sin

Symbolic sine function

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Syntax

sin(X)

Description

sin(X) returns the sine function of X.

example

Examples

Sine Function for Numeric and Symbolic Arguments

Depending on its arguments, sin returns floating-point or exact symbolic results.

Compute the sine function for these numbers. Because these numbers are not symbolic objects, sin returns floating-point results.

A = sin([-2, -pi, pi/6, 5*pi/7, 11])

A =
   -0.9093   -0.0000    0.5000    0.7818   -1.0000

Compute the sine function for the numbers converted to symbolic objects. For many symbolic (exact) numbers, sin returns unresolved symbolic calls.

symA = sin(sym([-2, -pi, pi/6, 5*pi/7, 11]))

symA =
[ -sin(2), 0, 1/2, sin((2*pi)/7), sin(11)]

Use vpa to approximate symbolic results with floating-point numbers:

vpa(symA)

ans =
[ -0.90929742682568169539601986591174,...
0,...
0.5,...
0.78183148246802980870844452667406,...
-0.99999020655070345705156489902552]

Plot Sine Function

Plot the sine function on the interval from $- 4 π$ to $4 π$ .

syms x
fplot(sin(x),[-4*pi 4*pi])
grid on

Figure contains an axes object. The axes object contains an object of type functionline.

Handle Expressions Containing Sine Function

Many functions, such as diff, int, taylor, and rewrite, can handle expressions containing sin.

Find the first and second derivatives of the sine function:

syms x
diff(sin(x), x)
diff(sin(x), x, x)

ans =
cos(x)
 
ans =
-sin(x)

Find the indefinite integral of the sine function:

int(sin(x), x)

ans =
-cos(x)

Find the Taylor series expansion of sin(x):

taylor(sin(x), x)

ans =
x^5/120 - x^3/6 + x

Rewrite the sine function in terms of the exponential function:

rewrite(sin(x), 'exp')

ans =
(exp(-x*1i)*1i)/2 - (exp(x*1i)*1i)/2

Evaluate Units with `sin` Function

sin numerically evaluates these units automatically: radian, degree, arcmin, arcsec, and revolution.

Show this behavior by finding the sine of x degrees and 2 radians.

u = symunit;
syms x
f = [x*u.degree 2*u.radian];
sinf = sin(f)

sinf =
[ sin((pi*x)/180), sin(2)]

You can calculate sinf by substituting for x using subs and then using double or vpa.

Input Arguments

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`X` — Input
symbolic number | symbolic scalar variable | symbolic matrix variable | symbolic expression | symbolic function | symbolic matrix function | symbolic vector | symbolic matrix

Input, specified as a symbolic number, scalar variable, matrix variable, expression, function, matrix function, or as a vector or matrix of symbolic numbers, scalar variables, expressions, or functions.

More About

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Sine Function

The sine of an angle, α, defined with reference to a right triangle is

$\sin (α) = \frac{opposite side}{hypotenuse} = \frac{a}{h} .$

Right triangle with vertices A, B, and C. The vertex A has an angle α, and the vertex C has a right angle. The hypotenuse, or side AB, is labeled as h. The opposite side of α, or side BC, is labeled as a. The adjacent side of α, or side AC, is labeled as b. The sine of α is defined as the opposite side a divided by the hypotenuse h.

The sine of a complex argument, α, is

$\sin (α) = \frac{e^{i α} - e^{- i α}}{2 i} .$

Version History

Introduced before R2006a

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R2022a: Compute sine of symbolic matrix functions

The sin function accepts an input argument of type symfunmatrix.

R2021a: Compute sine of symbolic matrix variables

The sin function accepts an input argument of type symmatrix.

sin