专题4.5函数y=Asin(ωx+φ)的图象及应用六大题型（原卷版）_11.4-2_2026-11-04

      
      
专题4.5 函数y=Asin(ωx+φ)的图象及应用【六大题型】

 1  
 y=Asin(ωx+φ) !"#$ ........................................................................................................2
 2  %&' !()
 y=Asin(ωx+φ)*+, ....................................................................................3
 3   !-./012 ........................................................................................................................5
 4  
3456789: ............................................................................................................6
 5  ;<
= ........................................................................................................................................7
 6  
 y=Asin(ωx+φ)-;<>?#$012 ................................................................................9
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 y=Asin(ωx+φ) !"2
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 y=Asin(ωx+φ)m;<
n
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 y=Asin(ωx+φ)
 !#$^"%&' !B
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^}@~€,‚Lƒ„…
.
†‡4 1  
 y=Asin(ωx+φ) !"#$
1ˆ
 y=Asin(ωx+φ) !‰Š
‰
 y=Asin(ωx+φ)(A>0Lω>0) !‹ŒŽ6Š
(1)4Š‰ 
“4Š”‰ y=Asin(ωx+φ)[ LzAm“”#•–$L— z=ωx+φL% z ˜
wB™š2 xL“”›œLEž™4Ÿ L¡4¢ž™ !Q
(2) !#$Š
%
 y=sin x  !“”#$ž£ y=Asin(ωx+φ) !¤Ž¥¦
“§¨©¢ª
«”-“§ª«¢¨©”.
2ˆ;<
 !#$:B*6Š
*Z;<
 !#$:Ž6Š'¬­§¨©¢ª«®§ª«¢¨©.¯*°±²Œ
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 !#$ž£¸³·
 !Q
      
      
(2)#¹º

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(3)6Š
½}#$6Š.
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*+,*6Š
1ˆ%&' !()
*+,6Š
% y=Asin(ωx+φ)(A>0Lω>0)³¾ !B¿*+,ÀLA ÁÂpÃ% ž™LămBÅ)Æ
 ω
® φL‹ŒŽ6Š
(1)ŒÇ !È(əMef T ÊË®“34”Ÿ LÌÍ%
½]B™ ωQ() φ ÀLÎRB
™ÏÐ4ÑtÒÓ !ÔÕ(֍×)34؟ 
LÙÚ
½]B™ φ.
(2)–Û4Ÿ ._³Ü݆4(Ñ4@ÑÞ4Ö34)Ÿ –Û*+,LßG1 €*™ ω ® φL
ÎV ALω à	Ö φ áâ¤ãäBL]åæ,#$ç¿à1AB.
†‡4 3  ;<
 !@./012*èé
1ˆêë
 y=Asin(ωx+φ)./ìí
êë y=Asin(ωx+φ)./À]µ ωx+φ î­³·ïIL_$ðŠ®
€G1ñòóô*.
2ˆ
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 !ö4·
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:QúmûJN:ü!õý
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6Šìí-þG
1ˆ
 y=Asin(ωx+φ)+k !¨©ÿĀ
“āĂÒăLÔÍă”.
2ˆ% y= sinωx £ y=sin(ωx+φ)(ω>0Lφ>0)#$
{ā¨©
·\Ąą„Ćć φ ·\Ąą„.
 1  
 y=Asin(ωx+φ) !"#$
K 152024·ĈĉĊ)·;=9µ
𝑓(𝑥) = sin 2𝑥−π
3  !{ā¨©
π
3·\Ąą„Lž£
𝑔(𝑥) 
!LÙ𝑔(𝑥) = 5    9
Aˆsin2𝑥
Bˆ−sin2𝑥
Cˆsin 2𝑥+ π
3
Dˆcos 2𝑥+ π
6
#, 1-152024·ċČČč·=ĎďĐ9µ
𝑓(𝑥) = sin 2𝑥−π
12  !{ā¨©
π
8·\Ąą„¢Lž£

𝑔(𝑥) !LÎ
𝑔(𝑥)øđĒ 0, 𝑎
3 ® 4𝑎, 7π
6 Ôē\ĔĕĖLÙJ
𝑎˜ėáâm5    9
Aˆ π
6 , 7π
24
Bˆ π
6 , π
2
Cˆ 7π
24 , π
2
Dˆ π
12 , 7π
24
      
