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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 55 "High School Modul
es > Trigonometry by Gregory A. Moore " }}{PARA 3 "" 0 "" {TEXT -1 4 "
    " }{TEXT 256 28 "Inverse Trig Function Graphs" }}{PARA 0 "" 0 "" 
{TEXT -1 88 "\nAn exploration of the graphs of the inverse sine, inver
se cosine, and inverse tangent.\n" }}{PARA 0 "" 0 "" {TEXT 258 153 "[D
irections : Execute the Code Resource section first. Although there wi
ll be no output immediately, these definitions are used later in this \+
worksheet.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "
" {TEXT -1 8 " 0. Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "re
start; with(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 388 "So
lidPlot := proc(f,a,b)\nlocal  box,i,n,x1,x2,xmid,delta,y1,y2,A,B, m,M
,slope, concav: \nn:= 60;   delta := (b-a)/n;   x2 := a;\nfor i from 1
 to n do\n\011\011x1 := evalf(x2);\011          y1 := evalf( f(x1));  \+
\n\011\011x2 := evalf(a + i*delta);\011 y2 := evalf( f(x2));\011\n  B[
i]:=polygonplot([[x1,0],[x1,y1],[x2,y2],[x2,0]],\n\011\011\011\011\011
color=gold, style=patchnogrid);\nod;\011\ndisplay(\{ seq( B[i],i=1..n \+
)  \} );\011\nend:\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 495 "
SolidPlotBound := proc(f,a,b, M)\nlocal  n,delta,x1,x2,y1,y2,B,i: \nn:
= 60;   delta := (b-a)/n;   x2 := a;\nfor i from 1 to n do\n\011\011x1
 := evalf(x2);\011          y1 := evalf( f(x1));\n\011\011x2 := evalf(
a + i*delta);\011 y2 := evalf( f(x2));\011\n  if(y1 > M) then y1 := M;
 else if (y1 < -M) then y1 := -M; fi;fi;\n  if(y2 > M) then y2 := M; e
lse if (y2 < -M) then y2 := -M;fi;fi;\n\n  B[i]:=polygonplot([[x1,0],[
x1,y1],[x2,y2],[x2,0]],\n\011\011\011\011\011color=red, style=patchnog
rid);\nod;\011\ndisplay(\{ seq( B[i],i=1..n )  \} );\011\nend:" }}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 27 " 1.  The Inverse Sine Graph" }}
{PARA 0 "" 0 "" {TEXT -1 182 "\nThe sine function is periodic, and the
refore not one-to-one. And since only one-to-one functions have invers
es, this makes a problem. Even the fundamental period is not one-to-on
e.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 344 "display( [ plot(sin
(x), x = -2*Pi..6*Pi, color = gold),\n         plot(sin(x), x = -0..2*
Pi,    color = orange, thickness = 4),\n         polygonplot( [[0,-1],
[2*Pi,-1],[2*Pi,1],[0,1]], \n         color =COLOR(RGB, 1, .8, .2), th
ickness = 0 ),\n         plot( [[0,-1],[2*Pi,-1],[2*Pi,1],[0,1]], colo
r = red, linestyle = 2)] );\n`Fundamental Period`;" }}}{PARA 0 "" 0 "
" {TEXT -1 127 "\n\nHowever, we can find a piece of the sine which IS \+
one-to-one, is continuous, and covers the full range of the sine funct
ion.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 424 "display( plot(sin
(x), x = -2*Pi..2*Pi, color = gold),\n         plot(sin(x), x = -Pi/2.
