results********************************</P>

<script type="text/javascript">
main();         // must call the main() function to run the program.
function main() // define the enveloping "main()" function.
{ //start main() function block

// **** Define and initialize variables ****
var x = "";                 // angle, in radians to evaluate, input by user.
var default_x = Math.PI/3;
var nterms = "";            // number of terms in the infinite series to evaluate, input by user.
var default_nterms = 8;
var sumOfTerms = 0.0;
// *****************************************

// **** Collect user input via popup boxes ****
x = prompt("Enter the angle to evaluate in radians", default_x);
nterms = prompt("Enter the number of terms of the infinite series to use", default_nterms);
// ********************************************

/***** Convert user text input into decimal numbers ****
The parseFloat("string") function returns a number,
where "string" is the string value you want to parse.

The parseInt("string",radix) function parses a string argument and returns an integer
of the specified radix (the base in mathematical numeral systems).  An integer between
2 and 36  represents the radix of the string.  Be careful—this does not default to 10!
Set radix = 10 (decimal). Leading whitespace in this argument is ignored. Ex: parseInt(n,10).
*******************************************************/
x = parseFloat(x);
nterms = parseInt(nterms,10);

// Data Validation - Ensure user input is > 0 for angle and terms and max terms = 10.
if((x <= 0) || (nterms <= 0) || (nterms > 10))
  { //start data validation if block
  document.write("<P>angle =", x, ", terms =" , nterms);
  document.write("<P>angle and terms must be greater than zero, max value for number of terms is 10");
  return // stops execution of main() function
  } //endof data validation if block

// Display the user input.
document.write("<P><b>angle =", x, " radians, terms =" , nterms,"</b></P>");
document.write("<P><b>Javascript value for Math.sin(x) =", Math.sin(x),"</b></P>");

// calculate each term in the infinite series for the sine
for (let i = 0; i <= nterms-1; i++)
    { // start nterm for loop
    // Math.pow(base,exponent) returns "base" raised to the power of "exponent".
    sumOfTerms = sumOfTerms + ( (Math.pow(-1,i)) * (1/(factorial(2*i + 1))) * Math.pow(x,2*i + 1) );
    document.write("<P>term = ", i + 1, " sin(x) = ", sumOfTerms );
	} // endof nterm for loop


// ***************************************************
// debugging code - comment out as needed
/*
console.log("isTriangle is ",isTriangle);
*/
// ***************************************************

} //endof main() function block

// *****************************************************************************
// function to return the factorial of any passed integer
// note: the set of integers is abbreviated "Z"
function factorial(_Z)  // Z is set of integers, _Z means variable is an argument
{ //start factorial function block
  var counter = _Z; // will use the integer argument as the starting value of a counter
  var factorialZ = 1;
  if(_Z === 0)
    {//start if _Z is 0 if block
	factorialZ = 1;
	return factorialZ;
    }//endof if _Z is 0 if block
  while (counter >= 1)
    {//start while block
    factorialZ = counter*factorialZ;
    counter = counter - 1;
    }//endof while block
  return factorialZ;
} //endof factorial function block
// *****************************************************************************

</script>

<p>
<b>
For more information, visit:<br>
<a href="https://en.smath.com/cloud/worksheet/UBKaAzZ3">Infinite Series for the Sine Function</a><br>
<a href="https://en.smath.com/cloud/worksheet/LppF8NmG">Infinite Series for the Cosine Functio</a><br>
<a href="https://en.smath.com/cloud/worksheet/WvhTsoGS">Infinite Series for e</a><br>
<a href="https://en.smath.com/cloud/worksheet/buqQbeTd">Infinite Series for Pi</a><br>
<a href="https://en.smath.com/cloud/worksheet/47rawg6N">Comparing the Infinite Series for Pi, e, Sine, and Cosine</a><br>
</b>
<i>All SM=Studio Math files by Mr.C.</i>
</p>

</body>
</html>
	<!doctype html>
	<html>
	<head>
	<meta charset="UTF-8">
	<title>Infinite Series for the Sine Function</title>
	</head>
	<body>

	<h2>Infinite Series for the Sine Function</h2>

	<P>This program will use one of the many infinite series formulas to calculate
	an approximate value for the sine function. This infinite series for the sine
	function is the Taylor series and for this case is the same as the Maclaurin series
	for the sine function.</P>

	<P>This particular infinite series converges rapidly, the first 8 tems
	are within about 0.000000000000001 of the actual value of the sine value. Reminder,
	the factorial symbol in math "!" means, for n!, to multiply by n(n-1)(n-2) until n(1),
	Note: 0! = 1 by definition. For example: 4! = 432*1 = 24.</P>

	<P>Javascript numbers are always double-precision 64 bit binary (base 2) numbers. There is
	no "double precision" type.<br>
	Reference: <a href="https://indepth.dev/here-is-what-you-need-to-know-about-javascripts-number-type/">https://indepth.dev/here-is-what-you-need-to-know-about-javascripts-number-type/</a></P>

