### (Rewrite Rules): Improve the example.

`(Simplifying Formulas): Explain use of the I and H flags for simplification.`
parent 550c8289
 2009-08-27 Jay Belanger * calc.texi (Rewrite Rules): Improve the example. (Simplifying Formulas): Explain use of the I and H flags for simplification. 2009-08-25 Michael Albinus * dbus.texi (Bus names): Add optional parameter TIMEOUT to dbus-ping. ... ...
 ... ... @@ -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::
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