      
#, 1-252024·ĘęĚč·=ĎďĐ9µ
𝑓(𝑥) = cos 2𝑥−π
6 !Ôã¤4{ā¨©
5π
6 ·\Ąą„L
ž£
𝑔(𝑥)  !LÙ5    9
Aˆ𝑔(𝑥) = cos 2𝑥−2π
3
Bˆ𝑔(𝑥)ø −π
3 , π
3 Ô\ĔĕĖ
Cˆ𝑔(𝑥)ø 0, π
3 ÔÑËė­ 3
2
DˆěĜ𝑥=
π
4m𝑔(𝑥) !³ĝV»Ğ
#, 1-352024·nğ·=ĎďĐ9݆
𝑓(𝑥) = sin(4𝑥+ 𝜑) |𝜑| < π
2 L§µ
𝑓(𝑥) !{Ò¨©
π
12
·\Ąą„Lßµ !Ôã¤4؟ ªą£Ðw 2 ĠLġŸ …#L½]ž£
𝑔(𝑥) !ˆÎ

𝑔(𝑥) !Ģ y ĞV»LÙ𝑓π
8 = 5    9
Aˆ
1
2
Bˆ−
1
2
Cˆ 3
2
Dˆ−3
2
 2  %&' !()
 y=Asin(ωx+φ)*+,
K 252024·ĘČģĤ·=ĎďĐ9
𝑓(𝑥) = 𝐴sin(𝜔𝑥+ 𝜑) 𝜔> 0,𝐴> 0,|𝜑|< π
2 ø³·efo !Œ
 ãĥLٍ›ĦŠħ(m5    9
Aˆ𝑓(𝑥) = 2sin 2𝑥+ 3π
4
Bˆ𝑓(𝑥) = 2sin 2𝑥−π
4
Cˆ𝑓(𝑥) !{Ò¨©
3π
8 ·\Ąą„¢ž£
mĨ
Dˆ𝑓(𝑥) !{Ò¨©
3π
8 ·\Ąą„¢ž£
mĩ
#, 2-152024·nğ·;=9݆
𝑓(𝑥) = 𝐴sin(𝜔𝑥+ 𝜑) 𝐴> 0,𝜔> 0,−𝜋
2 < 𝜑< 𝜋
2 &' ĪŒ ã
ĥLÎ𝑓(𝜃) =
1
3LÙ𝑓2𝜃+ 5𝜋
3
= 5    9
      
      
Aˆ−
2
9
Bˆ
2
9
Cˆ−
7
9
Dˆ
7
9
#, 2-252024·ċČčī·=ĎďĐ9݆
𝑓(𝑥) = cos(𝜔𝑥+ 𝜑) 𝜔> 0,|𝜑|< π
2 &' !Œ ãĥL
¿Ĥ𝐴π
9 ,1 L𝐵5π
18 ,0 LÙ5    9
Aˆ𝜔= 2
Bˆ𝜑= −
π
6
CˆěĜ𝑥=
5π
12m𝑓(𝑥) !³ĝV»Ğ
Dˆ 11π
18 ,0 m𝑓(𝑥) !³·V»ĤĬ
#, 2-352024·ċČĭĮ·=ĎďĐ9µ
𝑓(𝑥) !؟ #­Ðw 2 Ġ5ġŸ …#9Lį
¢ß{ā¨©
π
12·\Ąą„Lž£
𝑔(𝑥) = 𝐴sin(𝜔𝑥+ 𝜑) 𝐴> 0,𝜔> 0,|𝜑|< π
2 &' !Œ ãĥLÙ

𝑓(𝑥)*+,­5    9
Aˆ𝑓(𝑥) = 3sin 4𝑥+ π
2
Bˆ𝑓(𝑥) = 3sin 4𝑥+ π
6
Cˆ𝑓(𝑥) = 3sin 𝑥+ π
6
Dˆ𝑓(𝑥) = 3sin 𝑥+ π
2
      
      
 3   !-./012
K 352024·nğ·;=9Œ L
𝑓(𝑥) = 𝐴sin(𝜔𝑥+ 𝜑) 𝐴> 0,𝜔> 0,|𝜑| ≤π
2  Ī-𝑥Ğ¿ĤŽ·
ö4'¬­ ALBL- y ĞöĢ4 CLD ­Ĝ¾𝐵𝐶Ĥ4L|𝑂𝐵| =
3|𝑂𝐶|L|𝑂𝐴| = 2,|𝐴𝐷| = 2 21
3 Lٍ›Ħ
Šħ(m5    9
  
Aˆ𝑓(𝑥)ÑËħef­12π
Bˆ𝑓(𝑥) !ĢěĜ𝑥= 8V»
Cˆ𝑓(2) = 𝑓(−4)
Dˆ𝑓(−𝑥+ 2)­Ĩ
#, 3-152024·ċČČč·=ĎďĐ9݆
𝑓(𝑥) = 2sin(𝜔𝑥+ 𝜑)(𝜔> 0,|𝜑| < π)&' !Œ ãĥL
µ
𝑓(𝑥) !{ā¨©
π
6·\Ąą„¢ž£
𝑔(𝑥) !LÙø›đĒÔ
𝑔(𝑥)\ĔĕĖm
5    9
  
