诱导公式

• ${\displaystyle \sin(2k\pi +\alpha )=\sin \alpha ,k\in \mathbb {Z} }$ 
• ${\displaystyle \cos(2k\pi +\alpha )=\cos \alpha ,k\in \mathbb {Z} }$ 
• ${\displaystyle \tan(2k\pi +\alpha )=\tan \alpha ,k\in \mathbb {Z} }$ 
• ${\displaystyle \cot(2k\pi +\alpha )=\cot \alpha ,k\in \mathbb {Z} }$ 
• ${\displaystyle \sec(2k\pi +\alpha )=\sec \alpha ,k\in \mathbb {Z} }$ 
• ${\displaystyle \csc(2k\pi +\alpha )=\csc \alpha ,k\in \mathbb {Z} }$ 

• ${\displaystyle \sin(\pi +\alpha )=-\sin \alpha }$ 
• ${\displaystyle \cos(\pi +\alpha )=-\cos \alpha }$ 
• ${\displaystyle \tan(\pi +\alpha )=\tan \alpha }$ 
• ${\displaystyle \cot(\pi +\alpha )=\cot \alpha }$ 
• ${\displaystyle \sec(\pi +\alpha )=-\sec \alpha }$ 
• ${\displaystyle \csc(\pi +\alpha )=-\csc \alpha }$ 

• ${\displaystyle \sin(-\alpha )=-\sin \alpha }$ 
• ${\displaystyle \cos(-\alpha )=\cos \alpha }$ 
• ${\displaystyle \tan(-\alpha )=-\tan \alpha }$ 
• ${\displaystyle \cot(-\alpha )=-\cot \alpha }$ 
• ${\displaystyle \sec(-\alpha )=\sec \alpha }$ 
• ${\displaystyle \csc(-\alpha )=-\csc \alpha }$ 

• ${\displaystyle \sin(\pi -\alpha )=\sin \alpha }$ 
• ${\displaystyle \cos(\pi -\alpha )=-\cos \alpha }$ 
• ${\displaystyle \tan(\pi -\alpha )=-\tan \alpha }$ 
• ${\displaystyle \cot(\pi -\alpha )=-\cot \alpha }$ 
• ${\displaystyle \sec(\pi -\alpha )=-\sec \alpha }$ 
• ${\displaystyle \csc(\pi -\alpha )=\csc \alpha }$ 

• ${\displaystyle \sin \left({\frac {\pi }{2}}-\alpha \right)=\cos \alpha }$ 
• ${\displaystyle \cos \left({\frac {\pi }{2}}-\alpha \right)=\sin \alpha }$ 
• ${\displaystyle \tan \left({\frac {\pi }{2}}-\alpha \right)=\cot \alpha }$ 
• ${\displaystyle \cot \left({\frac {\pi }{2}}-\alpha \right)=\tan \alpha }$ 
• ${\displaystyle \sec \left({\frac {\pi }{2}}-\alpha \right)=\csc \alpha }$ 
• ${\displaystyle \csc \left({\frac {\pi }{2}}-\alpha \right)=\sec \alpha }$ 

• ${\displaystyle \sin \left({\frac {\pi }{2}}+\alpha \right)=\cos \alpha }$ 
• ${\displaystyle \cos \left({\frac {\pi }{2}}+\alpha \right)=-\sin \alpha }$ 
• ${\displaystyle \tan \left({\frac {\pi }{2}}+\alpha \right)=-\cot \alpha }$ 
• ${\displaystyle \cot \left({\frac {\pi }{2}}+\alpha \right)=-\tan \alpha }$ 
• ${\displaystyle \sec \left({\frac {\pi }{2}}+\alpha \right)=-\csc \alpha }$ 
• ${\displaystyle \csc \left({\frac {\pi }{2}}+\alpha \right)=\sec \alpha }$ 

• ${\displaystyle \sin \left({\frac {3\pi }{2}}-\alpha \right)=-\cos \alpha }$ 
• ${\displaystyle \cos \left({\frac {3\pi }{2}}-\alpha \right)=-\sin \alpha }$ 
• ${\displaystyle \tan \left({\frac {3\pi }{2}}-\alpha \right)=\cot \alpha }$ 
• ${\displaystyle \cot \left({\frac {3\pi }{2}}-\alpha \right)=\tan \alpha }$ 
• ${\displaystyle \sec \left({\frac {3\pi }{2}}-\alpha \right)=-\csc \alpha }$ 
• ${\displaystyle \csc \left({\frac {3\pi }{2}}-\alpha \right)=-\sec \alpha }$ 

• ${\displaystyle \sin \left({\frac {3\pi }{2}}+\alpha \right)=-\cos \alpha }$ 
• ${\displaystyle \cos \left({\frac {3\pi }{2}}+\alpha \right)=\sin \alpha }$ 
• ${\displaystyle \tan \left({\frac {3\pi }{2}}+\alpha \right)=-\cot \alpha }$ 
• ${\displaystyle \cot \left({\frac {3\pi }{2}}+\alpha \right)=-\tan \alpha }$ 
• ${\displaystyle \sec \left({\frac {3\pi }{2}}+\alpha \right)=\csc \alpha }$ 
• ${\displaystyle \csc \left({\frac {3\pi }{2}}+\alpha \right)=-\sec \alpha }$ 

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：

• ${\displaystyle \sin \left({\frac {k\pi }{2}}\pm \alpha \right),k\in \mathbb {Z} }$ 
• ${\displaystyle \cos \left({\frac {k\pi }{2}}\pm \alpha \right),k\in \mathbb {Z} }$ 
• ${\displaystyle \tan \left({\frac {k\pi }{2}}\pm \alpha \right),k\in \mathbb {Z} }$ 
• ${\displaystyle \cot \left({\frac {k\pi }{2}}\pm \alpha \right),k\in \mathbb {Z} }$ 
• ${\displaystyle \sec \left({\frac {k\pi }{2}}\pm \alpha \right),k\in \mathbb {Z} }$ 
• ${\displaystyle \csc \left({\frac {k\pi }{2}}\pm \alpha \right),k\in \mathbb {Z} }$ 

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${\displaystyle k}$ 为奇数时，${\displaystyle \sin }$ 变为${\displaystyle \cos }$ ，${\displaystyle \cos }$ 变为${\displaystyle \sin }$ ，${\displaystyle \tan }$ 变为${\displaystyle \cot }$ ，${\displaystyle \cot }$ 变为${\displaystyle \tan }$ ，${\displaystyle \sec }$ 变为${\displaystyle \csc }$ ，${\displaystyle \csc }$ 变为${\displaystyle \sec }$ ；而${\displaystyle k}$ 为偶数时，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 Sine & CoSecant（正弦以及余割为正）。
• 第三象限的 T 即是 Tangent & Cotangent（正切以及余切为正）。
• 第四象限的 C 即是 Cosine & SeCant（余弦以及正割为正）。

• 数学4 必修. 人民教育出版社. ISBN 978-7-107-20334-3.