00010001010008main.ACT0002020016circular functions.EAC010000006df7[[0.91707],[0.398726]]ffffg`&rYt`ffffg&rYtvrALCuxy@  H! F@6raCc @6''iDY YraCb H  B@6 V$9 raC`  @  G @    @6`H! A@6 D@6@   @   @    @6 ywR#Y`raCc !@  W2g@  @  H  E@6 V$6   @6 R5dY V$5Y @  @    @6  `C@6P@  @  @     @6 Y ywR# V$7 !!@   W2g"@   #@# $ " @6raC`Y V$9  %@! $#%$&5@! 'ˈ"@6 (%H  ')a  'Rsy5H &1WG7 ('  '$  & *+[[0.5],[0.866025]],-.-,fffff`0`ffffg)vrALCuxy/@" 0H# C@6t@ fS 1ʀ @62H! A@63H  B@6ACr y`'d 4%@# 215H" 3145123023100,*^t[Graphs and Properties[of Circular Functions\Author6Ma. Louise Antonette De Las Penas,Phd Associate Professor Ateneo de Manila UniversityQuezon City, Philippines e-mail:mlp@mathsci.math.admu.edu.ph [**The worksheet has[been modified by Prof.[ Wei-Chi Yang.[[ Objective: [To determine the basic [properties of sin t and [cos t from the circle [definitions of these [ functions.[[ Example 1:[Consider the coordinates [on the unit circle for [the following questions:[Determine the quadrants[ in which [ a) sin t >0[ b) cos t <0[c) cos t >0 & sin t <0[d) tan t >0 & sec t <0 [ Solution 1:[In the geometry strip, [we get a visual image of[the point (x,y) as it [moves along the unit [circle,where (x,y) is [the point where the [terminal side of the [angle with radian [measure t intersects [the unit circle. By [definition, (x,y)=[ (cost,sint).[Insert a geometry strip [& construct the unit [circle.Tap Draw-Circle.[Construct the circle [with center A & point B[on the circle.Click [measurement box-[equation to arrive at a [circle with equation [x2+y2-1= 0.[Highlight point B & the [circle.Click Edit-Animate-[Add Animation. [To see the coordinates [of the point as B moves[along the circle, enter [a geometry link,drag B [to get a matrix [representing the [coordinates. To activate [animation,tap Edit-Animate[-Go(once)or Go(repeat).\Geometry Strip[Geometry link:[Coordinates of point B[ given by xy=cos xsin x[changes upon animation:][By observing the signs [of the coordinates of [the point as it moves [from quadrant to [quadrant, we obtain the[following conclusions:[a)Quadrants I,II[b)Quadrants II,III[ c)Quadrant IV[d)Quadrant III [[ Solution 2:[Alternatively,draw the[unit circle in Graph strip[using parametric mode.[Insert Graph strip &enter[xt1=cos(t), yt1=sin(t) [in Graph editor.[Tap Type to change to [parametric mode.Click [Graph to draw the circle[then click Analysis-Trace[to allow point (x,y)=[(cost,sint) to move along[ the circle.\ Graph Strip7 Graph2Dh Graph3D| LISTSYS4NModify $ STATCALC NSTATSYS \NSequenceD, Sheetp| Sheet3D| SolveEqhSolveLwrl SolveUprx StupFLG1(StupListD StupPict$ ViewWind xt10Hyt1HH` l x                !, "8 #D $P %\ &h 't ( ) * + , - . / 0 1 2 3 4 5 E F H( I4 J@ KL LX Md Np O| P Q R S T ] ^ _ ` a b      system]listsystem^]=system_^system`_systema`systemb~ aseq_histbNewFolde system]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=system  0@PSheet1^Sheet2^Sheet3^Sheet4^Sheet5^SheetSheet3D 0@PSheet1^Sheet2^Sheet3^Sheet4^Sheet5^SheetSheet3D i ^t $  4tP(t) 4tL(t)`P`P`` `(10qy`#YuY X\(10qX\X\ 5 $0<HT`lx ,8DP\ht(4@LXdp| $0<HT`lx ,8DP\ht(4@LXdp| $0<HT`lx ,8DP\h@0# AG y SYa!BV  s`  b5wTGg 2r H10qb5wTGc 2t 7f76b5wTGY 2u P7 gp II`A `R#)Y W6 p(D)UY fHE% ' i1Y sv0 #(f5Y F0ht h6TqY )t&W aYYc fd  P%SYId 0f'`!'YEpRg @797TYuF1U P6TiYG!D@ `VS6YPq p !BqY'7#8  B"#@fY4PY  peY9$)!