专题4.2同角三角函数基本关系式及诱导公式五大题型（原卷版）_11.4-2_2026-11-04

      
      
专题4.2 同角三角函数基本关系式及诱导公式【五大题型】

 1   !
"#$%&' ........................................................................................................3
 2  ()%&' ....................................................................................................................................3
 3   !
%&*+,- ....................................................................................................................4
 4   ./%&01 ................................................................................................................................4
 5   !
"#$%2()%&34' ............................................................................5
1+ !
"#$%5()%
67,
89:
;<=
(1)> !
&"#
$%?
?
(2)@A()%?BC*D
'
2022 EFGH
I 13 ?5
<
2023 EJKLHM
N
I 14 ?5 <
2023 EJKLHM
N
I 13 ?5 <
2024 EOP I H
I 4 ?
5 <
2024 EJKLHM
N
I 9 ?5 <
 !
$%Q()%R
 !
*,-&"S?R

&TUVWX.YZ[E&;\
]^?_7`“abc”+ 
!
$%Q()%34/UV?
defgh?Xijk+lm&
n%op?qrst/uvw.
xy6 1   !
&"#$
1z !
&"#$
"#$%
{|}~
€$
X α &‚a+ƒa&€2/„ 1.
…
$
X α &‚a+ƒa&…/„ α &‚
b.
2z"#$%&†n%
      
      
xy6 2   !
"#$%&'‡ˆ
1z‚ƒac+abcj5“2”“‹”Œ&>‡ˆ
(1)Ž
jp &‚a+ƒa&c?Ž
j
p &abc.
(2)n‘
/’“”ab.
2z•%&–5†n'
.
3z'%—•€˜™š&'
›„
œ %?Ž
?jxX
,ž.
xy6 3  ()%&'&>Ÿ 
1z()%&¡'
(1),-
¢‚?£¤?¥¦f§¨.
(2)*
9X?9X©?©ªf§¨.
2z« 2π ¬
­&()%&'
®§¯°&&$x?±:²«³ 2π &¬
­& !
%t´µ¶ 2π &¬
­·¸¹º
“
”»².‘
.
xy6 4  $%2()%&34'&>Ÿ 
1z*+,-
Ž !
$%2()%,-u*—?¼R½,¾¿+ÀÁÂ&Ã$?ghÄ%
“”†n.•&Åƛ !
-Ç	&ÈÉ.
2z()%,-
()%,-—?7ʄË`ÌÍWÂ&$?Ž¬Îύ&™š*>И.ÑÒ&cƒ
$³
Q
?
Q
?
Q
/?ÑÒ&cÓ$
Q
?
Q
?
Q
/.
€Ô‡ˆQÕÀ
      
      
1z !
$%&цn
2z()%&Ö×ØÙ
“چÛ܆?Ç	^ÝÞ”?ßt&Ú+ÛRà
&Ú
­2Û
­?†Q܆à!
©á&†
.
3z !
$%&•âã
±Ž !
&€$—?äå€?7æç•èéÇ	.
 1   !
"#$%&'
ê 1M2024·ëì·íîïðNä𝛼∈(0,π)?ñcos𝛼−sin𝛼=
1
2?òtan𝛼= M    N
Az4+ 7
5
Bz4−7
5
Cz4+ 7
3
Dz4−7
3
†% 1-1M2023·óô·íîïðNõxsin𝛼−cos𝛼=
1
5,𝛼∈−π
2 , π
2 ?ò
sin𝛼cos𝛼
sin𝛼+cos𝛼= M    N
Az−
12
5
Bz
12
5
Cz−
12
35
Dz
12
35
†% 1-2M2023·JK·íîïðNõx𝛼&ö6Q÷6ø4?ù¯Q𝑥ú&û¢üúø4?§¯±´ý
𝑦=
1
3𝑥þ?òsin2𝛼−cos(2𝛼−π) = M    N
Az−
1
5u
7
5
Bz
1
5u−
7
5
Cz
7
5
Dz−
1
5
†% 1-3M2023·ÿôĀā· íNõx€˜sin2𝛼+ 2sin𝛼cos𝛼−2sin𝛼−4cos𝛼= 0?òcos2𝛼−sin𝛼cos𝛼=
M    N
Az−
4
5
Bz
3
5
Cz−
3
5
Dz
4
5
 2  ()%&'
ê 2M2024·Ăìā·íîïðNäsin 𝛼+ π
3 =
1
4?òcos 𝛼+ 5π
6
= M    N
Az
1
4
Bz−
1
4
Cz ±
1
4
Dz 15
4
†% 2-1M2024·JK·íîïðNõxtan π
2 + 𝜃=
1
2?ò
sin3𝜃+2cos3𝜃
sin(π+𝜃)
= M    N
Az
3
5
Bz
5
6
Cz−
5
6
Dz−
3
5
†% 2-2M2024·ăĄ·íîïðNõx𝛼+
π
3&ö6±ąP÷6?ù¯Q𝑥ú‚üúø4?§¯ĆÐ6
𝑃1
2 ,
3
2 ?òcos 𝛼−π
6 = M    N
      