.Pi/2,    color = orange, thickness = 4),\n         polygonplot( [[-Pi
/2,-1],[Pi/2,-1],[Pi/2,1],[-Pi/2,1],[-Pi/2,-1]], \n         color =COL
OR(RGB, 1, .8, .2), thickness = 0 ),\n         plot( [[-Pi/2,-1],[Pi/2
,-1],[Pi/2,1],[-Pi/2,1],[-Pi/2,-1]], \n                 color = orange
, linestyle = 2) );\n`Restricted Domain - which IS one-to-one`;" }}}
{PARA 0 "" 0 "" {TEXT -1 57 "\nThis is the graph of the restricted sin
e. Its domain is " }{XPPEDIT 18 0 "[-Pi/2,Pi/2]" "6#7$,$*&%#PiG\"\"\"
\"\"#!\"\"F)*&F&F'F(F)" }{TEXT -1 25 " and its range is [-1,1]." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f := x-> sin(x);\nSolidPlot(
 f, -Pi/2, Pi/2);" }}}{PARA 0 "" 0 "" {TEXT -1 162 "\n\nThe inverse si
ne is the mirror image of the restricted sine. Its domain is the range
 of the restricted sine, and its range is the domain of the restricted
 sine." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "display( plot( ar
csin(x), x = -Pi/2..Pi/2, color=blue, thickness=3, scaling=constrained
),\n         plot( sin(x), x = -Pi/2..Pi/2, color=red, thickness=3, sc
aling=constrained));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 450 "di
splay( plot( arcsin(x), x = -Pi/2..Pi/2, \n               color = blue
, thickness = 2, scaling = constrained),\n   SolidPlot( x-> sin(x), -P
i/2, Pi/2),\n   plot( [[-Pi/2,-1],[Pi/2,-1],[Pi/2,1],[-Pi/2,1],[-Pi/2,
-1]], \n                 color = orange, linestyle = 2) ,\n   plot( [[
-1,-Pi/2],[-1,Pi/2],[1,Pi/2],[1,-Pi/2],[-1,-Pi/2]], \n                \+
 color = green, linestyle = 2),\n   plot( [[-Pi/2,-Pi/2],[Pi/2,Pi/2]],
 color = violet, linestyle = 3 )\n);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 29 " 2.  The Inverse Cosine Grap
h" }}{PARA 0 "" 0 "" {TEXT -1 60 "\nThe cosine is also periodic, and t
herefore not one-to-one.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
224 "display( plot(cos(x), x = -2*Pi..6*Pi, color = blue),\n         p
lot(cos(x), x = -0..2*Pi,    color = red, thickness = 4),\n         pl
ot( [[0,-1],[2*Pi,-1],[2*Pi,1],[0,1]], color = red, linestyle = 2) );
\n`Fundamental Period`;" }}}{PARA 0 "" 0 "" {TEXT -1 149 "\n\nAs befor
e, we can find a one-to-one piece of the cosine which covers the entir
e range of [-1,1]. However, the restriction is different than before\n
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 266 "display( plot(cos(x), x
 = -2*Pi..6*Pi, color = blue),\n         plot(cos(x), x = 0..Pi,    co
lor = red, thickness = 4),\n         plot( [[0,-1],[Pi,-1],[Pi,1],[0,1
],[0,-1]], \n                 color = orange, linestyle = 2) );\n`Rest
ricted Domain - which IS one-to-one`;" }}}{PARA 0 "" 0 "" {TEXT -1 59 
"\nThis is the graph of the restricted cosine. Its domain is " }
{XPPEDIT 18 0 "[0,Pi]" "6#7$\"\"!%#PiG" }{TEXT -1 25 " and its range i
s [-1,1]." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f := x-> cos(x)
;\nSolidPlot( f, 0, Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 170 "\n\nThe inv
erse cosine is the mirror image of the restricted cosine. Its domain i
s the range of the restricted cosine, and its range is the domain of t
he restricted cosine." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "di
splay( plot( arccos(x), x = -1..Pi, color=blue, thickness=3, scaling=c
onstrained),\n   SolidPlot( x-> cos(x), 0, Pi)  );" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 385 "display( \n   plot( arccos(x), x = -1..Pi,
