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三角函数公式

诱导公式​

sin⁡(−a)=−sin⁡(a)\sin(-a) = -\sin(a) cos⁡(−a)=cos⁡(a) \cos(-a) = \cos(a) sin⁡(π2−a)=cos⁡(a) \sin\left(\frac{\pi}{2} - a\right) = \cos(a) cos⁡(π2−a)=sin⁡(a) \cos\left(\frac{\pi}{2} - a\right) = \sin(a) sin⁡(π2+a)=cos⁡(a) \sin\left(\frac{\pi}{2} + a\right) = \cos(a) cos⁡(π2+a)=−sin⁡(a) \cos\left(\frac{\pi}{2} + a\right) = -\sin(a) sin⁡(π−a)=sin⁡(a) \sin(\pi - a) = \sin(a) cos⁡(π−a)=−cos⁡(a) \cos(\pi - a) = -\cos(a) sin⁡(π+a)=−sin⁡(a) \sin(\pi + a) = -\sin(a) cos⁡(π+a)=−cos⁡(a) \cos(\pi + a) = -\cos(a) tan⁡(a)=sin⁡(a)cos⁡(a) \tan(a) = \frac{\sin(a)}{\cos(a)}

两角和与差的三角函数​

sin⁡(a+b)=sin⁡(a)cos⁡(b)+cos⁡(a)sin⁡(b)\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) cos⁡(a+b)=cos⁡(a)cos⁡(b)−sin⁡(a)sin⁡(b) \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) sin⁡(a−b)=sin⁡(a)cos⁡(b)−cos⁡(a)sin⁡(b) \sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b) cos⁡(a−b)=cos⁡(a)cos⁡(b)+sin⁡(a)sin⁡(b) \cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b) tan⁡(a+b)=tan⁡(a)+tan⁡(b)1−tan⁡(a)tan⁡(b) \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} tan⁡(a−b)=tan⁡(a)−tan⁡(b)1+tan⁡(a)tan⁡(b) \tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)}

三角函数和差化积公式​

sin⁡(a)+sin⁡(b)=2sin⁡(a+b2)cos⁡(a−b2)\sin(a) + \sin(b) = 2\sin\left(\frac{a + b}{2}\right)\cos\left(\frac{a - b}{2}\right) sin⁡(a)−sin⁡(b)=2cos⁡(a+b2)sin⁡(a−b2) \sin(a) - \sin(b) = 2\cos\left(\frac{a + b}{2}\right)\sin\left(\frac{a - b}{2}\right) cos⁡(a)+cos⁡(b)=2cos⁡(a+b2)cos⁡(a−b2) \cos(a) + \cos(b) = 2\cos\left(\frac{a + b}{2}\right)\cos\left(\frac{a - b}{2}\right) cos⁡(a)−cos⁡(b)=−2sin⁡(a+b2)sin⁡(a−b2) \cos(a) - \cos(b) = -2\sin\left(\frac{a + b}{2}\right)\sin\left(\frac{a - b}{2}\right)

积化和差公式​

sin⁡(a)sin⁡(b)=−12[cos⁡(a+b)−cos⁡(a−b)]\sin(a)\sin(b) = -\frac{1}{2} \left[\cos(a + b) - \cos(a - b)\right] cos⁡(a)cos⁡(b)=12[cos⁡(a+b)+cos⁡(a−b)] \cos(a)\cos(b) = \frac{1}{2} \left[\cos(a + b) + \cos(a - b)\right] sin⁡(a)cos⁡(b)=12[sin⁡(a+b)+sin⁡(a−b)] \sin(a)\cos(b) = \frac{1}{2} \left[\sin(a + b) + \sin(a - b)\right]

二倍角公式​

sin⁡(2a)=2sin⁡(a)cos⁡(a)\sin(2a) = 2\sin(a)\cos(a) cos⁡(2a)=cos⁡2(a)−sin⁡2(a)=2cos⁡2(a)−1=1−2sin⁡2(a) \cos(2a) = \cos^2(a) - \sin^2(a) = 2\cos^2(a) - 1 = 1 - 2\sin^2(a)

半角公式​

sin⁡2(a2)=1−cos⁡(a)2\sin^2\left(\frac{a}{2}\right) = \frac{1 - \cos(a)}{2} cos⁡2(a2)=1+cos⁡(a)2 \cos^2\left(\frac{a}{2}\right) = \frac{1 + \cos(a)}{2} tan⁡(a2)=1−cos⁡(a)sin⁡(a)=sin⁡(a)1+cos⁡(a) \tan\left(\frac{a}{2}\right) = \frac{1 - \cos(a)}{\sin(a)} = \frac{\sin(a)}{1 + \cos(a)}

万能公式​

sin⁡(a)=2tan⁡(a2)1+tan⁡2(a2)\sin(a) = \frac{2\tan\left(\frac{a}{2}\right)}{1 + \tan^2\left(\frac{a}{2}\right)} cos⁡(a)=1−tan⁡2(a2)1+tan⁡2(a2) \cos(a) = \frac{1 - \tan^2\left(\frac{a}{2}\right)}{1 + \tan^2\left(\frac{a}{2}\right)} tan⁡(a)=2tan⁡(a2)1−tan⁡2(a2) \tan(a) = \frac{2\tan\left(\frac{a}{2}\right)}{1 - \tan^2\left(\frac{a}{2}\right)}

其他公式​

asin⁡(a)+bcos⁡(a)=a2+b2sin⁡(a+c)a\sin(a) + b\cos(a) = \sqrt{a^2 + b^2}\sin(a + c)

其中，tan⁡(c)=ba\tan(c) = \frac{b}{a}

asin⁡(a)−bcos⁡(a)=a2+b2cos⁡(a−c)a\sin(a) - b\cos(a) = \sqrt{a^2 + b^2}\cos(a - c)

其中，tan⁡(c)=ab\tan(c) = \frac{a}{b}

1+sin⁡(a)=(sin⁡(a2)+cos⁡(a2))21 + \sin(a) = \left(\sin\left(\frac{a}{2}\right) + \cos\left(\frac{a}{2}\right)\right)^2 1−sin⁡(a)=(sin⁡(a2)−cos⁡(a2))2 1 - \sin(a) = \left(\sin\left(\frac{a}{2}\right) - \cos\left(\frac{a}{2}\right)\right)^2

其他非重点三角函数​

csc⁡(a)=1sin⁡(a)\csc(a) = \frac{1}{\sin(a)} sec⁡(a)=1cos⁡(a)\sec(a) = \frac{1}{\cos(a)}

双曲函数​

sinh⁡(a)=ea−e−a2 \sinh(a) = \frac{e^a - e^{-a}}{2} cosh⁡(a)=ea+e−a2\cosh(a) = \frac{e^a + e^{-a}}{2} tanh⁡(a)=sinh⁡(a)cosh⁡(a)\tanh(a) = \frac{\sinh(a)}{\cosh(a)}