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sin(x) represents the sine function.
Specify the argument
x in radians, not in
degrees. For example, use π to
specify an angle of 180o.
All trigonometric functions are defined for complex arguments.
Floating-point values are returned for floating-point arguments. Floating-point intervals are returned for floating-point interval arguments. Unevaluated function calls are returned for most exact arguments.
Translations by integer multiples of π are eliminated from the argument. Further, arguments that are rational multiples of π lead to simplified results; symmetry relations are used to rewrite the result using an argument from the standard interval . Explicit expressions are returned for the following arguments:
See Example 2.
rewrite expressions in terms of a specific target function. For example,
you can rewrite expressions involving the sine function in terms of
other trigonometric functions and vice versa. See Example 5.
The inverse function is implemented by
See Example 6.
The float attributes are kernel functions, thus, floating-point evaluation is fast.
When called with a floating-point argument,
sensitive to the environment variable
DIGITS which determines
the numerical working precision.
sin with the following exact and symbolic
sin(PI), sin(1), sin(5 + I), sin(PI/2), sin(PI/11), sin(PI/8)
sin(-x), sin(x + PI), sin(x^2 - 4)
Floating point values are computed for floating-point arguments:
sin(123.4), sin(5.6 + 7.8*I), sin(1.0/10^20)
Floating point intervals are computed for interval arguments:
sin(0...1), sin(20...30), sin(0...5)
Some special values are implemented:
sin(PI/10), sin(2*PI/5), sin(123/8*PI), sin(-PI/12)
Translations by integer multiples of π are eliminated from the argument:
sin(x + 10*PI), sin(3 - PI), sin(x + PI), sin(2 - 10^100*PI)
All arguments that are rational multiples of π are transformed to arguments from the interval :
sin(4/7*PI), sin(-20*PI/9), sin(123/11*PI), sin(-PI/13)
Arguments that are rational multiples of
rewritten in terms of hyperbolic functions:
sin(5*I), sin(5/4*I), sin(-3*I)
For other complex arguments, use
expand to rewrite the result:
sin(5*I + 2*PI/3), sin(PI/4 - 5/4*I), sin(-3*I + PI/2)
expand(sin(5*I + 2*PI/3)), expand(sin(5/4*I - PI/4)), expand(sin(-3*I + PI/2))
implements the addition theorems:
expand(sin(x + PI/2)), expand(sin(x + y))
uses these theorems in the other direction, trying to rewrite products
of trigonometric functions:
The trigonometric functions do not immediately respond to properties
assume(n, Type::Integer): sin(n*PI), sin((2*n + 1)*PI/2)
take such properties into account:
simplify(sin(n*PI)), simplify(sin((2*n + 1)*PI/2))
assume(n, Type::Odd): sin(n*PI + x), simplify(sin(n*PI + x))
y := sin(x + n*PI) + sin(x - n*PI); simplify(y)
delete n, y
obtain a representation in terms of a specific target function:
rewrite(sin(x)*exp(2*I*x), exp); rewrite(sin(x), cot)
The inverse function is implemented as
arcsin(sin(x)) does not necessarily
values with real parts in the interval :
arcsin(sin(3)), arcsin(sin(1.6 + I))
diff(sin(x^2), x), float(sin(3)*cot(5 + I))
limit(sin(x)/x, x = 0)
taylor(sin(x), x = 0)
Arithmetical expression or a floating-point interval