### (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 2009-08-25 Michael Albinus * dbus.texi (Bus names): Add optional parameter TIMEOUT to dbus-ping. * dbus.texi (Bus names): Add optional parameter TIMEOUT to dbus-ping. ... ...
 ... @@ -5287,25 +5287,25 @@ Suppose we want to simplify this trigonometric formula: ... @@ -5287,25 +5287,25 @@ Suppose we want to simplify this trigonometric formula: @smallexample @smallexample @group @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 group @end smallexample @end smallexample @noindent @noindent If we were simplifying this by hand, we'd probably replace the 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 @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} 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 algebra command will do the latter, but we'll do both with rewrite rules just for practice. rules just for practice. Rewrite rules are written with the @samp{:=} symbol. Rewrite rules are written with the @samp{:=} symbol. @smallexample @smallexample @group @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} 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: ... @@ -5335,7 +5335,7 @@ To merge over a common denominator, we can use another simple rule: @smallexample @smallexample @group @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} a r a/x + b/x := (a+b)/x @key{RET} ... @@ -5350,13 +5350,13 @@ denominators. ... @@ -5350,13 +5350,13 @@ denominators. Second, meta-variable names are independent from variables in the Second, meta-variable names are independent from variables in the target formula. Notice that the meta-variable @samp{x} here matches 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}. @samp{x}. And third, rewrite patterns know a little bit about the algebraic And third, rewrite patterns know a little bit about the algebraic properties of formulas. The pattern called for a sum of two quotients; properties of formulas. The pattern called for a sum of two quotients; Calc was able to match a difference of two quotients by matching 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] @c [fix-ref Algebraic Properties of Rewrite Rules] We could just as easily have written @samp{a/x - b/x := (a-b)/x} for We could just as easily have written @samp{a/x - b/x := (a-b)/x} for ... @@ -5368,14 +5368,14 @@ of Rewrite Rules}, for some examples of this.) ... @@ -5368,14 +5368,14 @@ of Rewrite Rules}, for some examples of this.) One more rewrite will complete the job. We want to use the identity 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 @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 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 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 latter rule has a more general pattern so it will work in many other situations, too. situations, too. @smallexample @smallexample @group @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 a r sin(x)^2 := 1 - cos(x)^2 @key{RET} a s ... @@ -5397,7 +5397,7 @@ having to retype it. ... @@ -5397,7 +5397,7 @@ having to retype it. ' a/x + b/x := (a+b)/x @key{RET} s t merge @key{RET} ' 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} ' 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 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}). ... @@ -22294,6 +22294,8 @@ turn the default simplifications off first (with @kbd{m O}). @noindent @noindent @kindex a s @kindex a s @kindex I a s @kindex H a s @pindex calc-simplify @pindex calc-simplify @tindex simplify @tindex simplify The @kbd{a s} (@code{calc-simplify}) [@code{simplify}] command applies The @kbd{a s} (@code{calc-simplify}) [@code{simplify}] command applies ... @@ -22317,6 +22319,23 @@ and rewrite rules. @xref{Rearranging with Selections}. ... @@ -22317,6 +22319,23 @@ and rewrite rules. @xref{Rearranging with Selections}. simplification occurs automatically. Normally only the ``default simplification occurs automatically. Normally only the ``default simplifications'' occur. 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 @menu * Default Simplifications:: * Default Simplifications:: * Algebraic Simplifications:: * Algebraic Simplifications::
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