      
Az−3
2
Bz−
1
2
Cz
1
2
Dz 3
2
†% 2-3M2024·JK·íîïðNõxcos 𝜃−2π
5
=
2
3?ò2sin 19π
10 −𝜃+ cos 𝜃+ 13π
5
= M    N
Az−2
Bz2
Cz−
2
3
Dz
2
3
 3   !
%&*+,-
ê 3M2023·ć죷íîïðNõxtan𝛼= 3?ò
cos3𝛼−cos𝛼
cos 𝛼+π
2
= M    N
Az−
3
4
Bz
3
4
Cz−
3
10
Dz
3
10
†% 3-1M2024·ĈĉĊċČ·žíNõx𝛼(0° < 𝛼< 360°)§¯þ𝐴6ąPf(sin310°,cos310°)?ò𝛼=
M    N
Az130°
Bz140°
Cz220°
Dz230°
†% 3-2M2024·ÿôčĎ·íîïðNõx
sin𝛼
sin π
3 −𝛼= 4?òtan𝛼= M    N
Az−2 3
Bz−3
Cz 3
2
Dz2 3
3
†% 3-3M2023·ăĄďĐ·íîïðNõx𝛼fIžÝÞ?äsin 2023π
2
−𝛼=
1
4,òtan𝛼= M    N
Az−15
Bz 15
Cz−15
15
Dz 15
15
 4   ./%&01
ê 4M23-24 X·JK·O¹đĒNētan 𝛼+ 8𝜋
7
= 𝑚.,0
sin 𝛼+15𝜋
7
+3cos 𝛼−13𝜋
7
sin −𝛼+20𝜋
7
−cos 𝛼+22𝜋
7
=
𝑚+3
𝑚+1.
†% 4-1M2024 X·JK·ĔĕN,0
(1)
sin𝛼−cos𝛼+1
sin𝛼+cos𝛼−1Ė
1+sin𝛼
cos𝛼ė
(2)2(sin6𝜃+ cos6𝜃)−3(sin4𝜃+ cos4𝜃) +1 = 0
      
      
†% 4-2M23-24 Xþ·JK·O¹đĒNM1N,0
tan(2𝜋−𝛼)sin(−2𝜋−𝛼)cos(6𝜋−𝛼)
sin(𝛼+3𝜋
2 )cos(𝛼+3𝜋
2 )
= −tan𝛼ė
M2Nētan(𝛼+
8𝜋
7 ) = 𝑚?,0
sin( 15𝜋
7 +𝛼)+3cos(𝛼−13𝜋
7 )
sin( 20𝜋
7 −𝛼)−cos(𝛼+22𝜋
7 ) =
𝑚+3
𝑚+1.
†% 4-3M23-24 X·JK·ĘęĔĕN,0
(1)sin4𝛼−cos4𝛼= sin2𝛼−cos2𝛼ė
(2)sin4𝑥+ sin2𝑥cos2𝑥+ cos2𝑥= 1ė
(3)
cos𝛼
1+sin𝛼−
sin𝛼
1+cos𝛼=
2(cos𝛼−sin𝛼)
1+sin𝛼+cos𝛼z
 5   !
"#$%2()%&34'
ê 5M24-25 Xþ·þë·O¹đĒNõx𝛼&§¯ĆÐ6𝑃(
4
5,−
3
5).
(1),
sin(π−𝛼)
sin(𝛼+π) ⋅
tan(𝛼−π)
cos(3π−𝛼)&-ė
(2),
sin3(π−𝛼)+5cos3(𝛼−3π)
3cos3(π−𝛼)+sin2(π−𝛼)cos(𝛼−2π)&-.
†% 5-1M23-24 Xw·GôĚěĜ·ĝtN±①4sin(2023π +𝛼) = 3cos(2024π +𝛼)ė②sin𝛼+ cos𝛼=
1
5ė③𝛼,𝛽&§¯„𝑥ú›á?Bñ4sin𝛽= 3cos𝛽œ ¾¿tĞX?Óğ±Ġýþ?BġĢģ.
õxIăÝÞ𝛼Ĥĥ__________?,wĦħ%&-.
      
