Math.Sin(Double) Method

Definition

Returns the sine of the specified angle.

public static double Sin (double a);

Parameters

a
Double

An angle, measured in radians.

Returns

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

Examples

The following example uses Sin 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
*/

Remarks

The angle, a, 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.

Applies to

Product Versions
.NET Core 1.0, Core 1.1, Core 2.0, Core 2.1, Core 2.2, Core 3.0, Core 3.1, 5, 6, 7, 8, 9
.NET Framework 1.1, 2.0, 3.0, 3.5, 4.0, 4.5, 4.5.1, 4.5.2, 4.6, 4.6.1, 4.6.2, 4.7, 4.7.1, 4.7.2, 4.8, 4.8.1
.NET Standard 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 2.0, 2.1
UWP 10.0