Aˆ π
6 , π
3
Bˆ 3π
2 , 5π
2
Cˆ 5π
6 , 7π
6
Dˆ π, 3π
2
#, 3-252024·İıýIJ·=ĎďĐ9݆
𝑓(𝑥) = 𝐴sin(𝜔𝑥+ 𝜑)(𝐴> 0,𝜔> 0,|𝜑|< π)&' !Œ
 ãĥLٍ›Gijħ(Ĵ	m5    9
①
𝑓(𝑥) !Ģ4 π
6 ,0 ýĤĬV»Q
      
      
②
𝑓(𝑥)*+,]^­𝑓(𝑥) = 2cos 2𝑥−2π
3 Q
③
𝑓(𝑥)ø π
12 , 13π
24 Ôėĵ­[0,2]Q
④Îû𝑓(𝑥) ĪÔã¤4؟ «Ķ­Ðw
2
3ĠLġŸ …#Lß{Ò¨©
π
12·\ĄLÙãž
m
𝑦= 2sin 3𝑥+ π
12 .
Aˆ①③
Bˆ②③
Cˆ③④
Dˆ①④
#, 3-352024·ķĸ·;=9݆
𝑓(𝑥) = 𝐴sin(𝜔𝑥+ 𝜑) 𝐴> 0,𝜔> 0,|𝜑|< π
2 L !Œ ãĥL›
ĦŠħ(m5    9
Aˆ
𝑓(𝑥)Ĺĺm2LĻšm
π
6
BˆÎ
𝑓(𝑥) !Ôã¤4{ā¨©
π
12¢LV2
­ĩ
LÙ𝜔= 2
CˆÎ
𝑓(𝑥)ø π
3 , π
2 Ô\ĔĕăLÙ𝜔˜ėáâ­ 2, 10
3
DˆÎ
𝑓(𝑥) !Ģ 7π
12 ,0 ĤĬV»LÙ
𝑓(𝑥)ÑËħef𝑇ÑËė­7π
 4  
3456789:
K 452024·ĘČģļ·ú=9µ
𝑓(𝑥) = 2sin 3𝑥+ π
4  !{Ò¨©𝜑5𝜑> 09·\Ąą„Lž£

𝑔(𝑥) !LÎ
𝑔(𝑥)øđĒ(0,𝜑)ÔĽ¤Ž·34LÙ𝜑˜ėáâm5    9
Aˆ 5π
12 , 3π
4
Bˆ 3π
4 , 13π
12
Cˆ 5π
12 , 3π
4
Dˆ 3π
4 , 13π
12
#, 4-152024·ĘČąľ·³=9݆
𝑓(𝑥) = 𝐴sin(𝜔𝑥+ 𝜑)(𝐴> 0,𝜔> 0,|𝜑| <
π
2)&' !Œ ã
ĥLÎ67𝑓(𝑥) = 𝑚ø[−
π
2,0]Ô¤Ž·…š?J
8LÙJ
 m ˜ėáâm5    9
      
      
Aˆ[−2,−3]
Bˆ(−2,−3]
Cˆ(−2,−1]
Dˆ[−2,−1]
#, 4-252024·ċČčī·=ĎďĐ9µ
𝑓(𝑥) = sin𝑥 !{ā¨©
π
6·\Ąą„Lßµãž
 
!Ôã¤4؟ #­Ðw
1
𝜔(𝜔> 0)ĠLġŸ …#Lž£
𝑔(𝑥) !LÎ
𝜑(𝑥) =
𝑥2 + 2𝑥,𝑥≤0
𝑔(𝑥),𝑥> 0
 
ø(−∞,3π]Ô¤ 5 ·34LÙ𝜔˜ėáâm5    9
Aˆ 17
18 , 23
18
Bˆ 17
18 , 23
18
Cˆ 11
18 , 17
18
Dˆ 11
18 , 17
18
#, 4-352024·ĿŀŁł·³=9µ
𝑓(𝑥) !؟ ªą­Ðw 2 ĠLß{ā¨©
π
3\ĄLž£

𝑔(𝑥) = sin(2𝑥+ 𝜑) 0 < 𝜑< π
2 &' !5Œ ãĥ9ˆVĢ∀𝑥1L𝑥2 ∈[𝑎,𝑏]LŃ𝑥1 ≠𝑥2LÎ𝑔(𝑥1) = 𝑔
(𝑥2)LIJ¤𝑔(𝑥1 + 𝑥2) =
3
2 ýńLٍ›GijĤ…ħ(m5    9
  
