专题4.1任意角和弧度制、三角函数的概念五大题型（原卷版）_11.4-2_2026-11-04

      
      
专题4.1 任意角和弧度制、三角函数的概念【五大题型】

 1   !"# ........................................................................................................................................4
 2  $%# ....................................................................................................................................................4
 3  &'()*+ ....................................................................................................................................5
 4  ,-#".#/
"01)+ ........................................................................................................6
 5  .#/
23	"40 ........................................................................................................................7
15,-#6&'(5.#/
"78
9:;
<=>
?@A
(1)BC,-#"786
&'(
(2)DEF&'G#'"
HIJKLM&'("
N:O
(3)PQRSTC.#
/
UVW5XW5VYZ
"01
2023 [\]^
_ 13 I5
@
2024 [\]^
_ 12 I5
@
,-#6&'(5.#/
"78
`.#/
"abI`
"N
cdef.ghi["?jklIm
:n,-#"785.#/
"78I
fopq5rs"tuvwIx
yz{R.
|}9 1  .#/
"a~78
1,-#
(1)#"78
#€plf‚ƒ„…†‡"ˆ9‰Š‹"Œt.
(2)#"Ž
Œ
‘’
ƒ„"“’S” OA•
–
ƒ„"—S” OB•
˜™9
ƒ„"ˆ9 O•
š›œ
Œ"#€›ž“# ”Ÿ“
”Ÿ“
AOB”.
      
      
2$%#G !"#
(1) !"#
 # ,
 !I¡‡¢"£ž
¤# "‰Š(¦§¨Ÿ©§¨)k(kªZ)«¬­#
.
fo®I¯¢°
‹°G#  !"#I±!# ²cI€³f´µ¶
I¬,fG#  !"#I·€pŽ# G¸
´«#"6. 
(2)$%#5¹„#
‘$%#5¹„#"78
²º»¼#½¾£I¿#"™9GÀ9Á¶I#"’G x ¹"ÂÃĹÁ¶.ÅÆI#"²
_i$%IÇÈÉ´#`_i$%#•¿#"²½¾¹ÊIÅÆÇ˞ɴ#ÌÍÎ,Ïf´$%I
ÐÉ´#ž¹„#.
    –$%#"µ¶Ž
$%#
#"µ¶Ž
_f$%#
_Ñ$%#
_.$%#
_Ò$%#
3#'(5&'("78
(1)#'(
#€p'žRSkEF'ÓI1 '"#ÕΫ#"
.ÉÖ'מRSk'Ó#"RS(ØÙ#
'(. 
(2)&'(" 78
‘1 &'"#
Ú'ÕÎÄÛÚ"T&‹Ü"TÝ#.
–&'(
01
p&'מRSk'Ó#"RS(.
›œ
&'RS3	 rad ŽIÞ×&'.
4,-#".#/
(1)ßRST01,-#".#/
à `f´,-#I ªRI‡" OP GRST áÎ9 P(x,y).
‘â9 P "ã½¾ y ØÙ "VW/
I›×
I¬ y=
•
–â9 P "ä½¾ x ØÙ "XW/
I›×
I¬ x=
•
      
      
˜â9 P "ã½¾Gä½¾"y2
ØÙ "VYI›×
I¬
=
 (x≠0). 
¯¢¤VW/
5XW/
6VY/
=О.#/
Iå椇¢›ž
VW/
XW/
VY/
(2)#"Ê"9"½¾Ž.#/
ŒIà `f´,-#I‡"Ê,-f9 P(ÌGÀ9 O Á¶)"½¾ž(x,y)I9 P GÀ9"çè
ž r.¡
=
I
=
I
=
.
    
