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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 24 "Calculus Single Variable
" }{TEXT 307 3 "  \n" }{TEXT 300 35 "Brian E. Blank and Steven G. Kran
tz" }{TEXT 301 46 "\n\nSection 6.6\nInverse Trigonometric Functions\n
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 35 "1. The Function Sin and its \+
Inverse" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
21 "Define the function  " }{XPPEDIT 18 0 "proc (x) options operator, \+
arrow; Sin(x) end proc;" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"-%$Sin
G6#F%F*F*F*" }{TEXT -1 6 "  by  " }{XPPEDIT 18 0 "Sin(x) = sin(x);" "6
#/-%$SinG6#%\"xG-%$sinG6#F'" }{TEXT -1 8 "  for   " }{XPPEDIT 18 0 "x \+
= -Pi/2 .. Pi/2;" "6#/%\"xG;,$*&%#PiG\"\"\"\"\"#!\"\"F+*&F(F)F*F+" }
{TEXT -1 72 ".   In other words, we define  Sin by restricting the dom
ain of sine to " }{XPPEDIT 18 0 "-Pi/2 .. Pi/2;" "6#;,$*&%#PiG\"\"\"\"
\"#!\"\"F)*&F&F'F(F)" }{TEXT -1 40 ".  Because the domain of Sin, name
ly    " }{XPPEDIT 18 0 "-Pi/2 .. Pi/2;" "6#;,$*&%#PiG\"\"\"\"\"#!\"\"F
)*&F&F'F(F)" }{TEXT -1 49 ", is not the same as the domain of sine, na
mely  " }{XPPEDIT 18 0 "-infinity .. infinity;" "6#;,$%)infinityG!\"\"
%)infinityG" }{TEXT -1 90 ", these two functions are different. The us
e of a new name, Sin, is therefore appropriate." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 46 "Observe that  Sin  is an invertible function. " }}{PARA 13 "" 
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0 2 9 1 4 1 1.000000 43.000000 81.000000 0 0 "Curve 1" "Curve 2" }}}
{PARA 0 "" 0 "" {TEXT -1 118 "\n                                      \+
                                                                      \+
         " }{TEXT 308 8 "Figure 1" }{TEXT -1 56 "\nThe inverse is deno
ted  arcsin.  By definition, we have" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 8 "        " }{XPPEDIT 18 0 "arcsin(sin(x))
 = x;" "6#/-%'arcsinG6#-%$sinG6#%\"xGF*" }{TEXT -1 13 "        for  " 
}{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 9 "   in    " }{XPPEDIT 18 0 "
[-Pi/2, Pi/2];" "6#7$,$*&%#PiG\"\"\"\"\"#!\"\"F)*&F&F'F(F)" }{TEXT -1 
19 ",    and  \n        " }{XPPEDIT 18 0 "sin(arcsin(y)) = y;" "6#/-%$
sinG6#-%'arcsinG6#%\"yGF*" }{TEXT -1 13 "        for  " }{XPPEDIT 18 
0 "y;" "6#%\"yG" }{TEXT -1 9 "   in    " }{XPPEDIT 18 0 "[-1, 1];" "6#
7$,$\"\"\"!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The plot \+
of  " }{XPPEDIT 18 0 "y = arcsin(x);" "6#/%\"yG-%'arcsinG6#%\"xG" }
{TEXT -1 25 "  is given in  Figure 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 16 "                " }}{PARA 13 "" 1 "" 
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$FhhlFhhl7$F]ilF]il7$FbilFbil7$FgilFgil7$F\\jlF\\jl7$FajlFajl7$FfjlFfj
l7$F[[mF[[m-F\\[l6&F^[l$\")=THvFa[lFi^mFi^m-%*LINESTYLEG6#\"\"%-Fe[l6#
Fhz-%%TEXTG6'7$$FhzF)$\"#9F)Q.y~=~arcsin(x)6\"%+ALIGNRIGHTG%+ALIGNBELO
WGF[[l-Fb_m6'7$$!#:F)$!\"'F)Q+y~=~Sin(x)Fi_mFj_mF[`mF][m-%+AXESLABELSG
6%Q\"xFi_mQ!Fi_m-%%FONTG6#%(DEFAULTG-%*AXESTICKSG6$\"\"$F`am-%(SCALING
G6#%,CONSTRAINEDG-%%VIEWG6$;$!0!\\zEjzq:!#9$\"0!\\zEjzq:F[bmFham" 1 2 
0 1 10 0 2 9 1 4 1 1.000000 44.000000 153.000000 0 0 "Curve 1" "Curve \+
2" "Curve 3" "Curve 4" "Curve 5" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 118 "                                        \+
                                                                      \+
        " }{TEXT 309 8 "Figure 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 311 9 "Exercise:
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Calcu
late    " }{XPPEDIT 18 0 "arcsin(1/2)" "6#-%'arcsinG6#*&\"\"\"F'\"\"#!