Aˆ𝑔(𝑥) = sin 2𝑥+ π
3
Bˆ𝑓(𝑥) = sin 4𝑥−π
3
Cˆ𝑔(𝑥)ø π, 3π
2 Ô\ĔĕĖ
Dˆ
𝑓(𝑥)ø 0, 4π
3 34­𝑥1,𝑥2,⋯,𝑥𝑛LÙ𝑥1 +2𝑥2 +2𝑥3 +⋯+ 2𝑥𝑛−1 + 𝑥𝑛=
85π
12
 5  ;<
=
K 552024·İıŅĘ·;=9ņĿŇm³ÊõŇňʼnŊaŋ—ŌLōŎŸøņĿŇŏŐőŒŒ
œŔÔõL]^sŕŖŗİeŘřˆŚņĿŇÑ4śÏœù„­ 120mLõŜ즭 110mL—ŝ 48
      
      
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30minˆŚōŎŸÔņĿŇŏŐ 10min ¢śÏœù„ŧ­5    9
Aˆ92.5m
Bˆ87.5m
Cˆ82.5m
Dˆ 55 3
2
+ 65 m
#, 5-152024·ĘČ·=ĎďĐ9Ś/4Ą©𝑦(cm)-ŨũÀĒ𝑥(s)Æ,­𝑦= sin(𝜔𝑥+ 𝜑)
(𝜔> 0,𝜑∈(−π,π))L¿ !Œ ãĥL !-𝑦Ğö4Ÿ ­ 0,−
3
2 L-ěĜ𝑦=
1
2šŪ;·ö4؟
 ūŬ­
π
6L
7π
18L
5π
6 Lٍ›ĦŠħ(m5    9
Aˆ𝜔= 4
Bˆ𝜑= −
2π
3
Cˆ/4ø 1, 3
2 soĄ© !­\Ĕĕă
Dˆ/4ø 0, 7π
18 soŭ”Ů7­(3−3)cm
#, 5-252023·hi·=ĎďĐ9ůŰűŲųŴ-ŵŶŷŸLŹŲųűźwźŻ£nî.ďE£ 2025
ghżŹűĖŽʼn•ž£ 111.2GWLĤiŽʼn•ſÁž£ƀƁl³.݆ŹŤƂ)ÀŹŲųűʼnƃƄâ
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ƋõĞĤĬśÏē­ 20 ƌLĻƍÀb𝐴ŕĢãøųűʼnõĞĤĬħÔ6L𝐵ŕĢãøųűʼnõĞĤĬħ
6LŃ𝐴®𝐵âƅƊƋųűʼnõĞĤĬƆţŤƇeŨũ.%ĢŽ·ųűʼnãŕĄŝŹŤ…¹L𝐴4õŤ­5
πm/sL𝐵4õŤ­8πm/sL^ÀĒ𝑡5\Ą
Ǝ9­Ƌ#•L𝐴®𝐵-ƊƋųűʼnõĞĤĬ„Ə­2#•L
'¬ž;<
𝑓(𝑡)-𝑔(𝑡)L›¶6,]^ç𝑓(𝑡)#­𝑔(𝑡)5    9
      
      
Aˆµ𝑓(𝑡) !Ôã¤4{Ò¨©π·\Ąą„Lߵ؟ ƐÊ£Ðw
8
5Ġ
Bˆµ𝑓(𝑡) !Ôã¤4{ā¨©π·\Ąą„Lߵ؟ «Ë£Ðw
5
8Ġ
Cˆµ𝑓(𝑡) !Ôã¤4؟ ƐÊ£Ðw
8
5ĠLß{ā¨©
5
2·\Ąą„
Dˆµ𝑓(𝑡) !Ôã¤4؟ «Ë£Ðw
5
8ĠLß{Ò¨©
5
2·\Ąą„
#, 5-352024·hi·=ĎďĐ9Œ L³·ƑƒŠšÀŢ6{õũˆ—ƑƒÔŚ·ƓƔƑ𝑊£Ɣù
śÏ­𝑑5\Ą
ƌ95øƔùLÙ𝑑­ƕ
9ˆÎ^ƓƔƑ𝑊ƖƗ™ƔùÀŞƍEÀĒL𝑑-ÀĒ𝑡
5\Ą
'Ƙ9ƙĒÆ­𝑑= 4sin 2𝑡−π
6 +2ˆŚÀb𝑡05\Ą
'Ƙ9ÀLƓƔƑ𝑊ø”4𝑂5𝑂­Ƒ
ƒĞĬ9ƚěěĜāÓLŃ£ƔùśÏ­ 5 ƌLÙßƛ”
π
6'Ƙ¢LƓƔƑ𝑊5    9
AˆøƔù
BˆøƔùÔ
CˆĽƜŞƍÛƔ
DˆĽƜŞƍ™Ɣ
 6  
 y=Asin(ωx+φ)-;<>?#$012
K 652024·ĈĉƝƔ·=ĎďĐ9݆
𝑓(𝑥) = sin 2𝑥+ 𝜋
3 + cos 2𝑥+ 𝜋
6 −2sin𝑥cos𝑥.
(1)B
𝑓(𝑥)ÑËħef"V»Ğ67Q
(2)µ
𝑦= 𝑓(𝑥) !{ā¨©
𝜋
12·\ĄLߵ㞠!ÔƊ4ġŸ …#､؟ ªą­Ðw 2 ĠL
ž£
𝑦= 𝑔(𝑥) !LB𝑦= 𝑔(𝑥)ø[0L2π]Ô\ĔĕăđĒ.
      
