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{SECT 0 {PARA 4 "" 0 "" {TEXT -1 23 "Module 4 : Trigonometry" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 17 "401 : Sine Grap
hs" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 9 "S E T U \+
P" }}{PARA 0 "" 0 "" {TEXT -1 250 "In this project we will use the fol
lowing command package. Type and execute this line before begining the
 project below. If you re-enter the worksheet for this project, be sur
e to re-excute this statement before jumping to any point in the works
heet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "restart; with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 
"Warning, the name changecoords has been redefined\n" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________________________
__________________________________________________________" }}{PARA 4 
"" 0 "" {TEXT -1 21 "A. Components of Sine" }}{PARA 0 "" 0 "" {TEXT 
-1 83 "_______________________________________________________________
____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 162 "We're going to conside
r functions of the form y = a sin(bs+c) + d. Each of the parameters ha
s a different affect on the graph. We will consider each individually.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 9 "AMPL
ITUDE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 220 
"The parameter a, affects the amplitude of the graph. y = a sin(x) has
 the same period, and passes through the origin just a sin(x) does. Ho
wever, the different values of a cause the graph to stretch or shrink \+
vertically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 181 "The gold graph is the standard y = sin(x). The blue is the res
ult of increasing the amplitude from 1 to 2, and the red graph is the \+
result of decreasing the amplitude from 1 down 0." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "display( pl
ot( sin(x), x = 0..2*Pi, color = gold), animate((1+t)*sin(x), \n      \+
   x = 0..2*Pi, t = 0..1, color = blue), animate( (1-t)*sin(x),       \+
            x=0..2*Pi, t = 0..(.95), color = red));" }}{PARA 13 "" 1 "
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257 "" 0 "" {TEXT -1 6 "PERIOD" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 188 "The parameter b affects the period. The \+
period of sin(x) is 2pi. The period of sin(bx) is 2pi/b. Here is what \+
it looks like as b changes from 1 to 2, and the period changes from 2p
i to pi." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
179 "The gold graph is the reference graph of y = sin(x). As the perio
d shortens the frequency increases. The resulting graph has two period
s in the space where it had only one period." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "display( plot( si
n(x), x = 0..2*Pi, color = gold), animate(sin(t*x), \n         x = 0..
2*Pi, t = 1..2, color = blue));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 
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-1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 13 "VERICAL SHIFT" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "The parameter d off
ers the verical position of the graph. y = sin(x) + d will be exactly \+
d units above y = sin(x)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 173 "The gold graph is the reference again. The blue g
raph shows how the graph changes when d goes from 0 to 3,and the red g
raph shows what happens when d goes from 0 down to -3." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "displa
y( plot( sin(x), x = 0..2*Pi, color = gold), animate((t + sin(x)), \n \+
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-1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 16 "HORIZONTAL SHIFT" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "The parameter c e
ffects the horizontal position of the graph. y = sin(x - c) will be mo
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"" }}{PARA 0 "" 0 "" {TEXT -1 151 "The gold graph is the reference aga
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0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 110 "Its slightly more complicated when there is a chang
e of period and  a horizontal translation at the same time." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "Clearly the per
iod of the new graph is 2pi/3, however the horizontal shift is not pi/
2!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
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1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "One way to understand \+
what the new horizontal shift is to solve the argument of the sine fun
ction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 
"This number is the actual horizontal shift!" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve( 3*x - Pi/2 \+
=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"'" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________
____________________________________________________________________" 
}}{PARA 4 "" 0 "" {TEXT -1 35 "B. Combining the Components of Sine" }}
{PARA 0 "" 0 "" {TEXT -1 83 "_________________________________________
__________________________________________" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
261 "We saw above how changes in the various parameters can individual
ly effect a sine graph. What happens when several or all of these chan
ges all occur at once? In this section we will create such graphs and \+
compare them to the original unaltered y = sin(x) graph." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "This creates the g
raph of y = sin(x). One fundamental period is shaded in yellow." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
37 "Phi := x -> x;  f:= x -> sin(Phi(x));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$PhiGR6#%\"xG6\"6$%)operatorG%&arrowGF(9$F(F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowG
F(-%$sinG6#-%$PhiG6#9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 457 "theta[1] := solve( Phi(x) = 0,x);\ntheta[2] := solve( Phi(x) = \+
2*Pi, x);\ntmid := (theta[1] + theta[2])/2:\nm := evalf(minimize(f(x))
) :\nM := evalf(maximize(f(x))):\nmid := (m + M) / 2:\ndisplay( plot( \+
f(x), x = -3*Pi..3*Pi, color = blue, thickness = 2), \n         plot( \+
sin(x), x = 0..2*Pi, color = black, thickness = 1),polygonplot(\{[[the
ta[1],m],[theta[1],M],[theta[2],M],[theta[2],m]],[[theta[1],mid],[thet
a[2],mid]],[[tmid,m],[tmid,M]]  \}, color = coral ));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>&%&thetaG6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%&thetaG6#\"\"#,$%#PiGF'" }}{PARA 13 "" 1 "" 
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