calc: Trigonometric and Hyperbolic Functions
9.2 Trigonometric/Hyperbolic Functions
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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. 
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’. 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.)
Info Catalog
calc: Logarithmic Functions
calc: Scientific Functions
calc: Advanced Math Functions