      
#, 6-152024·ƞęƞƟ·=ĎďĐ9݆
𝑓(𝑥) = 2sin𝑥cos𝑥−2 3sin2𝑥+
3.
(1)Î𝑥∈0, π
4 ÀL𝑚< 𝑓(𝑥)>ýńLBJ
𝑚˜ėáâQ
(2)µ
𝑓(𝑥) !؟ «Ë­Ðw
1
2LġŸ …#Lßµ¿{Ò¨©
π
6·\ĄLž£
𝑔(𝑥) !.
Î𝑥∈[0,𝑡]L
𝑔(𝑥)¤ŃƠ¤ 4 ·34LBJ
𝑡˜ėáâ.
#, 6-252024·ĘČơƢ·;=9݆
𝑓(𝑥) = 𝐴sin(𝜔𝑥+ 𝜑) 𝐴> 0,𝜔> 0,0 < 𝜑< π
2  !]%
𝑦= 3sin𝑥 !¨©ž£LŃĢěĜ𝑥=
π
3V»ˆ
(1)B𝑓7π
12 ėQ
(2)B
𝑔(𝑥) = 𝑓2𝑥+ π
6 +𝑓2𝑥−π
6 ,𝑥∈[0,π]\ĔĕĖđĒˆ
#, 6-352023·ƣƤƥƦƧƨ·;=9݆
𝑓(𝑥) = 2sin𝜔𝑥cos𝜑+ 2sin𝜑−4sin2𝜔𝑥
2 sin𝜑
(𝜔> 0,|𝜑|< π)L¿ !³ĝV»Ğ-šŪV»ĤĬ؟ šƏ
π
4L______Ls^Ž·ĝƩĤƪ³
·ƫƬøƭØĜĤ.①
𝑓(𝑥) !{ā¨©
π
3·\Ąą„¢ž£ !Ģ y ĞV»Ń𝑓(0) < 0Q②

𝑓(𝑥) !³·V»ĤĬ­
π
12 ,0 Ń𝑓π
6 > 0.
(1)B
𝑓(𝑥)*+,Q
(2)µ
𝑓(𝑥) !Ôã¤4؟ #­Ðw
1
𝑡(𝑡> 0)ĠLġŸ …#Lž£
𝑦= 𝑔(𝑥) !LÎ

𝑦= 𝑔(𝑥)øđĒ 0, π
3 ÔĽ¤ 3 ·34LB t ˜ėáâ.
      
      
³@\
1ˆ52024·ĘęƮƯ·;=9­Mž£ 𝑦= sin2𝑥+ cos2𝑥 !LưAû 𝑦=
2cos2𝑥 !Ôã¤4
5     9
Aˆ{Ò¨ô©ũ 
π
8 ·\Ąą„
Bˆ{ā¨ô©ũ 
π
8 ·\Ąą„
Cˆ{Ò¨ô©ũ 
π
4 ·\Ąą„
Dˆ{ā¨ô©ũ 
π
4 ·\Ąą„
2ˆ52024·ƱĉƲƳ·=ĎďĐ9Î
𝑓(𝑥) = sin(2𝑥+ 𝜑)50 < 𝜑< π9{āħ©𝜑·\Ą¢øđĒ 0, π
2 Ô
\ĔĕĖLÙ𝜑= 5    9
Aˆ
π
3
Bˆ
π
2
Cˆ
π
6
Dˆ
2π
3
3ˆ52024·İıƋƴ·;=9
𝑓(𝑥) = 𝐴sin(𝜔𝑥+ 𝜑)5𝜔> 0L|𝜑| <
π
29&' !Œ ãĥL𝑓(𝑥) !
- y ĞöĢ M 4L- x ĞöĢ C 4L4 N ø𝑓(𝑥) !ÔL4 M@N Ģ4 C V»L›ĦŠƵƶm
5    9
Aˆ
𝑓(𝑥)ÑËħefmπ
Bˆ
𝑓(𝑥) !Ģ4 5π
6 ,0 V»
Cˆ
𝑓(𝑥)ø −π
2 ,−π
6 \ĔĕĖ
Dˆ
𝑓(𝑥) !{Ò¨©
π
6¢Lž£
𝑔(𝑥) !LÙ𝑔(𝑥)­ĩ
4ˆ52024·oƷƸƹ®ƺƻ·ú=9݆
𝑓(𝑥) = 2 2sin𝑥cos 𝑥+ π
4 LƼ™›İ·ƽĤLħ(m
5    9
Aˆ
𝑓(𝑥)ÑËħefm2π
      