@  $`YA  QP'3Yf$3)   )GuyHYt4'XY0 GiYWtV2Y@ fyWYUThYP 6Ef)YPx2'hY`u3AYBRC)IYpH1qY)a@YwDYx'Y%# YvaY@YSd6 6YV$0yYt9FS3Y'qDY &!4YqWWrA6Y0yrY Y6tY@3(iY Q` sYPyWCY wSfQY`%&QY ic5Yp#f4bY 2WVAYt49E FXYQ#iB& E&bCYf!F2 XBtf1YwwBq0 %F2wY hQfq0 EFUrY0T7C6 2&tB"9Y@4i(u& rvDU`YPft) T%WY`uVXxQ 1&f7#Yp4q' PhUBYvYQ d`!yA8Y 'G0t@ sfdY `e yATY &8D& %'!Y  T 2 0@(PYdb"0 (10q` R[ [Suggested Exploratory[ Activity: [To reinforce the concept[that the values of the [trigonometric functions [at t are determined by [the coordinates of the [point P(x,y) on the unit [circle corresponding to t[locate the following [points on the unit circle:[1. (cos(3),sin(3)) or[cos(3)sin(3) [ x^2+y^2=1\ drag and drop8`wcEYg)`GW8s#6!2CWvrALCuxy9H #`@6geYgeY:H `@6fTxE  ;H `@6 fTxE <H  @6<;:9[2. (cos(6),sin(6)) or[cos(6)sin(6) [3. (cos(54),sin(54)) or[cos(54)sin(54) [[(Note: you can add [other values of t in [radians)[ Solution 1:[In the geometry strip,[draw a unit circle(follow[steps in Exercise 1).[Draw a point C on the [ unit circle.[Insert a geometry link,[drag point C to the link,[observe that coordinates[of C will appear.[To plot the point [(cos(3),sin(3 )) on the[unit circle,change the [coordinates on geometry[link. Shade the matrix,[then press EXE.[Consequently,point[C will be moved to the[new location on the [circle(first quadrant).Tap[point C to display the [ coordinates [(cos(3),sin(3)) ~[(0.5,0.866025)[To check the other pts[given in 2 and 3, drag[C to the link and repeat[ the process.[ Note that [(cos(6),sin(6))~ [(0.866025,0.5)[(cos(4),sin(4))~ [(0.707107,0.707107)[\Geometry Strip 2.[Enter coordinates[xy=cos tsin t of point C [in Geometry Link:]+[[ Solution 2:[In the same graph[window given in example[1 where xt1=cos(t), [yt1=sin(t),tap Analysis-[Trace then enter 1.[Give the corresponding[ angle, say 3 ;then the[pt will be located on[the unit circle giving the[coordinates of [(cos(3),sin(3)). [ Example 2:[Verify that sine, cosine[functions have period 2. [(See notes below to [recall the definition of [periodic functions.)[ Solution:[In the graph strip given[in Example 1,click table.[Then tap diamond-link.[As you move the cursor [along the rows of xt1 [values,you move along [the unit circle. On the [t column in the table,[experiment with values [of t, t+2,t+4 and so [on..For example,while[cursor is on t column,[enter 1,then type values[1,1+2,1+4. Note from [the table that the xt1,[yt1 values remain the [same and the position of[the point does not [change on the unit [circle.Try as many points [ as possible.[[The following parametric[graphs animate how the[sine and cosine functions[are constructed from the[unit circle.The values[of each function repeat[themselves after every[2 units or 1 revolution[of the circle, showing [that the sine and cosine[functions are periodic[with period 2.[[\x^2+y^2=1, sin(x)=Graph2D, Graph3D@ LISTSYSL4NModify $ STATCALC NSTATSYS \NSequence, Sheet4| Sheet3D| SolveEq,SolveLwr0 SolveUpr< StupFLG1H(StupListpD StupPict$ ViewWind     $ 0 < H T ` l x       ! " # $ % & ' ( ) *, +8 ,D -P .\ 0h 1t 2 3 4 5 E F H I J K L M N O P( Q4 R@ SL TX ]d ^h _l `p at bx |     system]listsystem^]=system_^system`_systema`systemb~ aseq_histbNewFolde system]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=system  0@PSheet1 ^Sheet2 ^Sheet3 ^Sheet4 ^Sheet5 ^SheetSheet3D 0@PSheet1 ^Sheet2 ^Sheet3 ^Sheet4 ^Sheet5 ^SheetSheet3D i ^t $ `P`P`p`p `(10qy`#YuY  xvs8 Y7Eyi@` O\x^2+y^2=1, cos(x)>Graph2D, Graph3D@ LISTSYSL4NModify $ STATCALC NSTATSYS \NSequence, Sheet4| Sheet3D| SolveEq,SolveLwr0 SolveUpr< StupFLG1H(StupListpD StupPict$ ViewWind     $ 0 < H T ` l x       ! " # $ % & ' ( ) *, +8 ,D -P .\ 0h 1t 2 3 4 5 E F H I J K L M N O P( Q4 R@ SL TX ]d ^h _l `p at bx |     system]listsystem^]=system_^system`_systema`systemb~ aseq_histbNewFolde system]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=system  0@PSheet1 ^Sheet2 ^Sheet3 ^Sheet4 ^Sheet5 ^SheetSheet3D 0@PSheet1 ^Sheet2 ^Sheet3 ^Sheet4 ^Sheet5 ^SheetSheet3D i ^t $ `P`P`p`p `(10qy`#YuY db"0 (10q` O[\ Notes?A function f is periodic if there exists a positive realnumber k such that f(t+k)=f(t) for every t in the domain of f. The least positive real number k, if it exists is the period of f. [eAct020008tide.EAC0100000018db^t[ Fit a tide[\find the trig function