我发现采用博客学习也是种不错的方式。

同角

  • sin⁡2α+cos⁡2α=1\sin^2\alpha+\cos^2\alpha=1
  • tan⁡α=sin⁡αcos⁡α\tan\alpha=\frac{\sin\alpha}{\cos\alpha}

正弦

正弦定理。

  • asin⁡A=bsin⁡B=csin⁡C=2R=D\frac a{\sin A}=\frac b{\sin B}=\frac c{\sin C}=2R=D

余弦

余弦定理。

  • a2=b2+c2−2bccos⁡A\begin{array}{l}a^2=b^2+c^2-2bc\cos A\end{array}
  • b2=a2+c2−2accos⁡B\begin{array}{l}b^2=a^2+c^2-2ac\cos B\end{array}
  • c2=a2+b2−2abcos⁡C\begin{array}{l}c^2=a^2+b^2-2ab\cos C\end{array}

万能

  • sin⁡2α=2tan⁡α1+tan⁡2α\sin2\alpha=\frac{2\tan\alpha}{1+\tan^2\alpha}
  • cos⁡2α=1−tan⁡2α1+tan⁡2α\cos2\alpha=\frac{1-\tan^2\alpha}{1+\tan^2\alpha}
  • tan⁡2α=2tan⁡α1−tan⁡2α\tan2\alpha=\frac{2\tan\alpha}{1-\tan^2\alpha}

倍数

  • sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β;sin⁡(α−β)=sin⁡αcos⁡β−cos⁡αsin⁡β\begin{array}{l}\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta;\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta\end{array}
  • cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β;cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\begin{array}{l}\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta;\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta\end{array}
  • sin⁡2α=2sin⁡αcos⁡α\begin{array}{l}\sin2\alpha=2\sin\alpha\cos\alpha\end{array}
  • cos⁡2α=cos⁡2α−sin⁡2α=2cos⁡2α−1=1−2sin⁡2α\begin{array}{l}\cos2\alpha=\cos^2\alpha-\sin^2\alpha=2\cos^2\alpha-1=1-2\sin^2\alpha\end{array}
  • sin⁡α2=±1−cos⁡α2\begin{array}{l}\sin\frac\alpha2=\pm\sqrt{\frac{1-\cos\alpha}2}\end{array}
  • cos⁡α2=±1+cos⁡α2\begin{array}{l}\cos\frac\alpha2=\pm\sqrt{\frac{1+\cos\alpha}2}\end{array}
  • tan⁡α2=1−cos⁡αsin⁡α=sin⁡α1−cos⁡α\begin{array}{l}\tan\frac\alpha2=\frac{1-\cos\alpha}{\sin\alpha}=\frac{\sin\alpha}{1-\cos\alpha}\end{array}

辅助

  • asin⁡θ+bcos⁡θ=a2+b2sin⁡(θ+φ)a\sin\theta+b\cos\theta=\sqrt{a^2+b^2}\sin(\theta+\varphi)