|}9 2  ,-#6&'("Céê
1 !"#"µ¶
ß !"#"µ¶€p;ë¶ìí‚î"#Iïœ`ðñvGÉ´#" !"‹°#"µ
¶Iòóåôµ¶"õ
 k(kªZ)ö2k;­‹÷"#.
2ø0
I
(kªN*)"S”"ïœ
ðñv
Ÿ
"ùúIòóûü k "€Dý2ø0
Ÿ
"‹²"S”.
3+&'(Cþÿ"iĀ:9
+&'(Cþÿ§+-
(1)ßāt"&Ú6»ĂuC§I:-#"RSNă`&'.
(2);āt»ĂĄĀ2"ÿ§IæŠžÑą/
"Ą2ÿ.
(3)²Cþ&Úÿ6āt»Ăÿ§I:¶®ßTÝ#‹²".#t.
|}9 3  .#/
"01)+"Céê
1.#/
01"+
(1)¼Ćß.#/
"01IćĈĉ0#"Êf´9"½¾I)É9ĈÀ9"çèIø0É´#
".#/
2.
(2)Ċ|#"ìf´.#/
2I€påô.#/
"01ċvČõ
"ïčI;õ
"2.
240.#/
2"3	"Céê
:40.#/
2"3	IĎ`:ďĐ.#/
"#`_i$%#IđûüV5XW/
2²Ē
$%"3	ø02"3	.¿ÌDø0#‹²$%IÅÇ:EF@ēĔĕ;C.
      
      
 1   !"#
Ė 1U2024·ėĘ·ęĚěĜZĝċG
7π
4 " !"#"ŽĞuIVø"`U    Z
A2𝑘π +
3π
4 (𝑘∈Z)
B2𝑘π +
7π
4 (𝑘∈Z)
C𝑘π−
π
4(𝑘∈Z)
D𝑘π +
5π
4 (𝑘∈Z)
ğu 1-1U23-24 fÊ·cĠġ·ĢģZ #𝛼G#−
2π
5 " !I¡𝛼€D`U    Z
A
12π
5
B−
10π
5
C
22π
5
D−
22π
5
ğu 1-2U23-24 fĝ·ĤĥĦħĨ·ĩĪīĬZ #𝛼"²¼„𝑦= 𝑥ÊI¡#𝛼"ý2µ¶ž
U    Z
A 𝛼∣𝛼= 𝑘⋅360∘+ 45∘，𝑘∈𝑍
B 𝛼∣𝛼= 𝑘⋅360∘+ 135∘，𝑘∈𝑍
C 𝛼∣𝛼= 𝑘⋅180∘−135∘，𝑘∈𝑍
D 𝛼∣𝛼= 𝑘⋅180∘−45∘，𝑘∈𝑍
ğu 1-3U23-24 fĝ·ĭĮįİ·ĩĪīĬZ¤#𝛼"…½¾À9 O ¦§¨‰Š 60°óG 130°#"
Á¶I¡G#𝛼 !"#"µ¶žU    Z
A{𝛽|𝛽= 𝑘× 180° + 90°,𝑘∈𝑍}
B{𝛽|𝛽= 𝑘× 360° + 90°,𝑘∈𝑍}
C{𝛽|𝛽= 𝑘× 180° + 150°,𝑘∈𝑍}
D{𝛽|𝛽= 𝑘× 360° + 70°,𝑘∈𝑍}
 2  $%#
Ė 2U2024·ėĘ·ęĚěĜZ 𝛼`_f$%#I¡ĝċĒ#`_.$%#"`U    Z
A90°−𝛼
B180°−𝛼
C270°−𝛼
D−𝛼
ğu 2-1U23-24 fÊ·Ĥ\ıIJ·ĢģZĊ|𝛼= 944°I¡𝛼`U    Z
A_f$%#
B_Ñ$%#
C_.$%#
D_Ò$%#
ğu 2-2U23-24 fĝ·Ĥĥ·ĩĪīĬZĊ|#𝛼p x ¹VŞ’Iijô9𝑃sin 2π
3 ,cos 2π
3 I
¡3π +𝛼`U    Z
A_f$%#
B_Ñ$%#
C_.$%#
D_Ò$%#
ğu 2-3U2024·Ĵĵ·ęĚěĜZ“𝛼`_Ò$%#”`“
𝛼
2`_џ_Ò$%#”"U    Z
AĶ@ÌN:‚î
BN:ÌĶ@‚î
CĶ:‚î
DķÌĶ@ĸÌN:‚î
      
