Sine
One of the trigonometric functions:
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The domain of definition is the whole real line and the range of values is the interval . The sine is an odd periodic function of period
. Sine and cosine are connected by the formula
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Sine and cosecant are connected by the formula
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The derivative of sine is:
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The indefinite integral of sine is:
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Sine has the following power series representation:
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The function inverse to sine is called arcsine.
The sine and cosine of a complex argument are related to the exponential function by Euler's formulas:
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and if is pure imaginary, then
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where is the hyperbolic sine.
Comments
Of course, can be defined by the Euler formulas or by its power series. A visual definition runs as follows. Consider the unit circle with centre at the origin
in a rectangular coordinate system and with a rotating radius vector
. Let
be the angle between
and
(being reckoned positive in the counter-clockwise direction) and let
be the projection of
on
. Then
is defined as the ratio
,
as the ratio
and
as the ratio
.
Figure: s085480a
Another, analytical, approach starts with the function defined on the closed interval
by
. For
this integral is improper, but convergent. It is easy to see that
is monotone increasing and continuous on the closed interval
and differentiable on the open interval
, and has values in
. So it has an inverse function, defined on
, with values in
. This function is called
, and it can be proved that the domain of definition of this function can be continued to the whole real axis. The function
is called arcsine.
The graph of is the sinusoid (see also Trigonometric functions).
References
[a1] | M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1972) pp. §4.3 |
Sine. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sine&oldid=31901