ࡱ> qs pn'` $bjbj{P{P 4::+YYYYZ@{\( .e.e.e.e f f f;z=z=z=z=z=z=z$|hD|az f f f f faz.e.ezMkMkMk f.e.e;zMk f;zMkMk:p,+q.e[ MRI`Yhbp r,{0@{pRjR+q+q f fMk f f f f fazazCk f f f@{ f f f f04%4 MTH 114 EXAM 3 REVIEW TOPICS INVERSES: Know the domain and range of y = arcos x, y=arcsinx and y=arctanx Know the graphs and key points of y = arcos x, y=arcsinx and y=arctanx Understand that the input and output must be of any trig or inverse trig function. Be able to evaluate EXACTLY the value of arctrig(#) where that # is on the unit circle. REMEMBER, angles MUST be in radian mode NEVER degrees, and in the correct quadrant for the range of the arctrig function. Be able to approximate the value of arctrig(#) using your calculator. Determine if trig(arctrig(#))=# or arctrig(trig(angle)=angle based on the domain and range Evaluate trig(arctrig(#)) or arctrig(trig(angle)) EXACTLY. Make sure you can do all of the problems from the inverse trig worksheet posted at HYPERLINK http://www.math.msu.edu/~giovanni/mth114/invTrigWorksheet.doc http://www.math.msu.edu/~giovanni/mth114/invTrigWorksheet.doc IDENTITIES: Make sure you have memorized all of the identities and recognize them in different forms! Be able to simplify expressions using algebra and trig IDs Given the value of one trig x = # and info about a quadrant, find the value of all other trig functions using trig IDs only (NO Triangles allowed). Be able to verify Identities. Remember you MUST WORK ON ONE SIDE ONLY, clearly show all steps. Make sure you NEVER drop your variables cos is not the same as cos x! Practice problems using all of the common techniques using ids, combining fractions, breaking up fractions, simplifying complex fractions, conjugates, expanding or combining sum formulas, double angle, half-angle ids, converting to all sines and cosines etc. Watch out for common algebra mistakes! Find the exact value of sin, cos or tan of an angle that is the sum or difference of 2 angles on the unit circle or half an angle on the unit circle. For example evaluate sin(13/12) or tan(17/12), cos(/8) etc. You may need to find the angles yourself! You will likely be required to simplify expressions completely. This may require simplifying complex fractions, radical expressions or rationalizing denominators etc. Given trig A = # and info about what quadrant A is in, and trig(B) = # and info about the quadrant B is in, find the EXACT value of Trig(A+/- B), Trig(-A), Trig(pi/2 A), Trig(2A), Trig (A/2), a different trig function of A etc. You may be required to simplify your answers completely. Make sure you remember to draw your triangles in the appropriate quadrants and analyze the sign values get the proper sign values