# Math.SinCos(Double) Method

## Definition

Returns the sine and cosine of the specified angle.

public:
static ValueTuple<double, double> SinCos(double x);
public static (double Sin, double Cos) SinCos (double x);
static member SinCos : double -> ValueTuple<double, double>
Public Shared Function SinCos (x As Double) As ValueTuple(Of Double, Double)

x
Double

#### Returns

The sine and cosine of x. If x is equal to NaN, NegativeInfinity, or PositiveInfinity, this method returns NaN.

## Examples

The following example uses SinCos to evaluate certain trigonometric identities for selected angles.

// Example for the trigonometric Math.Sin( double )
// and Math.Cos( double ) methods.
using System;

class SinCos
{
public static void Main()
{
Console.WriteLine(
"This example of trigonometric " +
"Math.Sin( double ), Math.Cos( double ), and Math.SinCos( double )\n" +
"generates the following output.\n" );
Console.WriteLine(
"Convert selected values for X to radians \n" +
"and evaluate these trigonometric identities:" );
Console.WriteLine( "   sin^2(X) + cos^2(X) == 1\n" +
"   sin(2 * X) == 2 * sin(X) * cos(X)" );
Console.WriteLine( "   cos(2 * X) == cos^2(X) - sin^2(X)" );
Console.WriteLine( "   cos(2 * X) == cos^2(X) - sin^2(X)" );

UseSineCosine(15.0);
UseSineCosine(30.0);
UseSineCosine(45.0);

Console.WriteLine(
"\nConvert selected values for X and Y to radians \n" +
"and evaluate these trigonometric identities:" );
Console.WriteLine( "   sin(X + Y) == sin(X) * cos(Y) + cos(X) * sin(Y)" );
Console.WriteLine( "   cos(X + Y) == cos(X) * cos(Y) - sin(X) * sin(Y)" );

UseTwoAngles(15.0, 30.0);
UseTwoAngles(30.0, 45.0);

Console.WriteLine(
"\nWhen you have calls to sin(X) and cos(X) they \n" +
"can be replaced with a single call to sincos(x):" );

UseCombinedSineCosine(15.0);
UseCombinedSineCosine(30.0);
UseCombinedSineCosine(45.0);
}

// Evaluate trigonometric identities with a given angle.
static void UseCombinedSineCosine(double degrees)
{
double angle = Math.PI * degrees / 180.0;
(double sinAngle, double cosAngle) = Math.SinCos(angle);

// Evaluate sin^2(X) + cos^2(X) == 1.
Console.WriteLine(
"\n                           Math.SinCos({0} deg) == ({1:E16}, {2:E16})",
degrees, sinAngle, cosAngle);
Console.WriteLine(
"(double sin, double cos) = Math.SinCos({0} deg)",
degrees );
Console.WriteLine(
"sin^2 + cos^2 == {0:E16}",
sinAngle * sinAngle + cosAngle * cosAngle );
}

// Evaluate trigonometric identities with a given angle.
static void UseSineCosine(double degrees)
{
double angle    = Math.PI * degrees / 180.0;
double sinAngle = Math.Sin(angle);
double cosAngle = Math.Cos(angle);

// Evaluate sin^2(X) + cos^2(X) == 1.
Console.WriteLine(
"\n                           Math.Sin({0} deg) == {1:E16}\n" +
"                           Math.Cos({0} deg) == {2:E16}",
degrees, Math.Sin(angle), Math.Cos(angle) );
Console.WriteLine(
"(Math.Sin({0} deg))^2 + (Math.Cos({0} deg))^2 == {1:E16}",
degrees, sinAngle * sinAngle + cosAngle * cosAngle );

// Evaluate sin(2 * X) == 2 * sin(X) * cos(X).
Console.WriteLine(
"                           Math.Sin({0} deg) == {1:E16}",
2.0 * degrees, Math.Sin(2.0 * angle) );
Console.WriteLine(
"    2 * Math.Sin({0} deg) * Math.Cos({0} deg) == {1:E16}",
degrees, 2.0 * sinAngle * cosAngle );

// Evaluate cos(2 * X) == cos^2(X) - sin^2(X).
Console.WriteLine(
"                           Math.Cos({0} deg) == {1:E16}",
2.0 * degrees, Math.Cos(2.0 * angle) );
Console.WriteLine(
"(Math.Cos({0} deg))^2 - (Math.Sin({0} deg))^2 == {1:E16}",
degrees, cosAngle * cosAngle - sinAngle * sinAngle );
}