\"\"" }{TEXT -1 11 "    and    " }{XPPEDIT 18 0 "arcsin(sin(3*Pi/2));
" "6#-%'arcsinG6#-%$sinG6#*(\"\"$\"\"\"%#PiGF+\"\"#!\"\"" }{TEXT -1 1 
"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 310 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 27 
"There are many values of   " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 
14 "  for which   " }{XPPEDIT 18 0 "sin(x) = 1/2;" "6#/-%$sinG6#%\"xG*
&\"\"\"F)\"\"#!\"\"" }{TEXT -1 25 "    but only one value,  " }
{XPPEDIT 18 0 "x = Pi/3;" "6#/%\"xG*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT -1 
22 ",    in the interval  " }{XPPEDIT 18 0 "[-Pi/2, Pi/2];" "6#7$,$*&%
#PiG\"\"\"\"\"#!\"\"F)*&F&F'F(F)" }{TEXT -1 10 ".  Thus,  " }{XPPEDIT 
18 0 "arcsin(1/2) = Pi/6;" "6#/-%'arcsinG6#*&\"\"\"F(\"\"#!\"\"*&%#PiG
F(\"\"'F*" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 27 "2. The Derivat
ive of arcsin" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 4 "For " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 18 " in the \+
interval  " }{XPPEDIT 18 0 "[-Pi/2, Pi/2];" "6#7$,$*&%#PiG\"\"\"\"\"#!
\"\"F)*&F&F'F(F)" }{TEXT -1 67 "   we have\n \n                       \+
                               " }{XPPEDIT 18 0 "arcsin(sin(x)) = x;" 
"6#/-%'arcsinG6#-%$sinG6#%\"xGF*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Therefore" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Diff(arcs
in(sin(x)),x) = Diff(x,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%Diff
G6$-%'arcsinG6#-%$sinG6#%\"xGF--F%6$F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "Th
e right side is 1 and, from an application of the Chain Rule,  the lef
t side becomes    " }{XPPEDIT 18 0 "D(arcsin)(sin(x))*Diff(sin(x),x);
" "6#*&--%\"DG6#%'arcsinG6#-%$sinG6#%\"xG\"\"\"-%%DiffG6$-F+6#F-F-F." 
}{TEXT -1 8 ". Thus,\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "'D(arcs
in)(sin(x))'*diff(sin(x),x) = 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*
&--%\"DG6#%'arcsinG6#-%$sinG6#%\"xG\"\"\"-%$cosGF-F/F/" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 31 "'D(arcsin)(sin(x))' = 1/cos(x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/--%\"DG6#%'arcsinG6#-%$sinG6#%\"xG*&\"\"\"F/-%$co
sGF,!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 7 "\nLet   " }{XPPEDIT 18 0 "y \+
= sin(x);" "6#/%\"yG-%$sinG6#%\"xG" }{TEXT -1 10 ".   Then  " }
{XPPEDIT 18 0 "cos(x)^2 = 1-y^2;" "6#/*$)-%$cosG6#%\"xG\"\"#\"\"\",&F+
F+*$)%\"yGF*F+!\"\"" }{TEXT -1 18 ".    Notice that  " }{XPPEDIT 18 0 
"0 <= cos(x);" "6#1\"\"!-%$cosG6#%\"xG" }{TEXT -1 11 "  because  " }
{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 21 " is in the interval  " }
{XPPEDIT 18 0 "[-Pi/2, Pi/2];" "6#7$,$*&%#PiG\"\"\"\"\"#!\"\"F)*&F&F'F
(F)" }{TEXT -1 58 ".   Therefore we use the positive root when we solv
e for  " }{XPPEDIT 18 0 "cos(x);" "6#-%$cosG6#%\"xG" }{TEXT -1 6 ":   \+
  " }{XPPEDIT 18 0 "cos(x) = sqrt(1-y^2);" "6#/-%$cosG6#%\"xG-%%sqrtG6
#,&\"\"\"F,*$)%\"yG\"\"#F,!\"\"" }{TEXT -1 48 ".    This results in th
e differentiation formula" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 12 "            " }}{PARA 0 "" 0 "" {TEXT -1 25 "     \+
                    " }{XPPEDIT 18 0 "D(arcsin)(y) = 1/sqrt(1-y^2);" "
6#/--%\"DG6#%'arcsinG6#%\"yG*&\"\"\"F,-%%sqrtG6#,&F,F,*$)F*\"\"#F,!\"
\"F4" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT 312 
9 "Exercise:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let  " }
{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 56 "  be a positive constant.  \+
Calculate the derivative of  " }{XPPEDIT 18 0 "proc (x) options operat
or, arrow; arcsin(x/a) end proc;" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG
6\"-%'arcsinG6#*&F%\"\"\"%\"aG!\"\"F*F*F*" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 313 9 "Solution:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "By the Ch
ain Rule" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "eqn := Diff(arcs
in(x/a),x) = subs(u = x/a, D(arcsin)(u))*Diff(x/a, x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$eqnG/-%%DiffG6$-%'arcsinG6#*&%\"xG\"\"\"%\"aG!
