诱导公式 是数学 三角函数 中将角度 比较大的三角函数利用角度 的周期 性，转换为角度比较小的三角函数 的变形公式。诱导公式分为以下六类：
sin
(
2
k
π
+
α
)
=
sin
α
,
k
∈
Z
{\displaystyle \sin(2k\pi +\alpha )=\sin \alpha ,k\in \mathbb {Z} }
cos
(
2
k
π
+
α
)
=
cos
α
,
k
∈
Z
{\displaystyle \cos(2k\pi +\alpha )=\cos \alpha ,k\in \mathbb {Z} }
tan
(
2
k
π
+
α
)
=
tan
α
,
k
∈
Z
{\displaystyle \tan(2k\pi +\alpha )=\tan \alpha ,k\in \mathbb {Z} }
cot
(
2
k
π
+
α
)
=
cot
α
,
k
∈
Z
{\displaystyle \cot(2k\pi +\alpha )=\cot \alpha ,k\in \mathbb {Z} }
sec
(
2
k
π
+
α
)
=
sec
α
,
k
∈
Z
{\displaystyle \sec(2k\pi +\alpha )=\sec \alpha ,k\in \mathbb {Z} }
csc
(
2
k
π
+
α
)
=
csc
α
,
k
∈
Z
{\displaystyle \csc(2k\pi +\alpha )=\csc \alpha ,k\in \mathbb {Z} }
sin
(
π
+
α
)
=
−
sin
α
{\displaystyle \sin(\pi +\alpha )=-\sin \alpha }
cos
(
π
+
α
)
=
−
cos
α
{\displaystyle \cos(\pi +\alpha )=-\cos \alpha }
tan
(
π
+
α
)
=
tan
α
{\displaystyle \tan(\pi +\alpha )=\tan \alpha }
cot
(
π
+
α
)
=
cot
α
{\displaystyle \cot(\pi +\alpha )=\cot \alpha }
sec
(
π
+
α
)
=
−
sec
α
{\displaystyle \sec(\pi +\alpha )=-\sec \alpha }
csc
(
π
+
α
)
=
−
csc
α
{\displaystyle \csc(\pi +\alpha )=-\csc \alpha }
sin
(
−
α
)
=
−
sin
α
{\displaystyle \sin(-\alpha )=-\sin \alpha }
cos
(
−
α
)
=
cos
α
{\displaystyle \cos(-\alpha )=\cos \alpha }
tan
(
−
α
)
=
−
tan
α
{\displaystyle \tan(-\alpha )=-\tan \alpha }
cot
(
−
α
)
=
−
cot
α
{\displaystyle \cot(-\alpha )=-\cot \alpha }
sec
(
−
α
)
=
sec
α
{\displaystyle \sec(-\alpha )=\sec \alpha }
csc
(
−
α
)
=
−
csc
α
{\displaystyle \csc(-\alpha )=-\csc \alpha }
y 轴的对称性）[ 编辑 ]
sin
(
π
−
α
)
=
sin
α
{\displaystyle \sin(\pi -\alpha )=\sin \alpha }
cos
(
π
−
α
)
=
−
cos
α
{\displaystyle \cos(\pi -\alpha )=-\cos \alpha }
tan
(
π
−
α
)
=
−
tan
α
{\displaystyle \tan(\pi -\alpha )=-\tan \alpha }
cot
(
π
−
α
)
=
−
cot
α
{\displaystyle \cot(\pi -\alpha )=-\cot \alpha }
sec
(
π
−
α
)
=
−
sec
α
{\displaystyle \sec(\pi -\alpha )=-\sec \alpha }
csc
(
π
−
α
)
=
csc
α
{\displaystyle \csc(\pi -\alpha )=\csc \alpha }
sin
(
π
2
−
α
)
=
cos
α
{\displaystyle \sin \left({\frac {\pi }{2}}-\alpha \right)=\cos \alpha }
cos
(
π
2
−
α
)
=
sin
α
{\displaystyle \cos \left({\frac {\pi }{2}}-\alpha \right)=\sin \alpha }
tan
(
π
2
−
α
)
=
cot
α
{\displaystyle \tan \left({\frac {\pi }{2}}-\alpha \right)=\cot \alpha }
cot
(
π
2
−
α
)
=
tan
α
{\displaystyle \cot \left({\frac {\pi }{2}}-\alpha \right)=\tan \alpha }
sec
(
π
2
−
α
)
=
csc
α
{\displaystyle \sec \left({\frac {\pi }{2}}-\alpha \right)=\csc \alpha }
csc
(
π
2
−
α
)
=
sec
α
{\displaystyle \csc \left({\frac {\pi }{2}}-\alpha \right)=\sec \alpha }
sin
(
π
2
+
α
)
=
cos
α
{\displaystyle \sin \left({\frac {\pi }{2}}+\alpha \right)=\cos \alpha }
cos
(
π
2
+
α
)
=
−
sin
α
{\displaystyle \cos \left({\frac {\pi }{2}}+\alpha \right)=-\sin \alpha }
tan
(
π
2
+
α
)
=
−
cot
α
{\displaystyle \tan \left({\frac {\pi }{2}}+\alpha \right)=-\cot \alpha }
cot