      
Bˆ
𝑓(𝑥)øđĒ π
8 , 5π
8 Ômă
Cˆ
𝑓(𝑥) !Ģ4 −π
8 ,0 V»
Dˆ
𝑓(𝑥) !]%
𝑦=
2sin2𝑥 !{Ò¨©
π
8·\ĄLß{¨© 1 ·\Ąž£
5ˆ52024·İıýIJ·;=9øĤLûIŻ£Ų5þmÉ{¨ƝĄŝ9ħÁĢƾÏŞ¨ƝĄŝ
śÏŨũ»­“[ƿŨũ”.ø¨ùě<Ÿ ÆLŚ·[ƿŨũ]^
𝑓(𝑥) = 𝐴sin(𝜔𝑥+ 𝜑)5𝐴> 0L
𝜔> 0L|𝜑| < π9wœĥL¿&' !Œ ãĥLٍ›Gijħ(Ĵ	m5    9
①
𝑓(𝑥) !Ģ4 π
6 ,0 ýĤĬV»Q
②
𝑓(𝑥)*+,]^­𝑓(𝑥) = 2cos 2𝑥−2π
3 Q
③
𝑓(𝑥)ø π
12 , 13π
24 Ôėĵ­[0,2]Q
④Îû𝑓(𝑥) !Ôã¤4؟ «Ķ­Ðw
2
3ĠLġŸ …#Lß{Ò¨©
π
12·\ĄLÙãž
m
𝑦= 2sin 3𝑥+ π
12
Aˆ①③
Bˆ②③
Cˆ③④
Dˆ①④
6ˆ52024·ċČčī·=ĎďĐ9µ
𝑓(𝑥) = sin𝑥 !{ā¨©
π
6·\Ąą„Lßûãž
 !؟
 #­Ðw
1
𝜔(𝜔> 0)ĠL]^ž£
𝑔(𝑥) !LÎ𝑔(𝑥)ø π
6 , π
2 Ôǀ¤34LÙ𝜔˜ėáâm5    9
Aˆ 0, 5
3
Bˆ 5
3 ,3
Cˆ(0,3]
Dˆ(3, + ∞)
7ˆ52024·ĘČģĤ·=ĎďĐ9Œ ãĥǁǂǃDŽDžĜLdžLJLjljNJDžĜ5ĪƱùÔÞNjnjËƯø
ƔĤǍW-Ƌǎ€ý €Lq]^€!œ»ƾ­ǍWDžĜ9LǏ”š¹ĒǐLƾĹĺǑ#³ŬL
є4𝑀3π
4 , 3
2 L¿V267­|𝑦| = 2−1
2
2𝑥
π
|sin𝜔𝑥|5𝑥≥0L1 < 𝜔< 39L¿Ĥ[𝑥]­…ǒ” x ÑÊ
ï
.ÎǓljNJDžĜÔ³4 N ؟ ­
7π
6 LÙ4 N ġŸ ­5    9
      
      
Aˆ ± 1
Bˆ ±
2
2
Cˆ ±
1
2
Dˆ ±
3
2
8ˆ52024·İıǔƬ·=ĎďĐ9µ
𝑓(𝑥) = sin 𝜔𝑥−π
6 (0 < 𝜔< 6) !{Ò¨©
π
12·\Ąą„¢ž£