      
(1)
4sin𝛼+5cos𝛼
cos𝛼−sin𝛼
(2)sin2𝛼+ 3sin𝛼cos𝛼+ 1
†% 5-2M23-24 Xw·ĨĐĩā·ĪīĔĕNõx!
𝑓(𝛼) =
sin(2π−𝛼)cos(π+𝛼)tan(2π−𝛼)
sin 9π
2 +𝛼tan(π−𝛼)
(1)*𝑓(𝛼)ė
(2)ä𝑓(𝛼) = −
1
5?,cos𝛼､tan𝛼&-ė
(3)ä𝛼∈−π
6 , π
3 ,𝑓𝛼+ π
6 =
1
3?,cos 2π
3 + 𝛼+2cos 5π
6 −𝛼&-.
†% 5-3M23-24 Xw·ĨĐ£Ĭ·ĪīĔĕN±DĭĮt?¦𝛼&§¯QDĭĮ°į„6𝑃𝑚,
3
2 ?
ĬµĮİ𝑂2𝑃ı¥IJý𝑂𝑃?¶IJý𝑂𝑃ij6𝑂Ĵ–—ĵ€ĶķŒ𝜃¹QDĭĮ°į„6𝐵?ßt𝜃∈0, π
2 .
(1),4sin3 𝛼+π
2 +2sin2 3π
2 −𝛼−4cos(𝛼+π)
2+2cos2(5π+𝛼)+cos(−𝛼)
&-ė
(2)Ö6𝐵&ĠąPf𝑓(𝜃)?ä𝑓𝜃−π
6 =
1
4?,cos 𝜃−π
3 + cos 𝜃−5π
6 &-.
X+D
1zM2024·Ăìĸā·íîïðNõxtan𝛼= 2?ò
5sin𝛼+cos𝑎
2sin𝛼−cos𝛼= M    N
      
      
Az
1
3
Bz
11
3
Cz
5
3
Dz2
2zM2024·ĹĺĻļ· íNõxsin𝜃+ cos𝜃=
7
13?òsin𝜃−cos𝜃&-fM    N
Az
17
13
Bz
7
13
Cz ±
17
13
Dz ±
7
13
3zM2024·FG·íîïðNõx𝛼∈0, π
2 ?sin 𝛼−π
10 =
1
3?òcos 𝛼+ 2π
5
= M    N
Az−2 2
3
Bz2 2
3
Cz−
1
3
Dz
1
3
4zM2024·óĽľč·íîïðNõxsin 3π
2 + 𝛼=
3
2 ñ
𝜋
2 < 𝛼< π?òtan𝛼= M    N
Az−3
Bz−3
3
Cz 3
3
Dz3
5zM2024·ĿĽŀŁ·íîïðNõxsin 2π
7 + 𝛼=
1
5?łŃtan 3π
14 −𝛼= M    N
Az−
1
5
Bz ± 2 6
Cz2 6
5
Dz2 6
6zM2024·ńëôĐ·žíNõxsin𝛼+ cos𝛼= 3cos𝛼tan𝛼?òcos2𝛼tan𝛼−1 = M    N
Az−
3
5
Bz−
4
5
Cz−
2
3
Dz−
1
3
7zM2024·ĨĐ· íNõxtan𝛼=
1
2?òsin 𝛼+π
2 −cos 3π
2 −𝛼
cos(−𝛼)−sin(π−𝛼) = M    N
Az−1
Bz1
Cz−3
Dz3
8zM2023·óô·íîïðNõx𝛼,𝛽,𝛾ŅR¦?ēsin𝛼cos𝛽+ sin𝛽cos𝛾+ sin𝛾cos𝛼&ņ£-ftan𝜃?òsin𝜃
(sin𝜃+ cos𝜃)=M    N
Az 3
Bz
15
13
Cz1
Dz
5
13
ž+Ň
9zM2024·GňÑļ·íîïðNõx 𝛼 &§¯QDĭĮį„6 3
5 ,𝑦0 ?ò
sin𝛼+2cos𝛼
3sin𝛼−cos𝛼= M    N
Az
10
9
Bz−
10
9
Cz−
2
15
Dz
1
5
10zM2024·Ĺìʼnā· íNwĦŊԂŋ&³M   N
Azä𝛼&§¯Ð6𝑃1
2 ,
3
2 ?ò𝛼&Ō4R 𝛼|𝛼= π
3 + 2𝑘π
 