of the missing sides. You MUST SHOW the Pythagorean step AND justification for the sign of the missing side to receive full credit. WATCH CAREFULLY that the legs of your triangle are correct for the quad you are in and the reference angle and sides are clearly labeled in your picture. When solving half angle problems, make sure you remember to carefully analyze the sign value and quadrants. The sign outside the radical comes from the sign in the quad of the angle A/2. You must show the work identifying what quadrant A/2 is in and why you chose the sign value this to receive full credit. The sign under the radical is the sign of cos A in the quad of the angle A. Watch sign values CAREFULLY! Find the exact value of Trig(U+/- V), Trig(-U), Trig(2U), Trig (U/2) where U and V are of the form invtrig(#). For example  EMBED Equation.3  or  EMBED Equation.3  Make sure you can do all of the problems on the multiple angle worksheet posted at HYPERLINK http://www.math.msu.edu/~giovanni/mth114/MultipleAngleWorksheet.doc http://www.math.msu.edu/~giovanni/mth114/MultipleAngleWorksheet.doc! EQUATIONS: Solving equations: Be able to solve trig equations both in [0, 2pi) and in general. Practice problems with a variety of techniques isolating the trig function, factoring, quadratic equation, using trig ids, taking square roots (DONT forget the +/-) etc. You should know how to solve them exactly if the # is on the unit circle and approximate on a calculator using reference angles if not. Make sure you understand how to do ALL of the book homework, worksheets graded homework and all examples in class. Make sure you can solve equations involving angles other than just x! Watch out that you do not invent trig id or algebra rules that do not exist! Solve trig( ) = # not on the unit circle. You will need to show all reference angle work! Make sure you can do all textbook problems, worksheet problems, lecture examples etc. EXTRA PROBLEMS FOR EXAM 3: 1. Given  EMBED Equation.2  , find the exact value of a) sin u b) cos(-v) c) sin(u+v) d) cos(2v) e) sin(2u) f) cos(u/2) g) sin(v/2) h) cos(u-v) I) csc(2v) j) tan(2u) k) tan(v/2) 2. Given  EMBED Equation.3 , EMBED Equation.3  Find the EXACT value of each of the following. Simplify completely. a)  EMBED Equation.3  b)  EMBED Equation.3  c)  EMBED Equation.3  d)  EMBED Equation.3  3. Solve the following trig equations algebraically in [0, 2( ) and in general. Give all answers EXACTLY if possible. Otherwise give answers accurate to 0.0001. a)  EMBED Equation  b)  EMBED Equation  c)2 cos(2x)+1 = 0 d)  EMBED Equation  e) 2sin(3x)cot(2x)=cot(2x) f)  EMBED Equation  g) 7cos(2x)+3=0 h) 2tan(3x)+1=0 i) csc(2x)+4=0 j) 4sin2x -1 = 0 k)  