𝑔(𝑥) !LÎ 0, π
𝜔m𝑔(𝑥)³·\ĔĕĖđĒLÙ5   9
Aˆ𝑓(𝑥)ÑËħef­𝜋
Bˆ
𝐹(𝑥) = 𝑓(𝑥) +𝑔(𝑥)ÑÊė­ 1
Cˆ𝑓(𝑥)ø π
2 , 2π
3 Ô\Ĕĕă
Dˆ67𝑓(𝑥) = −
1
2ø[0,π]Ô¤ 5 ·J
8
ú@Ǖ
9ˆ52024·ǖǗǘ·;=9݆
𝑓(𝑥) =
3sin𝑥cos𝑥−cos2𝑥+
1
2Lٍ›ĦŠħ(m5    9
Aˆ𝑓(𝑥) = sin 2𝑥−π
6
Bˆ
𝑓(𝑥)ÑËħef­2π
Cˆ𝑥=
π
3m
𝑓(𝑥) !³ĝV»Ğ
Dˆ
𝑓(𝑥) !]%𝑦= sin2𝑥 !{Ò¨©
π
12·\Ąą„ž£
10ˆ52024·ƞČǙƟ·=ĎďĐ9݆
𝑓(𝑥) = 𝐴sin(𝜔𝑥+ 𝜑)(𝐴> 0,𝜔> 0,0 < 𝜑< π)&' !Œ ãĥL
Ú𝑔(𝑥) = 𝑓(𝑥)−cos2𝑥Lٍ›ĦŠħ(¤5    9ˆ
Aˆ𝑔(𝑥)³·V»ĤĬ π
6 ,0
Bˆ𝑔(𝑥)V»Ğ67­𝑥=
𝑘
2π +
π
3(𝑘∈𝑍)
Cˆ𝑔(𝑥)ø 0, π
2 Ôėĵ­ −1, 1
2
      
      
Dˆ𝑔(𝑥)\ĔĕăđĒ­ 𝑘π−π
6 ,𝑘π + π
3 (𝑘∈𝑍)
11ˆ52024·ƞČǔĸ·³=9ņĿŇm³ÊõŇňʼnŊaŋ—ŌLōŎŸøņĿŇŏŐőŒŒœŔ
ÔõL]^sŕŖŗİeŘřˆŚņĿŇÑ4śÏœù„­ 110 ƌLõŜ즭 100 ƌLņĿŇ
ƇeÔēţœčŽM 36 ·ŏŐLōŎjsśÏœùÑtĄŝóŐLŞş¢ņĿŇŠšÀŢ6{ţŤťõL
Şƍõũ t 'Ƙ¢śÏœù„­ H ƌLǚ𝑡= 15ÀLōŎjůŐl³ŬõǛśÏœùÑǜŕˆŒ L^
ņĿŇĞĬ O ­Ð4L-œù¨ôěĜ­ x Ğańě<Ÿ ÆLÙ𝐻(𝑡) = 𝐴sin(𝜔𝑡+ 𝜑) +𝑏(𝐴> 0,𝜔> 0,
|𝜑| < π)L›ĦŠĤħ(m5    9
Aˆ𝐻Ģ𝑡
𝐻(𝑡)mĨ
BˆÎø𝑡1,𝑡2(𝑡1 ≠𝑡2)ÀbLōŎjśÏœù„š?LÙ𝑡1 + 𝑡2ÑËė­ 30
CˆņĿŇťõ³e”7ĤLōŎjśÏœù„…ÞĢ 85 ƌÀą­ 10 'Ƙ
DˆÎj@ǝŽōŎ'¬Ÿø𝑃,𝑄Ž·ŏŐőLюǞšǐ 5 ·ŏŐ5µŏŐî­ƇeÔ49LÙǟ
Ǡ𝑃𝑄Ǡą𝑙=
50π
3 ƌ
;@~
12ˆ52024·ƱǔǡǢ·;=9ǣǤǥǞǦǧøǨǩǪǫĤǬ£
“ǭǮǯǰŧDZDzL4ƔÞdzǴǵǶ”L¡Ƿ
MǩǪ4ƔF€LǩǪ4ƔYç¨ǸƔù€ýƔǹǺLǻ˜¿Ĥ³¾ƔǹǺL¿€ň]tǼĢ
𝑓(𝑥) = 𝐴sin(𝜔𝑥+ 𝜑) 𝐴> 0，𝜔> 0，|𝜑|< π
2  !w¡ǽLŒ ãĥLÙ𝑓(𝑥) =      
ˆ
13ˆ52024·čǾǿƟ·=ĎďĐ9ƑƒȀ»­“ƔõƑƒ”L³^ȁƔ­ũŲL˜ƔȂȃȄHLƑƒųÈ
ĢȅĆƓĢȆLśȇݤ1000ǕgȈȉ.Œ LȊ—øƔȁ•Ƃ)FvL³·ȋ¦­3ƌƑƒŠš
ÀŢ6{ƆǏ6'Ƙõ³ȌţŤƇeŨũLƑƒĞĬ𝑂śÏƔù𝐵𝐶„­1.5ƌL—ƑƒÔŚ·ƓƔ
Ƒ𝑃ĻƍĄŝ­4𝐷5Ɣù-ƑƒÒÓö49Ls°ŕŞƍEÀL𝑡'ƘÀLǓƓƔƑśƔùśÏ­
      
      
𝐻= 𝑓(𝑡) = 𝐴sin(𝜔𝑡+ 𝜑) +𝑏𝐴> 0,𝜔> 0,|𝜑|< π
2 ,𝑡≥0 LÙ𝑓(2023) =      
.
14ˆ52024·İıǔƬ·=ĎďĐ9µ
𝑓(𝑥) = sin 𝜔𝑥−π
6 (0 < 𝜔< 6) !{Ò¨©
π
12·\Ąą„¢ž£