      
 3  &'()*+
Ė 3U2024·Ĺĥ·fęZvĺÎĻĘļĽľĿ"“vŀŁłŃńŅņńŇ”UŒ 1Z"ŇňʼnŊŃŋŅI
JŌºIōā»ŎIņňŏŐsőŒ“𝑆”ŁłIœŔŕw:>ŖŇň»ĂUŗ'ŘêÌ>ZIĜ­Ē
ř
üUŒ 2Z
𝐴𝐵≈8cm,𝐴𝐷≈2cm,𝐴𝑂≈5cmI sin37° ≈
3
5,π ≈3.14I¡ŇňU¬ŚÒt𝐴𝐵𝐶𝐷Z»
ĂhśžU    Z
A6.8cm2
B9.8cm2
C14.8cm2
D22.4cm2
ğu 3-1U2024·ŜŝŞşŠ·.ęZšŢţ`fřJŤťŦŧŨŒ`fSšŢţũŨŪ²ūŬš
vŢţ"ŭŮIůŰ"Łű)ŲųhślfůŞʼn"“Ŵ”ijĜÓŵ§ŶŷŸÝeŮ"&Ú`
5𝜋
6 I“Ŵ”‹
²T"Ä۞ 1.25 ŹIÉSšŢţũŨŪŶŷŸÝeŮ"çèžU    ZŹ
A5 2
6
B5 2
4
C5 3
4
D5 3
6
ğu 3-2U2024·ĴĵĴź·.ęZŻżŽŖžſ`¯ĘġƀcdƁžƂƃ"
ƄƅI*ŻïƆſŽ
ĉv>Ŗ&Ɔ»Ă‹"ijƇuž
&Ɔ»Ă =
1
2U弦× 矢+ 矢
2ZI&ƆUŒZƈT&6*‹ÜW‹
úIu“W”ƉT&‹ÜWÚI“Ɗ”ÕÎÄÛÚGTÝĈW"çèeƋ.wĊ|&Ɔ»Ăž4 3 +2Iƌ
W`Ɗ"2 3ƍIƎƏÊƐijƇu>Ŗ‹­&Ɔ"&Ú`U    Z
      
      
  
A
π
3
B
2π
3
C
4π
3
D
8π
3
ğu 3-3U2023·ƑƒƓƔ·ÑęZ ƕƖ²ƗƘ 3 ƙƚIġƛƜƝƞƟŞƠơcƢÇƣąĜvB
®ƤÄÛ.ƟŞƠơcƢq²ƥƦÉfƝß!fƧƨ„Uij„Z"Ŷ´ĽƩUƪƫĽ6ƬƭIJĀĽZE
FƮĜIƯưźƱ¼ƒơƫĽìƲƳ𝑆§IƬƭIJĀĽìƴ𝐴"ưźƱ„G®»#𝜃= 82.8∘IƵ|ìƶƷ
ƸF§Ĝ­𝐴G𝑆"çè¬ƹ&𝐴𝑆"ڞ 5000 ġƛƜƺI T«ƻý 3.125I¡€Ƽ>®ƤÄÛƽžU    Z
A35000 ġƛƜƺ
B40000 ġƛƜƺ
C45000 ġƛƜƺ
D50000 ġƛƜƺ
 4  ,-#".#/
"01)+
Ė 4U2023·ƾƿƾĵ·ęĚěĜZĊ|#𝛼"™9²½¾À9I’G𝑥¹ÂÃĹÁ¶Icos𝛼=
5
5 ,𝑃(𝑚,2)
ž*Êf9I¡𝑚= U   Z
A−4
B4
C−1
D1
ğu 4-1U2024·ƒǀ·ÑęZĊ|#𝛼"ijô9𝑀( 2,1)I¡cos𝛼= U    Z
A 6
3
B 3
3
C 2
D 2
2
ğu 4-2U2023·ĤĥŰǁ·.ęZà α `_Ñ$%#IPUxI1Zž*Êf9Iƌcos𝛼=
1
3𝑥I¡ tanα=
U    Z
A−2
2
B−2
4
C 2
2
D 2
4
ğu 4-3U2023·Ĺĥǂź·ęĚěĜZĊ|#𝛼"™9Sκ»¼#½¾£𝑥𝑂𝑦"À9I’²𝑥¹"Â
ÃĹÊIGRST áÎ9 −
2
2 ,
2
2 I¡sin𝛼cos𝛼= U    Z
      