(
π
2
+
α
)
=
−
tan
α
{\displaystyle \cot \left({\frac {\pi }{2}}+\alpha \right)=-\tan \alpha }
sec
(
π
2
+
α
)
=
−
csc
α
{\displaystyle \sec \left({\frac {\pi }{2}}+\alpha \right)=-\csc \alpha }
csc
(
π
2
+
α
)
=
sec
α
{\displaystyle \csc \left({\frac {\pi }{2}}+\alpha \right)=\sec \alpha }
sin
(
3
π
2
−
α
)
=
−
cos
α
{\displaystyle \sin \left({\frac {3\pi }{2}}-\alpha \right)=-\cos \alpha }
cos
(
3
π
2
−
α
)
=
−
sin
α
{\displaystyle \cos \left({\frac {3\pi }{2}}-\alpha \right)=-\sin \alpha }
tan
(
3
π
2
−
α
)
=
cot
α
{\displaystyle \tan \left({\frac {3\pi }{2}}-\alpha \right)=\cot \alpha }
cot
(
3
π
2
−
α
)
=
tan
α
{\displaystyle \cot \left({\frac {3\pi }{2}}-\alpha \right)=\tan \alpha }
sec
(
3
π
2
−
α
)
=
−
csc
α
{\displaystyle \sec \left({\frac {3\pi }{2}}-\alpha \right)=-\csc \alpha }
csc
(
3
π
2
−
α
)
=
−
sec
α
{\displaystyle \csc \left({\frac {3\pi }{2}}-\alpha \right)=-\sec \alpha }
sin
(
3
π
2
+
α
)
=
−
cos
α
{\displaystyle \sin \left({\frac {3\pi }{2}}+\alpha \right)=-\cos \alpha }
cos
(
3
π
2
+
α
)
=
sin
α
{\displaystyle \cos \left({\frac {3\pi }{2}}+\alpha \right)=\sin \alpha }
tan
(
3
π
2
+
α
)
=
−
cot
α
{\displaystyle \tan \left({\frac {3\pi }{2}}+\alpha \right)=-\cot \alpha }
cot
(
3
π
2
+
α
)
=
−
tan
α
{\displaystyle \cot \left({\frac {3\pi }{2}}+\alpha \right)=-\tan \alpha }
sec
(
3
π
2
+
α
)
=
csc
α
{\displaystyle \sec \left({\frac {3\pi }{2}}+\alpha \right)=\csc \alpha }
csc
(
3
π
2
+
α
)
=
−
sec
α
{\displaystyle \csc \left({\frac {3\pi }{2}}+\alpha \right)=-\sec \alpha }
值得注意的是，公式一至八其实都存在着内在联系，可以写成以下形式：
sin
(
k
π
2
±
α
)
,
k
∈
Z
{\displaystyle \sin \left({\frac {k\pi }{2}}\pm \alpha \right),k\in \mathbb {Z} }
cos
(
k
π
2
±
α
)
,
k
∈
Z
{\displaystyle \cos \left({\frac {k\pi }{2}}\pm \alpha \right),k\in \mathbb {Z} }
tan
(
k
π
2
±
α
)
,
k
∈
Z
{\displaystyle \tan \left({\frac {k\pi }{2}}\pm \alpha \right),k\in \mathbb {Z} }
cot
(
k
π
2
±
α
)
,
k
∈
Z
{\displaystyle \cot \left({\frac {k\pi }{2}}\pm \alpha \right),k\in \mathbb {Z} }
sec
(
k
π
2
±
α
)
,
k
∈
Z
{\displaystyle \sec \left({\frac {k\pi }{2}}\pm \alpha \right),k\in \mathbb {Z} }
csc
(
k
π
2
±
α
)
,
k
∈
Z
{\displaystyle \csc \left({\frac {k\pi }{2}}\pm \alpha \right),k\in \mathbb {Z} }
可用如下口诀将联系记忆起来：“奇变偶不变，符号看象限”。意思为，当
k
{\displaystyle k}
奇数 时，
sin
{\displaystyle \sin }
cos
{\displaystyle \cos }
cos
{\displaystyle \cos }
sin
{\displaystyle \sin }
tan
{\displaystyle \tan }
cot
{\displaystyle \cot }
cot
{\displaystyle \cot }
tan
{\displaystyle \tan }
sec
{\displaystyle \sec }
csc
{\displaystyle \csc }
csc
{\displaystyle \csc }
sec
{\displaystyle \sec }
k
{\displaystyle k}
偶数 时，三角函数则不变换。对于正负号，则要看最后角所在的象限进行判断，可以使用如下口诀：CAST ，也可以使用ASTC (All Students Take Calculus) 用来记忆。
第一象限的 A 即是 All（全部皆正）。
第二象限的 S 即是 Sin e & CoS ecant（正弦 以及余割 为正）。
第三象限的 T 即是 Tan gent & Cot angent（正切 以及余切 为正）。
第四象限的 C 即是 Cos ine & SeC ant（余弦 以及正割 为正）。 
