# sin

Symbolic sine function

## Description

example

sin(X) returns the sine function of X.

## 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 $-4\pi$ to $4\pi$.

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

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

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### Sine Function

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

The sine of a complex argument, α, is

$\mathrm{sin}\left(\alpha \right)=\frac{{e}^{i\alpha }-{e}^{-i\alpha }}{2i}\text{\hspace{0.17em}}.$