      
A−
1
2
B−2
2
C
1
2
D 2
2
 5  .#/
23	"40
Ė 5U2024·Ĥĥ·ęĚěĜZĊ|𝛼`_Ñ$%#I¡9(cos(sin𝛼),sin(cos𝛼))‹²"$%`U    Z
A_f$%
B_Ñ$%
C_.$%
D_Ò$%
ğu 5-1U2023·ÒǃDŽDž·.ęZĊ|#𝛼"Êf9"½¾(𝑎,2)I* a `ÂdžLJ
I¡ĝċ.#
/
2LjžV"`U    Z
Acos𝛼tan𝛼
Bsin𝛼cos𝛼
Csin𝛼tan𝛼
Dtan𝛼
ğu 5-2U2023·Ĥĥ·ęĚěĜZĊ|𝛼`_Ñ$%#I¡9Ucos(−𝛼)Isin(−𝛼)Z‹²"$%`U    Z
A_f$%
B_Ñ$%
C_.$%
D_Ò$%
ğu 5-3U2024·\]ljNJ·fęZ²º»¼#½¾£𝑥𝑂𝑦I#𝛼p𝑂𝑥ž’I²_.$%.¡U    Z
Asin𝛼−cos𝛼≤tan𝛼
Bsin𝛼−cos𝛼≥tan𝛼
Csin𝛼⋅cos𝛼< tan𝛼
Dsin𝛼⋅cos𝛼> tan𝛼
f5R
1U23-24 .ĝ·Njnj·ĩĪīĬZµ¶𝐴= {𝛼∣𝛼= −2024° + 𝑘⋅180°,𝑘∈Z}"ĄĀÃ#𝛼žU   Z
A−2024°
B−224°
C−44°
D−24°
2U2024·Ĥ\ǍƲ·ęĚěĜZ“#𝛼,𝛽"²!f‚¼„Ê”`“sin(𝛼−𝛽) = 0”"U    Z
AĶ@ÌN:‚î
BN:ÌĶ@‚î
CĶ:‚î
DķÌĶ@ĸÌN:‚î
3U23-24 fĝ·Ĥĥ·ĩĪīĬZŒIǎ²Ǐǐų@UǑǒǓZ"#𝛼"µ¶`U    Z
A 𝛼| 5π
6 + 2𝑘π ≤𝛼≤(2𝑘+ 1)π,𝑘∈𝐙
B 𝛼| 5π
6 + 𝑘π ≤𝛼≤(𝑘+ 1)π,𝑘∈𝐙
C 𝛼|−7π
6 + 2𝑘π ≤𝛼≤(2𝑘−1)π,𝑘∈𝐙
D 𝛼|−π
6 + 2𝑘π ≤𝛼≤2𝑘π,𝑘∈𝐙
4U2024·IJǔ·ęĚěĜZĊ|#𝛼"™9G½¾À9Á¶I’G𝑥¹"ÂÃĹÁ¶Iijô9𝑃
      
      
sin π
3 ,cos π
3 I¡cos 𝛼+ π
6 = U    Z
A0
B
1
2
C 2
2
D 3
2
5U2024·ėĘ·ęĚěĜZǕő5ǖő5ǗőǘОƿǙ.őǚǛǜÚ"ǗőIƈǔ«ǝƯ5ǞƀǟǠ
ǗÕǡǢǣkIǤЧƀEMǥǦIt\]5Ɲǧ5IJǀ5Įĵ5Ǩǔ5ǩƥp)ǪǫǗőǬĀm:ǜ
ǫǪǫǗőǘО“ĥïeǭ”I`ĥï®ǮǗőǯž"ǰƀŽIǘǨDZũĈDzdz5Ǵǵ5ǶǷ5Ǹǹ
ÕƿǙŒU1Z`f´ǺǻǗőI*V»`f´āǼ𝐴𝐵𝐶𝐷IŒU2ZIǗőŗ'ž 6cmI𝐴𝐷= 80
cmI𝐶𝐷= 3𝐴𝐵I𝐶𝐷‹Ü"TÝ#ž¼#I¡ǽǺǻǗő"Ž»ĂžURS
cm2ZU   Z
    