,𝑘∈𝑍
Bzäcos 𝛼+ π
6 =
3
5?òsin 𝛼+ 2π
3
=
3
5
Czätan𝛼= 2?òsin2𝛼+ sin𝛼cos𝛼=
6
5
      
      
Dzäōn&Ŏŏf8cm?Įİf2rad?òŐōn&üőR4cm
11zM2024·ĨĐ·íîïðNē𝛼fIXÝÞ,cos 𝛼−π
8 =
1
3,òM    N
Azsin 5π
8 −𝛼= −
1
3
Bzcos 𝛼+ 7π
8
= −
1
3
Czsin 13π
8 −𝛼= −2 2
3
Dztan π
8 −𝛼= −2 2
 +lm
12zM2024·ĐŒœŔó·íîïðNõx𝛼&§¯ĆÐ6𝑃(−1,2)?òcos(π + 𝛼)&-f      
.
13zM2024·FGŕļ·íîïðNõx
sin𝜃−2cos𝜃
sin𝜃+cos𝜃= 2?ò sin3𝜃+cos𝜃
2sin𝜃+cos3𝜃=      
z
14zM2024·Ăĺ·XíNõx x RIžÝÞ?äcos(𝑥−70°) =
1
5?òsin(𝑥+ 110°) =      
z
ă+>Ģ
15zM2024·Ŗŗ 1·íîïðNõx𝑓(𝑥) =
sin(𝜋−𝑥)+cos(−𝑥)
sin 𝜋
2+𝑥−2cos(𝜋+𝑥).
M1N,𝑓𝜋
3 &-ė
M2Nä𝑓(𝛼) = 1?ñ𝛼RI ÝÞ?,tan𝛼,sin𝛼&-.
16zM23-24 Xw·ĂìŘā·ĪīĔĕN*,-.
(1)*:
1−2sin190°cos190°
cos170°+ 1−cos2170°;
(2)õx:tan𝛼= −
3
4?:²:2sin2𝛼+ 3sin𝛼cos𝛼−cos2𝛼.
      
      
17zM2024·JK·íîïðNõx−
𝜋
2 < 𝑥< 0,sin𝑥+ cos𝑥=
1
5
(1),sin𝑥−cos𝑥&-ė
(2),
3sin2 𝑥
2 −2sin 𝑥
2 cos 𝑥
2+cos2 𝑥
2
tan𝑥+cot𝑥
&-.
18zM2024·ŖŗŖļ·XíNõxsin𝛼+ cos𝛼= −
1
5
（1）,sin(
𝜋
2 +𝛼) ⋅cos(
𝜋
2−𝛼)&-ė
（2）ä
𝜋
2 < 𝛼< 𝜋，ñ𝛽§¯ĆÐ6𝑃(−3, 7)，,
1
sin(𝜋−𝛼) +
1
cos(𝜋+𝛼) +
2
cos(−𝛽−2𝜋)&-
19zM2023·Ăì· íNõx𝛼&§¯ĆÐ6𝑃(3𝑚,−6𝑚)(𝑚≠0).
M1N,
sin(𝛼+𝜋)+cos(𝛼−𝜋)
sin 𝛼+𝜋
2 +2cos 𝛼−𝜋
2 &-ė
M2Nä𝛼RIžÝÞ?,sin2 𝛼+ 3𝜋
2
+ sin(𝜋−𝛼)cos𝛼−cos 𝜋
2 + 𝛼&-.