EMBED Equation.3  l)  EMBED Equation.3  4. Given  EMBED Equation . Fill in the following table with exact values sin(u/2)cos(u/2)SinuCosusin(2u)cos(2u)Tan(2u)3/5-8/177/8-2/3 5. Find the EXACT value of a)  EMBED Equation.3  b)  EMBED Equation.3  6. Evaluate the following exactly: a)  EMBED Equation  b)  EMBED Equation  c)  EMBED Equation.3  d)  EMBED Equation.3  e)  EMBED Equation.3  f)  EMBED Equation.3  g)  EMBED Equation.3  h)  EMBED Equation.3  EMBED Equation.3  i)  EMBED Equation.3  j)  EMBED Equation.3  k)  EMBED Equation  7. a)  EMBED Equation.3 = ____ b)  EMBED Equation.3 = ____ c)  EMBED Equation.3 = _____ d) h)  EMBED Equation.3  e))  EMBED Equation.3 = _____ f)  EMBED Equation.3  g)  EMBED Equation.3  h)  EMBED Equation.3  i)  EMBED Equation.3  j)  EMBED Equation.3  8 Verify the following ID: a)  EMBED Equation  b)  EMBED Equation  c)  EMBED Equation  d)  EMBED Equation  e)  EMBED Equation  f)  EMBED Equation  g)  EMBED Equation  h)  EMBED Equation.3  i)  EMBED Equation.3  j)  EMBED Equation.3  k)  EMBED Equation.3  9. Fill in the following table with exact values usin ucos utan usin u/2cos u/2tan u/2sin(2u) cos(2u)tan(2u) EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 7(/423(/616(/313(/8SKIPSKIPSKIP 10. a) Solve  EMBED Equation.3  b) Solve  EMBED Equation.3  in [0, 2) c) Solve  EMBED Equation.3  d) Solve  EMBED Equation.3  in [0, 2) e) Find  EMBED Equation.3  Key to exam 3 extra problems: 1. a) -1/3 b) -4/5 c)  EMBED Equation  d) 7/25 e)  EMBED Equation  f)  EMBED Equation  g)  EMBED Equation  h)  EMBED Equation  i) -25/24 j)  EMBED Equation  k) 3 2 a)  EMBED Equation.3  b)  EMBED Equation.3  c) -2 d) -24/7 3. a) t=(-5/13)(4/5)+(12/13)(3/5) = 16/65 b) 0, 2(/3, 5(/6, (, 5(/3, 11(/6 GS: x=0+n, x=2/3+n, x=5(/6 + n c) (/3, 2(/3, 4(/3, 5(/3 GS: x=/3+n, x=2/3+n d) (/6, (/3, 7(/6, 4(/3 GS: x=/6+n, x=/3+n e) 0, (/2, (, 3(/2, (/18, 13(/18, 25(/18, 5(/18, 17(/18, 29(/18 GS: x=0+n/2, x=/18+2n/3, x=5/18+2n/3 f) 0, (, 3(/4, 5(/4 GS: : x=0+n, x=3/4+2n, : x=5/4+2n g) arccos(-3/7)/2 =1.0069, arccos(-3/7)/2 + =4.1484, (2-arccos(-3/7))/2=2.1347, (2-arccos(-3/7))/2 +  = 5.2763 GS: ) arccos(-3/7)/2 + n =1.0069 + n, (2-arccos(-3/7))/2 + n=2.1347 + n (2+arctan(-1/2))/3 =1.9398, (2+arctan(-1/2))/3 - /3=.8926, (2+arctan(-1/2))/3 + /3=2.9870, (2+arctan(-1/2))/3 + 2/3=4.0342, (2+arctan(-1/2))/3 + =5.0814, (2+arctan(-1/2))/3 + 4/3=6.1286 GS: (2+arctan(-1/2))/3 + n/3 i) arcsin(-1/4)/2 + =3.0153, arcsin(-1/4)/2 + 2=6.1568, (-arcsin(-1/4))/2=1.6971, (-arcsin(-1/4))/2+=4.8387 GS: ) arcsin(-1/4)/2 + n=3.0153 + n, (-arcsin(-1/4))/2+ n=1.6971+ n j) (/6, 5/6, 7(/6, 11(/6 GS: x=/6+n, x=5/6+n k) (/18, 13(/18, 25(/18, 11(/18, 23(/18, 35(/18 GS: x=/18+n/3, x=11/18+n/3 l)  EMBED Equation.3  GS:  EMBED Equation.3  4. sin(u/2)cos(u/2)SinuCosusin(2u)cos(2u)tan(2u)-3(10/10-(10/103/5-4/5-24/257/25-24/7-5(34/34-3(34/3415/17-8/17-240/289-161/289240) T {  T U V : (7Vtjfbb^Zh?Khkh!