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(Rewrite Rules): Improve the example.

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(Simplifying Formulas): Explain use of the I and H flags for simplification.
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Jay Belanger 2009-08-27 13:36:19 +00:00
parent 550c8289fd
commit 0ff2d6c28e
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@ -1,3 +1,9 @@
2009-08-27 Jay Belanger <jay.p.belanger@gmail.com>
* calc.texi (Rewrite Rules): Improve the example.
(Simplifying Formulas): Explain use of the I and H flags for
simplification.
2009-08-25 Michael Albinus <michael.albinus@gmx.de>
* dbus.texi (Bus names): Add optional parameter TIMEOUT to dbus-ping.

43
doc/misc/calc.texi
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@ -5287,25 +5287,25 @@ Suppose we want to simplify this trigonometric formula:
@smallexample
@group
1: 1 / cos(x) - sin(x) tan(x)
1: 2 / cos(x)^2 - 2 tan(x)^2
.
' 1/cos(x) - sin(x) tan(x) @key{RET} s 1
' 2/cos(x)^2 - 2tan(x)^2 @key{RET} s 1
@end group
@end smallexample
@noindent
If we were simplifying this by hand, we'd probably replace the
@samp{tan} with a @samp{sin/cos} first, then combine over a common
denominator. There is no Calc command to do the former; the @kbd{a n}
algebra command will do the latter but we'll do both with rewrite
denominator. The @kbd{I a s} command will do the former and the @kbd{a n}
algebra command will do the latter, but we'll do both with rewrite
rules just for practice.
Rewrite rules are written with the @samp{:=} symbol.
@smallexample
@group
1: 1 / cos(x) - sin(x)^2 / cos(x)
1: 2 / cos(x)^2 - 2 sin(x)^2 / cos(x)^2
.
a r tan(a) := sin(a)/cos(a) @key{RET}
@ -5335,7 +5335,7 @@ To merge over a common denominator, we can use another simple rule:
@smallexample
@group
1: (1 - sin(x)^2) / cos(x)
1: (2 - 2 sin(x)^2) / cos(x)^2
.
a r a/x + b/x := (a+b)/x @key{RET}
@ -5350,13 +5350,13 @@ denominators.
Second, meta-variable names are independent from variables in the
target formula. Notice that the meta-variable @samp{x} here matches
the subformula @samp{cos(x)}; Calc never confuses the two meanings of
the subformula @samp{cos(x)^2}; Calc never confuses the two meanings of
@samp{x}.
And third, rewrite patterns know a little bit about the algebraic
properties of formulas. The pattern called for a sum of two quotients;
Calc was able to match a difference of two quotients by matching
@samp{a = 1}, @samp{b = -sin(x)^2}, and @samp{x = cos(x)}.
@samp{a = 2}, @samp{b = -2 sin(x)^2}, and @samp{x = cos(x)^2}.
@c [fix-ref Algebraic Properties of Rewrite Rules]
We could just as easily have written @samp{a/x - b/x := (a-b)/x} for
@ -5368,15 +5368,15 @@ of Rewrite Rules}, for some examples of this.)
One more rewrite will complete the job. We want to use the identity
@samp{sin(x)^2 + cos(x)^2 = 1}, but of course we must first rearrange
the identity in a way that matches our formula. The obvious rule
would be @samp{@w{1 - sin(x)^2} := cos(x)^2}, but a little thought shows
would be @samp{@w{2 - 2 sin(x)^2} := 2 cos(x)^2}, but a little thought shows
that the rule @samp{sin(x)^2 := 1 - cos(x)^2} will also work. The
latter rule has a more general pattern so it will work in many other
situations, too.
@smallexample
@group
1: (1 + cos(x)^2 - 1) / cos(x) 1: cos(x)
. .
1: (2 + 2 cos(x)^2 - 2) / cos(x)^2 1: 2
. .
a r sin(x)^2 := 1 - cos(x)^2 @key{RET} a s
@end group
@ -5397,7 +5397,7 @@ having to retype it.
' a/x + b/x := (a+b)/x @key{RET} s t merge @key{RET}
' sin(x)^2 := 1 - cos(x)^2 @key{RET} s t sinsqr @key{RET}
1: 1 / cos(x) - sin(x) tan(x) 1: cos(x)
1: 2 / cos(x)^2 - 2 tan(x)^2 1: 2
. .
r 1 a r tsc @key{RET} a r merge @key{RET} a r sinsqr @key{RET} a s
@ -22294,6 +22294,8 @@ turn the default simplifications off first (with @kbd{m O}).
@noindent
@kindex a s
@kindex I a s
@kindex H a s
@pindex calc-simplify
@tindex simplify
The @kbd{a s} (@code{calc-simplify}) [@code{simplify}] command applies
@ -22317,6 +22319,23 @@ and rewrite rules. @xref{Rearranging with Selections}.
simplification occurs automatically. Normally only the ``default
simplifications'' occur.
There are some simplifications that, while sometimes useful, are never
done automatically. For example, the @kbd{I} prefix can be given to
@kbd{a s}; the @kbd{I a s} command will change any trigonometric
function to the appropriate combination of @samp{sin}s and @samp{cos}s
before simplifying. This can be useful in simplifying even mildly
complicated trigonometric expressions. For example, while @kbd{a s}
can reduce @samp{sin(x) csc(x)} to @samp{1}, it will not simplify
@samp{sin(x)^2 csc(x)}. The command @kbd{I a s} can be used to
simplify this latter expression; it will transform @samp{sin(x)^2
csc(x)} into @samp{sin(x)}. However, @kbd{I a s} will also perform some
``simplifications'' which may not be desired; for example, it will
transform @samp{tan(x)^2} into @samp{sin(x)^2 / cos(x)^2}.
Similar to the @kbd{I} prefix, the Hyperbolic prefix @kbd{H} will
replace any hyperbolic functions in the formula with the appropriate
combinations of @samp{sinh}s and @samp{cosh}s before simplifying.
@menu
* Default Simplifications::
* Algebraic Simplifications::