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三角函数

基本关系​

sin⁡acos⁡a=tan⁡a\Large \frac{\sin a}{\cos a} = \tan a

倒数关系​

  • sin⁡α⋅csc⁡α=1\sin \alpha \cdot \csc \alpha = 1
  • cos⁡α⋅sec⁡α=1\cos \alpha \cdot \sec \alpha = 1
  • tan⁡α⋅cot⁡α=1\tan \alpha \cdot \cot \alpha = 1

平方关系​

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

基本公式​

两角和的公式​

  • sin⁡(α±β)=sin⁡α⋅cos⁡β±cos⁡α⋅sin⁡β\sin (\alpha \pm \beta) = \sin \alpha \cdot \cos \beta \pm \cos \alpha \cdot \sin \beta
  • cos⁡(α±β)=cos⁡α⋅cos⁡β−(+)sin⁡α⋅sin⁡β\cos (\alpha \pm \beta) = \cos \alpha \cdot \cos \beta -(+) \sin \alpha \cdot \sin \beta

倍角公式​

  • sin⁡2α=2sin⁡α⋅cos⁡α\sin 2\alpha = 2 \sin \alpha \cdot \cos \alpha
  • cos⁡2α=cos⁡2α−sin⁡2α=2cos⁡2α−1=1−2sin⁡2α\cos 2\alpha = \cos^2 \alpha - \sin^2 \alpha = 2 \cos^2 \alpha - 1 = 1 - 2\sin^2 \alpha

降幂公式​

  • sin⁡2α=12(1−cos⁡2α)\sin^2 \alpha = \frac{1}{2} (1 - \cos 2 \alpha)
  • cos⁡2α=12(1+cos⁡2α)\cos^2 \alpha = \frac{1}{2} (1 + \cos 2 \alpha)