𝑔(𝑥) !LÎ 0, π
𝜔m𝑔(𝑥)³·\ĔĕĖđĒLÙ67𝑓(𝑥) = −
1
2ø[0,π]ÔJ
8·
­      
.
İ@*ȍ
15ˆ52024·ȎǔDžȏ·=ĎďĐ9݆
𝑓(𝑥) = sin 2𝑥−π
6 .
(1)ȐȑùœȒȓ‰™
𝑓(𝑥)ø[0,π]Ô !
2𝑥−π
6
−π
6
0
π
11π
6
𝑥
5π
6
𝑓(𝑥)
1
(2)µ
𝑓(𝑥) !{Ò¨
π
3·\Ą¢ß{Ô¨© 1 ·\Ąž£𝑔(𝑥) !L*…?,𝑔(𝑥) ≥
1
2.
16ˆ52024·Ȕȕ·³=9Œ L<𝛼(𝛼∈𝑅)ƍƈ­𝑥ĞćƕȋĞLȖƈ-\ĄƇöĢ4𝑃L”4𝑃‰𝑦Ğȗ
ĜLȗȘ­𝑀,𝑀£ěĜ𝑂𝑃śÏ­|𝑀𝑁|.ε|𝑀𝑁|Ģ<𝛼
Æș­𝑦= 𝑓(𝑥).
      
      
  
(1)B𝑦= 𝑓(𝑥)*+,Q
(2)µ𝑓(𝑥) !Ôã¤4؟ «Ķ­Ðw
1
25ġŸ …#9Lߵ㞠!{ā¨©
π
6·\Ąą„Lž
£
𝑔(𝑥) !LB𝑔(𝑥)ø 0, π
2 \ĔĕĖđĒ.
17ˆ52023·ȚǔțěȜȝȞ\Ą·=ĎďĐ9Œ ­
𝑓(𝑥) = 2cos(𝜔𝑥+ 𝜑) 𝜔> 0,|𝜑|< π
2 &' !L
Ń|𝐶𝐷| =
π
4L𝐴−5π
12 ,−2 ˆ
(1)B𝜔L𝜑ėQ
(2)µ𝑓(𝑥) !Ôã¤4؟ ƐÊ£Ðw 3 Ġ5ġŸ …#9Lß{Ò¨©
3π
4 ·\Ąą„Lž£

𝑔(𝑥) !Lȟij
𝑦= 𝑔(𝑥)−𝑎øđĒ −π, π
2 34·
ˆ
      
      
18ˆ52024·ĉȠęļ·ú=9݆
𝑓(𝑥) = sin(𝜔𝑥+ 𝜑) 𝜔> 0,0 < 𝜑< π
2 &' !Œ ãĥ.
(1)B𝜔ėQ
(2)s›;·ĝƩĤ}³·‰­Ý†Lç
𝑓(𝑥)ȡøLȓB
𝑓(𝑥)ø 0, π
2 ÔÑÊė®ÑËė.
ĝƩ①

𝑓𝑥+ 5π
12 mĩ
Q
ĝƩ②
µ
𝑓(𝑥) !{Ò¨©
π
12·\Ąą„¢ž£𝑦= sin𝜔𝑥 !Q
ĝƩ③
𝑓(0) = 𝑓2π
3 .

ŒÇ}ĝƩ…à1ABLl529:ž 0 'QŒÇ}Ǖ·à1ABĝƩ'¬*ȍLŠl³·*
ȍE'.
19ˆ52024·čǾ1Ȣ·=ĎďĐ9Śĭȣ3ȤđȥȦLŒ LÐ3Ȥđmđĵ𝑂𝐷𝐵𝐶LȥȦÀ]_&'­
ȧ€đĵ𝑂𝐴𝐷L݆∠𝑂𝐶𝐵= ∠𝐶𝑂𝐴=
π
2L𝑂𝐶= 10 3ƌL𝐵𝐶= 10ƌLđĵ𝑂𝐵𝐶­;<€Lđĵ𝑂𝐴𝐵m^𝑂𝐴
­ȋ¦ȧ€LŃ∠𝐴𝑂𝐷=
π
6ˆ
(1)Îäøđĵ𝑂𝐴𝐵𝐶ȨŇȩœùȪƞȫȬLBƞȫȬþą„Q
(2)øđĵ𝑂𝐴𝐷ĤL—ŝȭ€đĵ𝐻𝐺𝐼𝐹‰­ȮȯŴĥđLBȮȯŴĥđùȰ𝑆ÑÊėˆ