算法和数据结构笔记

三角恒等式

1. 概念[1][2]

1.1 勾股定理

  • sin⁡θ=1/csc⁡θ\sin \theta = 1 / \csc \theta 或者 csc⁡θ=1/sin⁡θ\csc \theta = 1/\sin \theta
  • cos⁡θ=1/sec⁡θ\cos \theta = 1 / \sec \theta 或者 sec⁡θ=1/cos⁡θ\sec \theta = 1/\cos \theta
  • tan⁡θ=1/cot⁡θ\tan \theta = 1 / \cot \theta 或者 cot⁡θ=1/tan⁡θ\cot \theta = 1/\tan \theta

一个重要的等式：

asin⁡x+bcos⁡x=a2+b2sin⁡(x+arctan⁡ba) a\sin x + b\cos x = \sqrt{a^2 + b^2}\sin\left(x + \arctan\frac{b}{a}\right)

1.2 半角公式

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

1.3 二倍角公式

sin⁡2α=2sin⁡αcos⁡α \sin 2\alpha = 2\sin\alpha\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

tan⁡2α=2tan⁡α1−tan⁡2α \tan 2\alpha = \frac{2\tan\alpha}{1 - \tan^2\alpha}

1.4 三倍角公式

sin⁡3α=3sin⁡α−4sin⁡3α=4sin⁡(60°−α)sin⁡αsin⁡(60°+α) \sin 3\alpha = 3\sin\alpha - 4\sin^3\alpha = 4\sin\left(60\degree - \alpha\right)\sin\alpha\sin\left(60\degree + \alpha\right)

cos⁡3α=4cos⁡3α−3cos⁡α=4cos⁡(60°−α)cos⁡αcos⁡(60°+α) \cos 3\alpha = 4\cos^3\alpha - 3\cos\alpha = 4\cos\left(60\degree - \alpha\right)\cos\alpha\cos\left(60\degree + \alpha\right)

1.5 和差化积

sin⁡α+sin⁡β=2sin⁡α+β2cos⁡α−β2 \sin\alpha + \sin\beta = 2\sin\frac{\alpha + \beta}{2}\cos\frac{\alpha - \beta}{2}

sin⁡α−sin⁡β=2cos⁡α+β2sin⁡α−β2 \sin\alpha - \sin\beta = 2\cos\frac{\alpha + \beta}{2}\sin\frac{\alpha - \beta}{2}

cos⁡α+cos⁡β=2cos⁡α+β2cos⁡α−β2 \cos\alpha + \cos\beta = 2\cos\frac{\alpha + \beta}{2}\cos\frac{\alpha - \beta}{2}

cos⁡α−cos⁡β=−2sin⁡α+β2sin⁡α−β2 \cos\alpha - \cos\beta = -2\sin\frac{\alpha + \beta}{2}\sin\frac{\alpha - \beta}{2}

1.6 积化合差

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

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

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

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

1.7 万能公式

sin⁡2α=2tan⁡α1+tan⁡2α \sin 2\alpha = \frac{2\tan\alpha}{1 + \tan^2\alpha}

cos⁡2α=1−tan⁡2α1+tan⁡2α \cos 2\alpha = \frac{1 - \tan^2\alpha}{1 + \tan^2\alpha}

tan⁡2α=2tan⁡α1−tan⁡2α \tan 2\alpha = \frac{2\tan\alpha}{1 - \tan^2\alpha}

  1. 三角恒等式，维基百科，https://zh.wikipedia.org/wiki/三角恒等式在新窗口打开 ↩︎

  2. https://byjus.com/maths/trigonometric-identities/在新窗口打开 ↩︎