&hhh0jhd56&h/f>*B*CJOJQJ^JaJph/jhh/fCJOJQJU^JaJh/fCJOJQJ^JaJ#jh/fCJOJQJU^JaJh/f h/f5h/fhl:5hl:hdh0jhPUh0jhPU56h]4h!Kh(O5h!Kh]45#)k ! | : ; gddh^hgdPU & Fgd]4 & FgdPUgdPUgd]4$a$gd]4$ (< $UY 'WY29QR\]h/fCJOJQJ^JaJ#jh/fCJOJQJU^JaJh/f h/f5h/fh0j5jh7Uhl:EHUj։N hl:CJUVaJjh7Uhl:EHUjN hl:CJUVaJjhl:Uhl:h!&h0jhd+|XY GHcd  gd2gd3Tgd]4gddDGHbno )ճyoy`QoyMh2j?hlhlCJEHUj:hTO hlCJUVaJjh3TCJU h3TCJ h3T5CJh!Khhhl:hdhPUh]4h0jhd56h0j h/fhl:h/fCJOJQJ^JaJ&h/f>*B*CJOJQJ^JaJph#jh/fCJOJQJU^JaJ/j.hh/fCJOJQJU^JaJ)*=>?@ABUVWXbgȻ~obj7hkth2EHUj˹QO h2CJUVaJjhkth2EHUjN h2CJUVaJjShkth2EHUj?N h2CJUVaJ h25j hph2EHUjljN h2CJUVaJj hph/fEHUjQO h/fCJUVaJh2jh2U$    LMO^-23D˼˲ˣ˲˅v˼˲gX˼˲jYhhCJEHUj:N hCJUVaJjhhCJEHUjN hCJUVaJjhhCJEHUj#N hCJUVaJjh3TCJU hCJ jph3TCJ h3TCJ h2CJh2jh2Ujhph2EHUjQO h2CJUVaJ".h$).6>F $Ifgd3T^gdgFgd3TDEFGJW[gkvwxyz˾˴˥wh_Y h2CJhhgFCJj&hTcqhTcqCJEHUj3cTO hTcqCJUVaJjhTcqCJU hTcqCJjq#h3ThCJEHUjAK hCJUVaJjhCJU hCJH* hgFCJ hCJ h3TCJjh3TCJUj hhCJEHUj0N hCJUVaJ")*wx     Ͻϰսyj[j1hRJhRJCJEHUjQO hRJCJUVaJj /hMrhMrCJEHUjfK hMrCJUVaJjhMrCJUj,hMAh3TCJEHUj~G h3TCJUV hMrCJ h2CJ h0jCJ h3TCJjh3TCJUj]8 h3TCJEHU$j]8 h3TCJUVmHnHuFGHIM0''' $Ifgd3Tkd($$Ifl    ֞v gXI:+"0    4 layt3TMNOPQ $Ifgd3TQRSTU0''' $Ifgd3Tkd;)$$Ifl    ֞v gXI:+"0    4 layt3TU[\]^ $Ifgd3T^_`ae0''' $Ifgd3Tkd)$$Ifl    ֞v gXI:+"0    4 layt3Tefghi $Ifgd3Tijklm0''' $Ifgd3Tkd*$$Ifl    ֞v gXI:+"0    4 layt3Tmrstu $Ifgd3Tuvw0+++gd3TkdQ+$$Ifl    ֞v gXI:+"0    4 layt3T> 1!2!!!I"d"e"f""""#m##########$ $Ifgd3Th^hgdRJgd3T  ! & ' : ; < = > B C V W X Y ^ _ r s t u z { ϶šsdUFjG K h CJUVaJj9h h CJEHUj! 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Cos "1 (2)=undefined FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjfObjInfoEquation Native >"(>// 2>1 FMicrosoft Equation 3.0 DS Equation Equation.39q/R/ "4_1330757028FOI`OI`Ole CompObjfObjInfoEquation Native :_1330757050FOI`OI`Ole CompObjf FMicrosoft Equation 3.0 DS Equation Equation.39q8;/ / 23 FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoEquation Native :_1330757074FOI`OI`Ole CompObjfObjInfoEquation Native :_1330757100FOI`OI`/R/ "4 FMicrosoft Equation 3.0 DS Equation Equation.39q/̶/ 3 FMicrosoft Equation 3.0 DS EqOle CompObjfObjInfoEquation Native 6_1330757156FOI`OI`Ole CompObjfObjInfouation Equation.39q0/_0 58 FMicrosoft Equation 3.0 DS Equation Equation.39q^/}/ Sin "1Equation Native :_1330756884(FOI`OI`Ole CompObjfObjInfoEquation Native z_1330757175FOI`OI`Ole  (sin( 118 )) FMicrosoft Equation 3.0 DS Equation Equation.39q"X /_0 "38Lb {\hCompObjfObjInfoEquation