 color=blue, thickness=3, scaling=constrained),\n   SolidPlot( x-> cos
(x), 0, Pi),\n   plot( [[0,-1],[Pi,-1],[Pi,1],[0,1],[0,-1]], \n       \+
          color = orange, linestyle = 2) ,\n   plot( [[-1,0],[-1,Pi],[
1,Pi],[1,0],[-1,0]], \n                 color = green, linestyle = 2),
\n   plot( [[-Pi/2,-Pi/2],[Pi,Pi]], color = violet, linestyle = 3 )\n)
;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 30 " 3.  The Inverse Tangent Graph" 
}}{PARA 0 "" 0 "" {TEXT -1 32 "\nThe tangent, too, is periodic.\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 282 "M := 15:\ndisplay( plot(tan
(x), x = -2*Pi..3*Pi, y = -M..M,color = blue, discont = true),\n      \+
   plot(tan(x), x = -Pi/2..Pi/2, y = -M..M,color = red, thickness = 4,
 discont = true),\n         plot( [[-Pi/2,-M],[Pi/2,-M],[Pi/2,M],[-Pi/
2,M],[-Pi/2,-M]], color = red, linestyle = 2) );" }}}{PARA 0 "" 0 "" 
{TEXT -1 163 "\n\nThe fundamental period of the tangent IS one-to-one \+
(except at the places it's undefined of course). Therefore, we can res
trict the tangent to the open interval " }{XPPEDIT 18 0 "(-Pi/2,Pi/2)
" "6$,$*&%#PiG\"\"\"\"\"#!\"\"F(*&F%F&F'F(" }{TEXT -1 126 " to get a r
estricted version of the tangent which is one-to-one,continous, and co
vers the complete range of all real numbers.\n" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 323 "M := 15:\ndisplay( plot(tan(x), x = -Pi..2*Pi, \+
y = -M..M,color = blue, discont = true),\n         plot(tan(x), x = -P
i/2..Pi/2, y = -M..M,color = red, thickness = 4, discont = true),\n   \+
      plot( [[-Pi/2,-M],[Pi/2,-M],[Pi/2,M],[-Pi/2,M],[-Pi/2,-M]], colo
r = red, linestyle = 2) );\n`Restricted Domain - which IS one-to-one`;
" }}}{PARA 0 "" 0 "" {TEXT -1 97 "\nThis is the graph of the restricte
d tangent. Its domain is [-Pi/2,Pi/2] and its range is [-1,1]." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SolidPlotBound( x->tan(x), -
Pi/2+.1, Pi/2-.1, M);" }}}{PARA 0 "" 0 "" {TEXT -1 165 "\n\nThe invers
e tangent is the mirror image of the restricted sine. Its domain is th
e range of the restricted sine, and its range is the domain of the res
tricted sine." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "M := 10:\n
display( plot( arctan(x), x = -M..M, color=blue, thickness=3, scaling=
constrained),\n  SolidPlotBound( x->tan(x), -Pi/2+.03, Pi/2-.03, M)  )
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 400 "display( \n   plot( a
rctan(x), x = -M..M, color=blue, thickness=3, scaling=constrained),\n \+
  SolidPlotBound( x->tan(x), -Pi/2+.03, Pi/2-.03, M), \n   plot( [[-Pi
/2,-M],[Pi/2,-M],[Pi/2,M],[-Pi/2,M],[-Pi/2,-M]], \n            color =
 red, linestyle = 2),\nplot( [[-M,-Pi/2],[-M,Pi/2],[M,Pi/2],[M,-Pi/2],
[-M,-Pi/2]],color = blue, linestyle = 2),\n   plot( [[-M,-M],[M,M]], c
olor = violet, linestyle = 3 )   );" }}}{PARA 0 "" 0 "" {TEXT -1 216 "
\nNotice that this variation of the tangent maps the real line into th
e open interval (0,1) as a one-to-one function - showing that there ar
e same number of real numbers as there are real numbers in between 0 a
nd 1.\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "f := x -> ((2/Pi
)*arctan(x) + 1)/2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot
( f(x), x = -30..30);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}}{PARA 0 "" 0 "" {TEXT 259 34 " \n       \251 2002 Waterloo Maple In
c" }}}{MARK "0 1" 42 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 
0 1 2 33 1 1 }