// Evaluate trigonometric identities that are functions of two angles.
static void UseTwoAngles(double degreesX, double degreesY)
{
double  angleX  = Math.PI * degreesX / 180.0;
double  angleY  = Math.PI * degreesY / 180.0;

// Evaluate sin(X + Y) == sin(X) * cos(Y) + cos(X) * sin(Y).
Console.WriteLine(
"\n        Math.Sin({0} deg) * Math.Cos({1} deg) +\n" +
"        Math.Cos({0} deg) * Math.Sin({1} deg) == {2:E16}",
degreesX, degreesY, Math.Sin(angleX) * Math.Cos(angleY) +
Math.Cos(angleX) * Math.Sin(angleY));
Console.WriteLine(
"                           Math.Sin({0} deg) == {1:E16}",
degreesX + degreesY, Math.Sin(angleX + angleY));

// Evaluate cos(X + Y) == cos(X) * cos(Y) - sin(X) * sin(Y).
Console.WriteLine(
"        Math.Cos({0} deg) * Math.Cos({1} deg) -\n" +
"        Math.Sin({0} deg) * Math.Sin({1} deg) == {2:E16}",
degreesX, degreesY, Math.Cos(angleX) * Math.Cos(angleY) -
Math.Sin(angleX) * Math.Sin(angleY));
Console.WriteLine(
"                           Math.Cos({0} deg) == {1:E16}",
degreesX + degreesY, Math.Cos(angleX + angleY));
}
}

/*
This example of trigonometric Math.Sin( double ) and Math.Cos( double )
generates the following output.

Convert selected values for X to radians
and evaluate these trigonometric identities:
sin^2(X) + cos^2(X) == 1
sin(2 * X) == 2 * sin(X) * cos(X)
cos(2 * X) == cos^2(X) - sin^2(X)

Math.Sin(15 deg) == 2.5881904510252074E-001
Math.Cos(15 deg) == 9.6592582628906831E-001
(Math.Sin(15 deg))^2 + (Math.Cos(15 deg))^2 == 1.0000000000000000E+000
Math.Sin(30 deg) == 4.9999999999999994E-001
2 * Math.Sin(15 deg) * Math.Cos(15 deg) == 4.9999999999999994E-001
Math.Cos(30 deg) == 8.6602540378443871E-001
(Math.Cos(15 deg))^2 - (Math.Sin(15 deg))^2 == 8.6602540378443871E-001

Math.Sin(30 deg) == 4.9999999999999994E-001
Math.Cos(30 deg) == 8.6602540378443871E-001
(Math.Sin(30 deg))^2 + (Math.Cos(30 deg))^2 == 1.0000000000000000E+000
Math.Sin(60 deg) == 8.6602540378443860E-001
2 * Math.Sin(30 deg) * Math.Cos(30 deg) == 8.6602540378443860E-001
Math.Cos(60 deg) == 5.0000000000000011E-001
(Math.Cos(30 deg))^2 - (Math.Sin(30 deg))^2 == 5.0000000000000022E-001

Math.Sin(45 deg) == 7.0710678118654746E-001
Math.Cos(45 deg) == 7.0710678118654757E-001
(Math.Sin(45 deg))^2 + (Math.Cos(45 deg))^2 == 1.0000000000000000E+000
Math.Sin(90 deg) == 1.0000000000000000E+000
2 * Math.Sin(45 deg) * Math.Cos(45 deg) == 1.0000000000000000E+000
Math.Cos(90 deg) == 6.1230317691118863E-017
(Math.Cos(45 deg))^2 - (Math.Sin(45 deg))^2 == 2.2204460492503131E-016

Convert selected values for X and Y to radians
and evaluate these trigonometric identities:
sin(X + Y) == sin(X) * cos(Y) + cos(X) * sin(Y)
cos(X + Y) == cos(X) * cos(Y) - sin(X) * sin(Y)

Math.Sin(15 deg) * Math.Cos(30 deg) +
Math.Cos(15 deg) * Math.Sin(30 deg) == 7.0710678118654746E-001
Math.Sin(45 deg) == 7.0710678118654746E-001
Math.Cos(15 deg) * Math.Cos(30 deg) -
Math.Sin(15 deg) * Math.Sin(30 deg) == 7.0710678118654757E-001
Math.Cos(45 deg) == 7.0710678118654757E-001

Math.Sin(30 deg) * Math.Cos(45 deg) +
Math.Cos(30 deg) * Math.Sin(45 deg) == 9.6592582628906831E-001
Math.Sin(75 deg) == 9.6592582628906820E-001
Math.Cos(30 deg) * Math.Cos(45 deg) -
Math.Sin(30 deg) * Math.Sin(45 deg) == 2.5881904510252085E-001
Math.Cos(75 deg) == 2.5881904510252096E-001
*/

// Example for the trigonometric Math.Sin( double )
// and Math.Cos( double ) methods.
// In F#, the sin and cos functions may be used instead.
open System

// Evaluate trigonometric identities with a given angle.
let useSineCosine degrees =
let angle = Math.PI * degrees / 180.
let sinAngle = Math.Sin angle
let cosAngle = Math.Cos angle

// Evaluate sin^2(X) + cos^2(X) = 1.
printfn $""" Math.Sin({degrees} deg) = {Math.Sin angle:E16} Math.Cos({degrees} deg) = {Math.Cos angle:E16}""" printfn$"(Math.Sin({degrees} deg))^2 + (Math.Cos({degrees} deg))^2 = {sinAngle * sinAngle + cosAngle * cosAngle:E16}"

// Evaluate sin(2 * X) = 2 * sin(X) * cos(X).
printfn $" Math.Sin({2. * degrees} deg) = {Math.Sin(2. * angle):E16}" printfn$"    2 * Math.Sin({degrees} deg) * Math.Cos({degrees} deg) = {2. * sinAngle * cosAngle:E16}"

// Evaluate cos(2 * X) = cos^2(X) - sin^2(X).
printfn $" Math.Cos({2. * degrees} deg) = {Math.Cos(2. * angle):E16}" printfn$"(Math.Cos({degrees} deg))^2 - (Math.Sin({degrees} deg))^2 = {cosAngle * cosAngle - sinAngle * sinAngle:E16}"

// Evaluate trigonometric identities that are functions of two angles.
let useTwoAngles degreesX degreesY =
let angleX = Math.PI * degreesX / 180.
let angleY = Math.PI * degreesY / 180.

// Evaluate sin(X + Y) = sin(X) * cos(Y) + cos(X) * sin(Y).
printfn $""" Math.Sin({degreesX} deg) * Math.Cos({degreesY} deg) Math.Cos({degreesX} deg) * Math.Sin({degreesY} deg) = {Math.Sin angleX * Math.Cos angleY + Math.Cos angleX * Math.Sin angleY:E16}""" printfn$"                           Math.Sin({degreesX + degreesY} deg) = {Math.Sin(angleX + angleY):E16}"

// Evaluate cos(X + Y) = cos(X) * cos(Y) - sin(X) * sin(Y).
printfn
$""" Math.Cos({degreesX} deg) * Math.Cos({degreesY} deg) - Math.Sin({degreesX} deg) * Math.Sin({degreesY} deg) = {Math.Cos angleX * Math.Cos angleY - Math.Sin angleX * Math.Sin angleY:E16}""" printfn$"                           Math.Cos({degreesX + degreesY} deg) = {Math.Cos(angleX + angleY):E16}"

// Evaluate trigonometric identities with a given angle.
let useCombinedSineCosine degrees =
let angle = Math.PI * degrees / 180.
let struct(sinAngle, cosAngle) = Math.SinCos angle

// Evaluate sin^2(X) + cos^2(X) = 1.
printfn $"\n Math.SinCos({degrees} deg) = ({sinAngle:E16}, {cosAngle:E16})" printfn$"(double sin, double cos) = Math.SinCos({degrees} deg)"
printfn \$"sin^2 + cos^2 = {sinAngle * sinAngle + cosAngle * cosAngle:E16}"

printfn
"""This example of trigonometric
Math.Sin( double ), Math.Cos( double ), and Math.SinCos( double )
generates the following output.

Convert selected values for X to radians
and evaluate these trigonometric identities:
sin^2(X) + cos^2(X) = 1\n   sin(2 * X) = 2 * sin(X) * cos(X)
cos(2 * X) = cos^2(X) - sin^2(X)
cos(2 * X) = cos^2(X) - sin^2(X)

"""

useSineCosine 15.
useSineCosine 30.
useSineCosine 45.

printfn """
Convert selected values for X and Y to radians
and evaluate these trigonometric identities:
sin(X + Y) = sin(X) * cos(Y) + cos(X) * sin(Y)
cos(X + Y) = cos(X) * cos(Y) - sin(X) * sin(Y)
"""

useTwoAngles 15. 30.
useTwoAngles 30. 45.

printfn """
When you have calls to sin(X) and cos(X) they
can be replaced with a single call to sincos(x):"""

useCombinedSineCosine 15.
useCombinedSineCosine 30.
useCombinedSineCosine 45.

// This example of trigonometric Math.Sin( double ) and Math.Cos( double )
// generates the following output.
//
// Convert selected values for X to radians
// and evaluate these trigonometric identities:
//    sin^2(X) + cos^2(X) = 1
//    sin(2 * X) = 2 * sin(X) * cos(X)
//    cos(2 * X) = cos^2(X) - sin^2(X)
//
//                            Math.Sin(15 deg) = 2.5881904510252074E-001
//                            Math.Cos(15 deg) = 9.6592582628906831E-001
// (Math.Sin(15 deg))^2 + (Math.Cos(15 deg))^2 = 1.0000000000000000E+000
//                            Math.Sin(30 deg) = 4.9999999999999994E-001
//     2 * Math.Sin(15 deg) * Math.Cos(15 deg) = 4.9999999999999994E-001
//                            Math.Cos(30 deg) = 8.6602540378443871E-001
// (Math.Cos(15 deg))^2 - (Math.Sin(15 deg))^2 = 8.6602540378443871E-001
//
//                            Math.Sin(30 deg) = 4.9999999999999994E-001
//                            Math.Cos(30 deg) = 8.6602540378443871E-001
// (Math.Sin(30 deg))^2 + (Math.Cos(30 deg))^2 = 1.0000000000000000E+000
//                            Math.Sin(60 deg) = 8.6602540378443860E-001
//     2 * Math.Sin(30 deg) * Math.Cos(30 deg) = 8.6602540378443860E-001
//                            Math.Cos(60 deg) = 5.0000000000000011E-001
// (Math.Cos(30 deg))^2 - (Math.Sin(30 deg))^2 = 5.0000000000000022E-001
//
//                            Math.Sin(45 deg) = 7.0710678118654746E-001
//                            Math.Cos(45 deg) = 7.0710678118654757E-001
// (Math.Sin(45 deg))^2 + (Math.Cos(45 deg))^2 = 1.0000000000000000E+000
//                            Math.Sin(90 deg) = 1.0000000000000000E+000
//     2 * Math.Sin(45 deg) * Math.Cos(45 deg) = 1.0000000000000000E+000
//                            Math.Cos(90 deg) = 6.1230317691118863E-017
// (Math.Cos(45 deg))^2 - (Math.Sin(45 deg))^2 = 2.2204460492503131E-016
//
// Convert selected values for X and Y to radians
// and evaluate these trigonometric identities:
//    sin(X + Y) = sin(X) * cos(Y) + cos(X) * sin(Y)
//    cos(X + Y) = cos(X) * cos(Y) - sin(X) * sin(Y)
//
//         Math.Sin(15 deg) * Math.Cos(30 deg) +
//         Math.Cos(15 deg) * Math.Sin(30 deg) = 7.0710678118654746E-001
//                            Math.Sin(45 deg) = 7.0710678118654746E-001
//         Math.Cos(15 deg) * Math.Cos(30 deg) -
//         Math.Sin(15 deg) * Math.Sin(30 deg) = 7.0710678118654757E-001
//                            Math.Cos(45 deg) = 7.0710678118654757E-001
//
//         Math.Sin(30 deg) * Math.Cos(45 deg) +
//         Math.Cos(30 deg) * Math.Sin(45 deg) = 9.6592582628906831E-001
//                            Math.Sin(75 deg) = 9.6592582628906820E-001
//         Math.Cos(30 deg) * Math.Cos(45 deg) -
//         Math.Sin(30 deg) * Math.Sin(45 deg) = 2.5881904510252085E-001
//                            Math.Cos(75 deg) = 2.5881904510252096E-001


## Remarks

The angle, x, must be in radians. Multiply by Math.PI/180 to convert degrees to radians.

This method calls into the underlying C runtime, and the exact result or valid input range may differ between different operating systems or architectures.