\"\"F-*&,&F.F.*&F-\"\"#F/!\"#F0#F0F4-F'6$F,F-F." }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "We have only to tidy up:
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 53 "lhs(eqn) = simplify(value(rhs(eqn)))  assuming a > 0;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%'arcsinG6#*&%\"xG\"\"\"%
\"aG!\"\"F+*&F,F,*$,&*$)F-\"\"#F,F,*$)F+F4F,F.#F,F4F." }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 
"" 0 "" {TEXT -1 3 "3. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 
"" 0 "" {TEXT -1 3 "4. " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 3 "5. " }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 3 "6. " }}{PARA 0 "" 0 "" {TEXT -1 1 
"\n" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 4 "Code" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The code for Figure 1:" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "SinePlot := plot(sin(x), x
 = -Pi/2 .. Pi/2, scaling = constrained, thickness = 2, color = PLUM, \+
view = [-Pi/2..Pi/2,-Pi/2..Pi/2]):\nlegend := plots[textplot]([0.2, 1,
 `y = Sin(x)`], align=\{ABOVE,RIGHT\}, color = PLUM):\nplots[display](
SinePlot,legend, tickmarks=[3,3])" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 21 "The code for Figure 2" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 664 "arcsinPlot := plot(arcsin(x), x = -1 .. 1, \+
scaling = constrained, thickness = 2, color = NAVY, view = [-Pi/2..Pi/
2,-Pi/2..Pi/2]):\nSinePlot := plot(sin(x), x = -Pi/2 .. Pi/2, scaling \+
= constrained, thickness = 2, color = PLUM, view = [-Pi/2..Pi/2,-Pi/2.
.Pi/2]):\nlinePlot := plot(x, x = -Pi/2..Pi/2, scaling = constrained, \+
linestyle = 4, thickness = 1, color = GRAY, view = [-Pi/2..Pi/2,-Pi/2.
.Pi/2]):\nlegend1 := plots[textplot]([0.1, 1.4, `y = arcsin(x)`], alig
n=\{BELOW,RIGHT\}, color = NAVY):\nlegend2 := plots[textplot]([-1.5, -
.6, `y = Sin(x)`], align=\{BELOW,RIGHT\}, color = PLUM):\nplots[displa
y](arcsinPlot, SinePlot, linePlot, legend1, legend2, tickmarks=[3,3]);
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 32 "Copyright and Author Information" }}
{PARA 0 "" 0 "" {TEXT -1 101 "\nWorksheet Title: BlankKrantz-06_6-R8.m
ws     A Maple  Release 8 worksheet.\n\nAuthor:  Brian E. Blank " }}
{PARA 0 "" 0 "" {TEXT -1 30 "Date Created:  26 January 2000" }}{PARA 
0 "" 0 "" {TEXT -1 485 "Date Last Revised: 29 August 2007\n\nThis docu
ment may not be distributed by any medium,\nincluding print, disk, and
 electronic transfer, without\nprior written permission of the author.
\n\nFor more information,  please contact the author:\n    \n     Depa
rtment of Mathematics, \n     Washington University in St. Louis\n    \+
 St. Louis, MO   63130\n   \n     Telephone:    (314) 935-6763\n      \+
       e-mail:    brian@math.wustl.edu\n\nCopyright:  \251 2000-2007 B
rian E. Blank,  All Rights Reserved.\n" }}{PARA 3 "" 0 "" {TEXT -1 0 "
" }}}}{MARK "8 3 0" 446 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }