The shift-`S` (`calc-sin`

) [`sin`

] command computes the sine
of an angle or complex number. If the input is an HMS form, it is interpreted
as degrees-minutes-seconds; otherwise, the input is interpreted according
to the current angular mode. It is best to use Radians mode when operating
on complex numbers.

Calc’s “units” mechanism includes angular units like `deg`

,
`rad`

, and `grad`

. While ‘`sin(45 deg)`’ is not evaluated
all the time, the `u s` (`calc-simplify-units`

) command will
simplify ‘`sin(45 deg)`’ by taking the sine of 45 degrees, regardless
of the current angular mode. See Basic Operations on Units.

Also, the symbolic variable `pi`

is not ordinarily recognized in
arguments to trigonometric functions, as in ‘`sin(3 pi / 4)`’, but
the default algebraic simplifications recognize many such
formulas when the current angular mode is Radians *and* Symbolic
mode is enabled; this example would be replaced by ‘`sqrt(2) / 2`’.
See Symbolic Mode. Beware, this simplification occurs even if you
have stored a different value in the variable ‘`pi`’; this is one
reason why changing built-in variables is a bad idea. Arguments of
the form ‘`x`’ plus a multiple of ‘`pi/2`’ are also simplified.
Calc includes similar formulas for `cos`

and `tan`

.

Calc’s algebraic simplifications know all angles which are integer multiples of
‘`pi/12`’, ‘`pi/10`’, or ‘`pi/8`’ radians. In Degrees mode,
analogous simplifications occur for integer multiples of 15 or 18
degrees, and for arguments plus multiples of 90 degrees.

With the Inverse flag, `calc-sin`

computes an arcsine. This is also
available as the `calc-arcsin`

command or `arcsin`

algebraic
function. The returned argument is converted to degrees, radians, or HMS
notation depending on the current angular mode.

With the Hyperbolic flag, `calc-sin`

computes the hyperbolic
sine, also available as `calc-sinh`

[`sinh`

]. With the
Hyperbolic and Inverse flags, it computes the hyperbolic arcsine
(`calc-arcsinh`

) [`arcsinh`

].

The shift-`C` (`calc-cos`

) [`cos`

] command computes the cosine
of an angle or complex number, and shift-`T` (`calc-tan`

) [`tan`

]
computes the tangent, along with all the various inverse and hyperbolic
variants of these functions.

The `f T` (`calc-arctan2`

) [`arctan2`

] command takes two
numbers from the stack and computes the arc tangent of their ratio. The
result is in the full range from *-180* (exclusive) to *+180*
(inclusive) degrees, or the analogous range in radians. A similar
result would be obtained with `/` followed by `I T`, but the
value would only be in the range from *-90* to *+90* degrees
since the division loses information about the signs of the two
components, and an error might result from an explicit division by zero
which `arctan2`

would avoid. By (arbitrary) definition,
‘`arctan2(0,0)=0`’.

The `calc-sincos`

[`sincos`

] command computes the sine and
cosine of a number, returning them as a vector of the form
‘`[ cos, sin]`’.
With the Inverse flag [

`arcsincos`

], this command takes a two-element
vector as an argument and computes `arctan2`

of the elements.
(This command does not accept the Hyperbolic flag.)
The remaining trigonometric functions, `calc-sec`

[`sec`

],
`calc-csc`

[`csc`

] and `calc-cot`

[`cot`

], are also
available. With the Hyperbolic flag, these compute their hyperbolic
counterparts, which are also available separately as `calc-sech`

[`sech`

], `calc-csch`

[`csch`

] and `calc-coth`

[`coth`

]. (These commands do not accept the Inverse flag.)