A3200π
B480π +960
C6880π +960
D3680π +960
6U2024·Òǃ··ęĚěĜZ²º»¼#½¾£I#𝛼"™9GÀ9Á¶I’G𝑥¹"ÂÃĹÁ¶I
ijô9𝑃(3,4)I¡
sin𝛼+2cos𝛼
cos𝛼−sin𝛼= U    Z
A11
B−10
C10
D−11
7U2024·Ƒƒ·ÑęZġǾâVW/
5XW/
5VY/
5XY/
5Vǿ/
5Xǿ/
5VƊ/

5XƊ/
ÉȀÖ.#/
"/
„¶ÐžȀ„.*XY/
cot𝜃=
1
tan𝜃IVǿ/
sec𝜃=
1
cos𝜃IXǿ/

csc𝜃=
1
sin𝜃IVƊ/
𝑣𝑒𝑟sin𝜃= 1−cos𝜃IXƊ/
𝑣𝑒𝑟cos𝜃= 1−sin𝜃.Œ#𝜃’ž𝑥¹"ÂÃĹI*
GRSTá9𝑃I𝐴5𝐵@ȁ`RSTG𝑥¹6𝑦¹VĹ"á9Iô9𝑃×𝑃𝑀Ȃ¼𝑥¹I×𝑃𝑁Ȃ¼𝑦¹I
Ȃȃ@ȁž𝑀5𝑁Iô9𝐴×𝑥¹"Ȃ„Iô9𝐵×𝑦¹"Ȃ„@ȁá𝜃"Î𝑇5𝑆I*𝐴𝑀5𝑃𝑆5𝐵𝑆5𝑁𝐵
ž°Ȅ„ĪIĝċŽVø"`U    Z
  
      
      
A𝑣𝑒𝑟sin𝜃= 𝐴𝑀
Bcsc𝜃= 𝑃𝑆
Ccot𝜃= 𝐵𝑆
Dsec𝜃= 𝑁𝐵
8U2024·IJǔȅȆ·fęZ2024 [ 2 ȇ 4 ȈI“łFȉ——Ȋȋł[ȌȍĀȎǢ”²IJǔȏƧȐȑ
ȒFIǢȓ"Ơî·°“ł”"ƗȔŸŒȕvĺÎĻĘļĽľĿ"“vŀŁłŃńŅņńŇ”UŒ 1ZÇ
`ÉȖfîȗȘńŇŇňʼnŊŃŋŅIJŌºIōā»ŎIŇňŏșsőŒ“S”ŁłIœŔŕw:
>ŖŇň»ĂUŗ'ŘêÌ>ZIĜ­Ēř
üUŒ 2Z
𝐴𝐵≈8cmI𝐴𝐷≈2cmI𝐴𝑂≈5cmI sin37° ≈
3
5Iπ ≈3.14I¡ŇňU¬ŚÒt ABCDZ»ĂhśžU    Z
A6.8cm2
B9.8cm2
C14.8cm2
D22.4cm2
Ñ5Ơ
9U2023·ĴĵȚ1·ęĚěĜZĝċȜVø"`U    Z
A sin𝛼= sin𝛽I¡𝛼G𝛽` !"#
B #𝛼"ô9𝑃(3𝑘,4𝑘)(𝑘≠0)I¡sin𝛼=
4
5
C āt"«Úž 3IÄ۞ 1I¡*TÝ#"Āțž 1 &'
D sin𝛼⋅cos𝛼> 0I¡#𝛼"²_f$%Ÿ_.$%
10U2023·Ĥ\ǕƞȜ·fęZ²º»¼#½¾£I#𝛼"™9G½¾À9Á¶I’G𝑥¹"ÂÃĹÁ
¶Iijô9𝑃
5
2 ,𝑥Iƌsin𝛼=
2
3𝑥I¡𝑥"2€p`U    Z
A ±
2
B ± 1
C0
D ± 2
11U2023·ȝȞ·ÑęZŒIAIB `²RSTÊũŨ"Ŷ´ȟ9.Ƞ’§ŊIȟ9 A ²U1I0ZƴIȟ9
B ²_f$%Iƌ∠𝐴𝑂𝐵=
𝜋
6.ȟ9 A p
𝜋
6rad/s"#ȡ'Ǝ©§¨ïȄũŨIȟ9 B !§p
𝜋
12rad/s"#ȡ'Ǝ
¦§¨ïȄũŨI¡U    Z
      