Native >_941364135  FOI`OI`Ole PIC  LMETA PICT   :b {O n  .1  @`&  & MathType` "- - Q ^] Times New Roman- 2 44 2 l2 2  2 2 ~2 2 2 2 }2 Times New Roman- 2 <2p 2 I2p 2  2p 2 2p 2 2pTimes New Roman4- 2 $sink 2 cos 2 2 (~ 2  sink 2  cos 2 p)~ 2 nsin(k~ 2 o)~ 2 msink Times New Roman- 2 Gxb 2 Txb 2 xb 2 xb 2 xbTimes New Roman4- 2 uxSymbol- 2 >= 2 P= 2 O= & "Systemn-:  " " 1" e" ~" dPPNTTimes New Roman,Times New Roman .+ 4)A2) 2)F2)2)2dPPNTTimes New Roman (2)2)42)2)82dPPNTTimes New Roman ( sin)cos)*() sin)cos)))sin() ))sindPPNTTimes New Roman (x)x)3x)x)8xdPPNTTimes New Roman +-xdPPNTSymbol, Symbol( :=)Q=)7=dPPNT"System FMathType Equation Equation CompObj\ObjInfo Ole10Native_941364184 FOI`OI`Equation9q 4sin x2 cos x2 =2(2sin x2 cos x2 )=2sin(2 x2 )=2sinxL;{!hOle PIC LMETA hPICT P      "%&'(),./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXZ]^_`abdefghilnopqrstuvwxyz{|}~;{W   .1  @6&5 & MathTypeP Times New RomanL- 2 4(sin~k 2 cos 2 )~ 2 o sink 2  sink 2  cos 2 Ocos 2 k(sin~k 2 !cos 2 B%)~ 2 (sink 2 *cos 2 N1sin(k~ 2 05)~Times New Roman- 2 x 2 x 2 ! x 2 (x 2 @x 2 -x 2 x 2 $x 2 *x 2 1-x 2 v4xSymbol- 2 + 2 A = 2 + 2 9+ 2 == 2  + 2  &+ 2 A.= 2 80+ Times New RomanL- 2 L2p 2 @ 2p 2 L2p 2 2p 2 #2pTimes New Roman- 2 02 2 !'2 2 W/1 2 32 & "Systemn-P dPPNTTimes New Roman ,Times New Roman .+ (sin)$cos)))sin).sin)cos)"cos))(sin)*cos)))sin)cos)2sin() )dPPNTTimes New Roman( x)"x)1x)(x)x)'x),x)'x)-x)x):xdPPNTSymbol, Symbol( +),=)'+)A+)(=)++)++)B=)+dPPNTTimes New Roman (B2) 2)h2),2)'2dPPNTTimes New Roman ( z2)2)B1)"2dPPNT"System FMathType Equation Equation Equation9q@ (sinx+cosx) 2 =sin 2 x+2sinxcosx+cos 2 x=(sin 2 x+cos 2 x)+2sinxcosx=1+sin(CompObj!\ObjInfo#Ole10Native$D_941364262FOI`OI`2x)PT1d|$ #adxpr  #a"#a currentpoint ",Times .+ c))) cos()4)t)), Symbol)=) 2)cos (WOle *PIC +dMETA  PICT - |$    #a .  & Times zww 0wt f-!c Times ww 0wt f-!) !cos( !4 #Times {ww 0wt f-!t *Times ww 0wt f-!) .PSymbol |ww 0wt f-!= 5Times ww 0wt f-!2 ?!cos( FTimes }ww 0wt f-!2WTimes ww 0wt f-!( [!2 `Times ~ww 0wt f-!t fTimes ww 0wt f-!) jPSymbol ww 0wt f-!- pTimes ww 0wt f-!1 xPSymbol ww 0wt f-!= Times ww 0wt f-!2 !( !2 !cos( Times ww 0wt f-!2Times ww 0wt f-!t PSymbol ww 0wt f-!- Times ww 0wt f-!1 !) Times ww 0wt f-!2PSymbol ww 0wt f-!- Times ww 0wt f-!1 PSymbol ww 0wt f-!=Times ww 0wt f-!2 !(!4!cos(Times ww 0wt f-!4.Times ww 0wt f-!t4PSymbol ww 0wt f-!-:Times ww 0wt f-!4C!cos(KTimes ww 0wt f-!t\PSymbol ww 0wt f-!+bTimes ww 0wt f-!1j!)oPSymbol ww 0wt f-!-vTimes ww 0wt f-!1~PSymbol ww 0wt f-!=Times ww 0wt f-!8!cos(Times ww 0wt f-!4Times ww 0wt f-!tPSymbol ww 0wt f-!-Times ww 0wt f-!8!cos(Times ww 0wt f-!tPSymbol ww 0wt f-!+Times ww 0wt f-!2PSymbol ww 0wt f-!-Times ww 0wt f-!1PSymbol ww 0wt f-!=Times ww 0wt f-!8!cos(Times ww 0wt f-!4Times ww 0wt f-!t$PSymbol ww 0wt f-!-+Times ww 0wt f-!83!cos(:Times ww 0wt f-!tKPSymbol ww 0wt f-!+RTimes ww 0wt f-!1Z & '2 +()2)t)))-)1) =) 2)()2)cos (2 +t)-)1)) (2 +-)1(=) 2)()4)cos (.4 +t)-) 4)cos)t)+)1)))-)1)=) 8)cos (4 +t)-) 8)cos)t)+) 2) -)1)=) 8)cos (4 +t)-)8)cos)t)+)1/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 11296 div 1120 3 -1 roll exch div scale currentpoint translate 64 58 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (c) -12 324 sh (t) 1280 324 sh (t) 3206 324 sh (t) 5668 324 sh (t) 1603 963 sh (t) 2883 963 sh (t) 5501 963 sh (t) 6753 963 sh (t) 9305 963 sh (t) 10557 963 sh 384 /Times-Roman f1 (\)) 138 324 sh (cos\() 430 324 sh (\)) 1409 324 sh (cos) 2202 324 sh (\() 2873 324 sh (\)) 3335 324 sh (\() 4587 324 sh (cos) 4959 324 sh (\)) 6299 324 sh (\() 513 963 sh (cos) 891 963 sh (cos) 2338 963 sh (\)) 3516 963 sh (cos) 4789 963 sh (cos) 6208 963 sh (cos) 8593 963 sh (cos) 10012 963 sh 384 /Times-Roman f1 (4) 1083 324 sh (2) 1965 324 sh (2) 3008 324 sh (1) 3795 324 sh (2) 4393 324 sh (2) 4722 324 sh (1) 6129 324 sh (1) 6923 324 sh (2) 319 963 sh (4) 655 963 sh (4) 2102 963 sh (1) 3346 963 sh (1) 3976 963 sh (8) 4565 963 sh (8) 5984 963 sh (2) 7247 963 sh (1) 7780 963 sh (8) 8369 963 sh (8) 9788 963 sh (1) 11020 963 sh 224 ns (2) 2723 152 sh (2) 5480 152 sh (2) 6434 152 sh (4) 1416 791 sh (4) 5314 791 sh (4) 9118 791 sh 384 /Symbol f1 (=) 1642 324 sh (-) 3536 324 sh (=) 4070 324 sh (-) 5870 324 sh (-) 6664 324 sh (=) -4 963 sh (-) 1805 963 sh (+) 3086 963 sh (-) 3717 963 sh (=) 4251 963 sh (-) 5703 963 sh (+) 6956 963 sh (-) 7521 963 sh (=) 8055 963 sh (-) 9507 963 sh (+) 10760 963 sh end MTsave restore dqMATHeOE c)cos(4t)=2cos 2 (2t)-1=2(2cos 2 t-1) 2 -1=2(4cos 4 t-4cost+1)-1=8cos 4 t-8cost+2-1=8cos 4 t-8cost+1chFMicrosoft Equation Editor 2.0DNQE Equation.2 cos(4t)=2cos 2 (2t)-1=2(2cos 2 t-1) 2 -1=2(4cos 4 t-4cost+1)-1=8cos 4 t-8CompObjY\ObjInfo[Ole10Native\Equation Native ccost+2-1=8cos 4 t-8cost+1e c)cos(4t)=2cos 2 (2t)-1=2(2cos 2 t-1) 2 -1=2(4cos 4 t-4cost+1)-1=8cos 4 t-8cost+2-1=8cos 4 t-8cost+1L3{`h3{L   .1  @/&. & MathTypeP Times _941364399" FOI`OI`Ole jPIC !$kLMETA mHNew Romanl- 2 4sin(k~ 2 )~ 2 sin(k~ 2 f )~ 2  cos(~ 2 )~ 2 (~ 2 ?sink 2 cos 2 )(~~ 2 ^sink 2 )~ 2 "sink 2 $%cos 2 '(~ 2 !+sink 2 E.)~ 2 ~4 2 2 2  2 2  2 2 2 2 O2 2 y1 2 n2 2 !4 2 <(1 2 1*2 Times New Roman- 2 /2p 2 ,2pTimes New Romanl- 2 Dtk 2  tk 2 gtk 2 /tk 2 tk 2 tk 2 p$tk 2 @'tk 2 -tkSymbol- 2 = 2 = 2 Y- 2 b = 2 )- & "Systemn-0x xdPPNTTimes New Roman x,Times New Roman .+ sin()))sin()))cos()))() sin)cos))()sin)))sin)cos)()sin))( 4)2)2)$2)2) 2):1)2)24)51)2dPICT #&0CompObj\ObjInfo%'Ole10Native$PPNTTimes New Roman (2)w2dPPNTTimes New Roman ( t)5t)$t)6t)t)8t),t)t)4tdPPNTSymbol, Symbol( %=)Z=)T-)0=)F-dPPNT"System FMathType Equation Equation Equation9q  sin(4t)=2sin(2t)cos(2t)=2(2sintcost)(1-2sin 2 t)=4sintcost(1-2sin 2 t)L)Ht)vM 6  _1330756897*FOI`OI`Ole PIC ),LMETA  .1   @%&% & MathType "-E EEZEvm  #9Pv {   Times New Roman - 2 4e 2 tk 2  tk 2 tk 2 1"tk 2 tk 2 tk 2 tk 2 {#tk 2 Xtk 2 X8tk Times New Roman- 2 t>Times New Roman - 2 )~ 2 &sink 2 (sin~k 2 p )~ 2 ( (~ 2 )~ 2 8(~ 2 cos(~ 2 f)~ 2 Ecos 2  (~ 2 "))~~ 2 T(~ 2 cos(~ 2  )~ 2 K (~ 2 cos(~ 2 ))~~ 2 (~ 2 cos(~ 2 e)~ 2 . cos(~ 2 $))~~ 2 XYcos(~ 2 X0)~ 2 X cos(~ 2 X)~ Times New Roman- 2 cos(bpWI 2 )I 2 4p 2  2p 2  2p 2 1p 2 A2p 2 i2p 2 +2p 2 !w1p 2 s4p 2 B 2p 2 y1p 2 i4p 2 y 1p 2 i 2p 2 y@1p 2 i<4p 2 y3p 2 i2p 2 ym1p 2 ii2p 2 3p 2 8p 2 1p 2 2p 2 1 1p 2 - 8pTimes New Roman - 2 1 2 2 2 2 2 k!2 2 1 2 2 2 22 2  1 2 ;4 2 2 2 2 2 "4 2 X2 2 Xr 4Symbol- 2 = 2  = 2  = 2 - 2 /+ 2 <= 2 - 2 K + 2  + 2 = 2 - 2 .+ 2 X<= 2 XZ- 2 X+ Symbol- 2 3-{ & "Systemn- FMicrosoft Equation 3.0 DS Equation Equation.39q e)sin 4 t=(sin 2 t) 2 =( 1-cos(2t)2 ) 2 = 1CompObj+.fObjInfoOle10Native-/Equation Native 4 (1-2cos(2t)+cos 2 (2t))= 14 (1-2cos(2t)+ 12 (1+cos(4t))= 14 ( 32 -2cos(2t)+ 12 cos(4t))= 38 - 12 cos(2t)+ 18 cos(4t)P DeskJet 850C/85(>// e)sin 4 t=(sin 2 t) 2 =( 1"cos(2t)2 ) 2 = 14 (1"2cos(2t)+cos 2 (2t))= 14 (1"2cos(2t)+ 12 (1+cos(4t))= 14 ( 32 "2cos(2t)+ 12 cos(4t))= 38 " 12 cos(2t)+ 18 cos(4t)LY!{hY!{N4 '  _9413647112 FOI`MRI`Ole PIC 14LMETA h.1  @@& & MathTypeP Times New Roman- 2 42 2 2 2  4 2 2 2 2 2 2 31 2 (2 2 2 2 X1 Times New Roman- 2 2p 2 2p 2 2pTimes New Roman- 2 $sink 2 (~ 2 n)~ 2 Mcos(~ 2 $ )~ 2 "sink 2 (~ 2 l)~ 2 sink 2 (~ 2 b)~Times New Roman- 2 tk 2  tk 2 tk 2 tkSymbol- 2 7+ 2  = 2 5+ 2 - 2 B= & "Systemn- dPPNTTimes New Roman ,Times New Roman .+ 2)2).4)2)2)1)2) 2)1dPPNTTimes New Roman (2)h2)G2dPPNTTimes New Roman ( sin)()))cos()))sin)()))&sin)())dPPNTTimes New Roman( 't).t):t)HtdPPNTSymbol, Symbol(PICT 36CompObj\ObjInfo57Ole10Native 2+).=):+)-)9=dPPNT"System FMathType Equation Equation Equation9q 2sin 2 (2t)+cos(4t)=2sin 2 (2t)+1-2sin 2 (2t)=1_9413647570\: FMRI`MRI`Ole PIC 9<LMETA       !"#$%&()*+,-./0123469:;<=>?@CFGHIJKLMNOPQRSVYZ[\]^_`abcfijklmnopqrstuvwxyz}L><x#><^M e  .1  8&8k & MathType Times New Roman4- 2 Lg 2 rx 2  x 2  x 2 Yx 2 x 2 ox 2 {x 2 K$x 2 +x 2 /x 2 4x 2 /7x 2 rx 2  x 2 x 2 x 2 x 2 ^x 2 e#x 2 )x 2 $.x 2 3xTimes New Roman0- 2 )~ 2 bsin(k~ 2 ,)~ 2 :sin(k~ 2 C)~ 2 Qsink 2 =cos(~ 2 \)~ 2 ;cos 2 Ssin(k~ 2 5 )~ 2 C"sink 2 %(~ 2 \(sink 2 +)~ 2 -cos 2 0(~ 2 2sink 2 4cos 2 7)~ 2 jsink 2 osink 2  sink 2 (~ 2 ;sink 2 )~ 2 sink 2 sink 2 ]!sink 2 b&sink 2 ,sink 2 !1sink 2 3 2  2 2 2 2 2 2 w%1 2 l'2 2 12 2 2 2 " 2 2 F1 2 2 2 m 2 2 r%2 2 D+3 2 104 Times New Roman4- 2 -*2p 2 q@3p 2 q 2p 2 q3p 2 q3(3p 2 q23pSymbol- 2  = 2  + 2 #= 2 %+ 2 != 2 W&- 2 ,+ 2 <= 2 j- 2  + 2 &- 2 = 2 - 2 W+ 2 ]$- 2 *= 2 /- & "Systemn-v& &dPPNTTimesPICT ;>'vCompObj5\ObjInfo=?7Ole10Native8$ New Roman &,Times New Roman .+ g)*x))x)x)'x)"x)&x)!x)&x)6x)'x)"x)x(!x)-x)(x).x)'x)-x)(x)-x))x)-xdPPNTTimes New Roman( )) sin()))sin()0))sin)cos()!))cos)sin()))sin)()sin)))cos)() sin)cos))(! sin)(sin).sin)()sin)))sin)(sin)-sin)(sin).sin)(sin( %3)@2)I2)G2)71)2)N2(!,2)-2)!1)\2)-2))2).3)(4dPPNTTimes New Roman (Q2(B3)V2)T3)V3)V3dPPNTSymbol, Symbol ( 8=)%+)=)H+)H=)*-)2+(!=)!-)-+)1-)+=)"-)-+)(-).=)(-dPPNT"System FMathType Equation Equation Equation9q  g)sin(3x)=sin(x+2x)=sinxcos(2x)+cosxsin(2x)=sinx(1-2sin 2 x)+cosx(2sinxcosx)=sinx-2sin 3 x+2sinx(1-sin 2 x)=sinx-2sin 3 x+2sinx-2sin 3 x=3sinx-4sin 3 x, FMicrosoft Equation 3.0 DS Equation Equation.39q| cot(2)"tan(2)sec(2)csc(2)=cos(_1267470636EBFMRI`MRI`Ole ACompObjACBfObjInfoDDEquation Native E_1267470850GFMRI`MRI`Ole TCompObjFHUf2)sin(2)"sin(2)cos(2)1cos(2)"1sin(2)=cos(2)sin(2)"sin(2)cos(2)1cos(2)"1sin(2)" sin(2)cos(2)sin(2)cos(2)=cos 2 (2)"sin 2 (2)1=cos(2(2))=cos(4) FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoIWEquation Native X_1330420556OLFMRI`MRI`Ole dD      !"#$%&'()*+,-./0123456789:;<=>?@ABoGEFIKdJLNMOQPRTSUVWYXZ[\^]_`abecfghijklmpqrstuvwxyz{|}~0_ cot 2 (3)"1csc 2 (3)=cos 2 (3)sin 2 (3)"11sin 2 (3)=cos 2 (3)sin 2 (3)"11sin 2 (3)" sin 2 (3)sin 2 (3)=cos 2 (3)"sin 2 (3)1=cos(2(3))=cos(6) FMicrosoft Equation 3.0 DS Equation Equation.39q  644444 la4yt 0$$If4!v h55595#55555 05 5 `5 5 t#v#v#v9#v##v#v#v#v#v 0#v #v `#v #v t:V l0    65 4a4yt kd$$$Ifl    " r P _ !'0    644444 la4yt 0$$If4!v h55595#55555 05 5 `5 5 t#v#v#v9#v##v#v#v#v#v 0#v #v `#v #v t:V l0    65 4a4yt kd$$Ifl    " r P _ !'0    644444 la4yt \Dd Llb H c $AH? ?3"`?42ik> }̟fX `!zik> }̟fXH HxڕQ1K@}6 ҡ8SU[#L* 8ttSJ`.Px};"`l P\&W&Ȑ4Ml!946-kkH.wF# x2ؓig CWU+z6%Zf4=C~\>oǝY]AܹBey12K4īIQM8|$<\;}"Sf'#OnѰ [T ?5$ s?'9pٛ_ԃAgSM h$qQ}u<(~U0=/^4[Dd Llb G c $AG? ?3"`?52#Eh*3"zUB `!y#Eh*3"zUH GxڕQJA};%Q* ^UIa'x!" ŕVbimO'(v ow/W4.efyoW m6ŕ "!R$(Ib5=+d"^bc/b ehl=v]y<~q{@OBˌe,6떎I'h}qw00/릧Y5A,eEU(y&@|(8\/ξJHz,EqN~*s?'39p;;_dM jDݞ%ʘIj{_ eDd <lb I c $AI? ?3"`?62{Lt4JNGC `!{Lt4JNGCh` QxڕQMK@}ۯBDXTz &Z (0t:4*0>I1lW;HQ%ǨB L_A|M8? 7,"K+ssŸV\EL`s>]2B7ƿfH!0q" '\x(.v0o8+KRsePdk)YO`~d}Dd $lb K c $AK? ?3"`?82XGU0Ai `!XGU0AHD ixcdd``a 2 ĜL0##0KQ* Wy A?d6@=P5< %!@5 @_L ĺE^X@%V;35V ZZ QӡV'-b&#.#' ~3 o1c +aꃘA|J ɳ pwlaeMa`2 2w^]lE1Ykns1˄l.½$̽P: Cpf^*4p8 0y{)I)$5!d2P"CXHgt_0ܰ&Dd b @ c $A@? ?3"`?92; Qx!BEp `!; Qx!BEp h|xڝRJA=w&.*bU$XIaaHHE׈ ȂhegXH>!.bgu mawb?" 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