	<P>The infinite series for sin(x) where x is in radians and x is a Real number is:</P>

	<P>sin(x) = x<sup>1</sup>/1! - x<sup>3</sup>/3! + x<sup>5</sup>/5! - x<sup>7</sup>/7! + x<sup>9</sup>/9!...the pattern continues to infinity.</P>

	<P>This series is an alternating series, that is, the terms alternate between
	plus and minus.</P>

	<P>This program is written in JavaScript.</P>

	<P>By Mr. C. March, 2020</P>

	<P>Enter angle in radians, default is pi/3 ~ 1.047197551</P>

	<P>Enter number of terms to sum, max is 10, default is 8.</P>

	<P>*******************results******************************</P>

	<script type="text/javascript">
	main(); // must call the main() function to run the program.
	function main() // define the enveloping "main()" function.
	{ //start main() function block

	// ** Define and initialize variables **
	var x = ""; // angle, in radians to evaluate, input by user.
	var default_x = Math.PI/3;
	var nterms = ""; // number of terms in the infinite series to evaluate, input by user.
	var default_nterms = 8;
	var sumOfTerms = 0.0;
	// *****************************************

	// ** Collect user input via popup boxes **
	x = prompt("Enter the angle to evaluate in radians", default_x);
	nterms = prompt("Enter the number of terms of the infinite series to use", default_nterms);
	// ********************************************

	/*** Convert user text input into decimal numbers **
	The parseFloat("string") function returns a number,
	where "string" is the string value you want to parse.

	The parseInt("string",radix) function parses a string argument and returns an integer
	of the specified radix (the base in mathematical numeral systems). An integer between
	2 and 36 represents the radix of the string. Be careful—this does not default to 10!
	Set radix = 10 (decimal). Leading whitespace in this argument is ignored. Ex: parseInt(n,10).
	*******************************************************/
	x = parseFloat(x);
	nterms = parseInt(nterms,10);

	// Data Validation - Ensure user input is > 0 for angle and terms and max terms = 10.
	if((x <= 0) \|\| (nterms <= 0) \|\| (nterms > 10))
	{ //start data validation if block
	document.write("<P>angle =", x, ", terms =" , nterms);
	document.write("<P>angle and terms must be greater than zero, max value for number of terms is 10");
	return // stops execution of main() function
	} //endof data validation if block

	// Display the user input.
	document.write("<P><b>angle =", x, " radians, terms =" , nterms,"</b></P>");
	document.write("<P><b>Javascript value for Math.sin(x) =", Math.sin(x),"</b></P>");

	// calculate each term in the infinite series for the sine
	for (let i = 0; i <= nterms-1; i++)
	{ // start nterm for loop
	// Math.pow(base,exponent) returns "base" raised to the power of "exponent".
	sumOfTerms = sumOfTerms + ( (Math.pow(-1,i)) * (1/(factorial(2i + 1))) Math.pow(x,2*i + 1) );
	document.write("<P>term = ", i + 1, " sin(x) = ", sumOfTerms );
	} // endof nterm for loop


	// ***************************************************
	// debugging code - comment out as needed
	/*
	console.log("isTriangle is ",isTriangle);
	*/
	// ***************************************************

	} //endof main() function block

	// *****************************************************************************
	// function to return the factorial of any passed integer
	// note: the set of integers is abbreviated "Z"
	function factorial(_Z) // Z is set of integers, _Z means variable is an argument
	{ //start factorial function block
	var counter = _Z; // will use the integer argument as the starting value of a counter
	var factorialZ = 1;
	if(_Z === 0)
	{//start if _Z is 0 if block
	factorialZ = 1;
	return factorialZ;
	}//endof if _Z is 0 if block
	while (counter >= 1)
	{//start while block
	factorialZ = counter*factorialZ;
	counter = counter - 1;
	}//endof while block
	return factorialZ;
	} //endof factorial function block
	// *****************************************************************************

	</script>

	<p>
	<b>
	For more information, visit:<br>
	<a href="https://en.smath.com/cloud/worksheet/UBKaAzZ3">Infinite Series for the Sine Function</a><br>
	<a href="https://en.smath.com/cloud/worksheet/LppF8NmG">Infinite Series for the Cosine Functio</a><br>
	<a href="https://en.smath.com/cloud/worksheet/WvhTsoGS">Infinite Series for e</a><br>
	<a href="https://en.smath.com/cloud/worksheet/buqQbeTd">Infinite Series for Pi</a><br>
	<a href="https://en.smath.com/cloud/worksheet/47rawg6N">Comparing the Infinite Series for Pi, e, Sine, and Cosine</a><br>
	</b>
	<i>All SM=Studio Math files by Mr.C.</i>
	</p>

	</body>
	</html>