      
Aijô 1sóIāt AOB "»Ăž
5π
12
Bijô 2sóIƹ&𝐴𝐵"ڞ
2π
3
Cijô 6sóIȟ9 B "½¾ž −
3
2 , 1
2
Dijô
22
3 sóIȟ9 AIB ²RSTÊ_fą Ȣ
.5rs
12U2024·ȣƥ·ÑęZĄŕ
ȤȥŷŽÊ"§¨Ú'` 1 ȦŹI¡§¨4hU§ZŠv"āt»Ă`
      
ºïȦŹ
13U2024·ėĘ·ęĚěĜZĊ|#𝜃"™9ž½¾À9I’ž𝑥¹"ÂÃĹ 𝑃(𝑚,2)`#𝜃Êf
9Iƌcos𝜃= −3 10
10 I¡𝑚=      

14U2023·ǨǔȧIJ·fęZ 9𝐴(cos𝜃,sin𝜃)ÎÀ9ÜÐ9ž𝐵cos 𝜋
6 −𝜃,sin 𝜋
6 −𝜃
Iñv𝜃"f´ý
2ž      
.
Ò5CȨ
15U2024 fĝ·ėĘ·īĬZĊ|#𝛼"²_Ò$%Iø0ĝċĒ#‹²"$%
(1)
𝛼
2•
(2)
𝛼
3•
16U23-24 f·ėĘ·ȩȪīĬZñvGĝċĒ# !"#"µ¶Iȫ⵶ë¶ÌÕu−720∘≤𝛽<
360∘"ƗȔ𝛽ñvk
      
      
(1)60∘•
(2)−45∘•
(3)1303∘18′•
(4)−225∘
17U23-24 f·ėĘ·ȩȪīĬZßRSTI;ë¶ĝċ‚î"# α "µ¶
(1)cos𝛼= −2
2 •
(2)sin𝛼≤
1
2
18U23-24 fÊ·ŋĥȬǤ·ĩĪīĬZ²º»¼#½¾£𝑥𝑂𝑦IRST𝑥2 + 𝑦2 = 1G x ¹"VĹ)Ã
Ĺ@ȁáÎ9 AIBI#𝛼"’ž x ¹"ÂÃĹIGRSTáÎ x ¹ĝïf9 P
(1)ŒI ∠𝑃𝑂𝐵= 120°I;9 P "½¾•
(2) 9 P "ä½¾ž−3
3 I;sin𝛼"2
      
      
19U2024·Êljņȭ·ÑęZìȮȯȰÙf´ȱȲȮȯǡǢȳ"ȴȵIȴȵ"ȶ»tŎ`Œ‹"ātǼ
»Uƈāt OAD ȷȸāt OBC ó³"Z.Ċ|𝑂𝐴= 10I𝑂𝐵= 𝑥(0 < 𝑥< 10)I„Ī BAICD G𝐵𝐶I
⌢
𝐴𝐷
"Ú'e6ž 30ITÝ#ž𝜃&'.
(1);𝜃Î x "/
ŽĞu•
(2)›ȴȵ"ȶ»»Ăž yIxÿ x ýÏ2§Iy "2ĄĀȹȫ;vĄĀ2.