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里面是错误的知识

三角函数​

诱导公式​

sin⁡(−α)=−sin⁡α\sin(-\alpha) = -\sin \alpha cos⁡(−α)=cos⁡α\cos(-\alpha) = \cos \alpha sin⁡(π2−α)=cos⁡α\sin(\frac{\pi}2 - \alpha) = \cos \alpha cos⁡(π2−α)=sin⁡α\cos(\frac{\pi}2 - \alpha) = \sin \alpha sin⁡(π2+α)=sin⁡α\sin(\frac{\pi}2 + \alpha) = \sin \alpha cos⁡(π2+α)=cos⁡α\cos(\frac{\pi}2 + \alpha) = \cos \alpha sin⁡(π−α)=sin⁡α\sin(\pi - \alpha) = \sin \alpha cos⁡(π−α)=−cos⁡α\cos(\pi - \alpha) = -\cos \alpha sin⁡(π+α)=−sin⁡α\sin(\pi + \alpha) = -\sin \alpha cos⁡(π+α)=−cos⁡α\cos(\pi + \alpha) = -\cos \alpha

平方关系​

1+tan⁡2α=sec⁡2α1 + \tan^2 \alpha = \sec^2 \alpha 1+cot⁡2α=csc⁡2α1 + \cot^2 \alpha = \csc^2 \alpha sin⁡2α+cos⁡2α=1\sin^2 \alpha + \cos^2 \alpha = 1

两角和与差的三角函数​

sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta sin⁡(α−β)=sin⁡αcos⁡β−cos⁡αsin⁡β\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta tan⁡(α+β)=tan⁡α+tan⁡β1−tan⁡αtan⁡β\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} tan⁡(α−β)=tan⁡α−tan⁡β1+tan⁡αtan⁡β\tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}

积化和差公式​

cos⁡αcos⁡β=12[cos⁡(α+β)+cos⁡(α−β)]\cos \alpha \cos \beta = \frac{1}{2}[\cos(\alpha + \beta) + \cos(\alpha - \beta)] cos⁡αsin⁡β=12[sin⁡(α+β)−sin⁡(α−β)]\cos \alpha \sin \beta = \frac{1}{2}[\sin(\alpha + \beta) - \sin(\alpha - \beta)] sin⁡αcos⁡β=12[sin⁡(α+β)+sin⁡(α−β)]\sin \alpha \cos \beta = \frac{1}{2}[\sin(\alpha + \beta) + \sin(\alpha - \beta)] sin⁡αsin⁡β=−12[cos⁡(α+β)−cos⁡(α−β)]\sin \alpha \sin \beta = -\frac{1}{2}[\cos(\alpha + \beta) - \cos(\alpha - \beta)]

和差化积公式​

sin⁡α+sin⁡β=2sin⁡(α+β2)cos⁡(α−β2)\sin \alpha + \sin \beta = 2 \sin \left( \frac{\alpha + \beta}{2} \right) \cos \left( \frac{\alpha - \beta}{2} \right) sin⁡α−sin⁡β=2cos⁡(α+β2)sin⁡(α−β2)\sin \alpha - \sin \beta = 2 \cos \left( \frac{\alpha + \beta}{2} \right) \sin \left( \frac{\alpha - \beta}{2} \right) cos⁡α+cos⁡β=2cos⁡(α+β2)cos⁡(α−β2)\cos \alpha + \cos \beta = 2 \cos \left( \frac{\alpha + \beta}{2} \right) \cos \left( \frac{\alpha - \beta}{2} \right) cos⁡α−cos⁡β=−2sin⁡(α+β2)sin⁡(α−β2)\cos \alpha - \cos \beta = -2 \sin \left( \frac{\alpha + \beta}{2} \right) \sin \left( \frac{\alpha - \beta}{2} \right)

倍角公式​

sin⁡2α=2sin⁡αcos⁡α\sin 2\alpha = 2 \sin \alpha \cos \alpha cos⁡2α=cos⁡2α−sin⁡2α=1−2sin⁡2α=2cos⁡2α−1\cos 2\alpha = \cos^2 \alpha - \sin^2 \alpha = 1 - 2 \sin^2 \alpha = 2 \cos^2 \alpha - 1 tan⁡2α=2tan⁡α1−tan⁡2α\tan 2\alpha = \frac{2 \tan \alpha}{1 - \tan^2 \alpha} cot⁡2α=2cot⁡αcot⁡2α−1\cot 2\alpha = \frac{2 \cot \alpha}{\cot^2 \alpha - 1}

半角公式​

sin⁡2α2=1−cos⁡α2\sin^2 \frac{\alpha}{2} = \frac{1 - \cos \alpha}{2} cos⁡2α2=1+cos⁡α2\cos^2 \frac{\alpha}{2} = \frac{1 + \cos \alpha}{2} sin⁡α2=±1−cos⁡α2\sin \frac{\alpha}{2} = \pm \sqrt{\frac{1 - \cos \alpha}{2}} cos⁡α2=±1+cos⁡α2\cos \frac{\alpha}{2} = \pm \sqrt{\frac{1 + \cos \alpha}{2}} tan⁡α2=sin⁡α1+cos⁡α=1−cos⁡αsin⁡α\tan \frac{\alpha}{2} = \frac{\sin \alpha}{1 + \cos \alpha} = \frac{1 - \cos \alpha}{\sin \alpha} cot⁡α2=sin⁡α1+cos⁡α=1+cos⁡αsin⁡α\cot \frac{\alpha}{2} = \frac{\sin \alpha}{1 + \cos \alpha} = \frac{1 + \cos \alpha}{\sin \alpha}

万能公式​

sin⁡α=2tan⁡α21+tan⁡2α2\sin \alpha = \frac{2 \tan \frac{\alpha}{2}}{1 + \tan^2 \frac{\alpha}{2}} cos⁡α=1−tan⁡2α21+tan⁡2α2\cos \alpha = \frac{1 - \tan^2 \frac{\alpha}{2}}{1 + \tan^2 \frac{\alpha}{2}}

反三角函数恒等式​

arcsin⁡x+arccos⁡x=π2\arcsin x + \arccos x = \frac{\pi}{2} arctan⁡x+arccotx=π2\arctan x + \text{arccot} x = \frac{\pi}{2} sin⁡(arccos⁡x)=1−x2\sin(\arccos x) = \sqrt{1 - x^2} cos⁡(arcsin⁡x)=1−x2\cos(\arcsin x) = \sqrt{1 - x^2} sin⁡(arcsin⁡x)=x\sin(\arcsin x) = x arcsin⁡(sin⁡x)=x\arcsin(\sin x) = x arccos⁡(cos⁡x)=x\arccos(\cos x) = x arccos⁡(−x)=π−arccos⁡x\arccos(-x) = \pi - \arccos x