1. 二倍角的正弦、余弦、正切公式:
\[\sin 2α = 2 \sin α \cos α\]
\[\cos 2α = \cos^2 α - \sin^2 α = 1 - 2 \sin^2 α = 2 \cos^2 α - 1\]
\[\tan 2α = \frac{2 \tan α}{1 - \tan^2 α}\]
//1.1.升幂缩角
\[1+\cos 2α = 2\cos^2 α\]
\[1-\cos 2α = 2\sin^2 α\]
//1.2.降幂扩角
\[\cos^2 α = \frac{1+\cos 2α}{2}\]
\[\sin^2 α = \frac{1-\cos 2α}{2}\]
1.3.平方公式
\[1 + \sin 2α = (\sin α + \cos α)^2\]
\[1 - \sin 2α = (\sin α - \cos α)^2\]
2. 半角公式:
\[\sin \frac{α}{2} = ± \sqrt \frac{1 - \cos α}{2}\]
\[\cos \frac{α}{2} = ± \sqrt\frac{1 + \cos α}{2}\]
\[\tan \frac{α}{2} = \frac{1 - \cos α}{\sin α} = \frac{\sin α}{1 + \cos α} = ± \sqrt \frac{1 - \cos α}{1 + \cos α}\]
3. 万能公式:
\[\sin α = \frac{2 \tan \frac{α}{2}}{1 + \tan^2 \frac{α}{2}}\]
\[\cos α = \frac{1 - \tan^2 \frac{α}{2}}{1 + \tan^2 \frac{α}{2}}\]
\[\tan α = \frac{2 \tan \frac{α}{2}}{1 - \tan^2 \frac{α}{2}}\]
4. 积化和差公式:
\[\sin α \cos β = \frac{1}{2} [\sin (α + β) + \sin (α - β)]\]
\[\cos α \sin β = \frac{1}{2} [\sin (α + β) - \sin (α - β)]\]
\[\cos α \cos β = \frac{1}{2} [\cos (α + β) + \cos (α - β)]\]
\[\sin α \sin β = - \frac{1}{2} [\cos (α + β) - \cos (α - β)]\]
5. 和差化积公式:
\[\sin α + \sin β = 2 \sin \frac{α + β}{2} \cos \frac{α - β}{2}\]
\[\sin α - \sin β = 2 \cos \frac{α + β}{2} \sin \frac{α - β}{2}\]
\[\cos α + \cos β = 2 \cos \frac{α + β}{2} \cos \frac{α - β}{2}\]
\[\cos α - \cos β = -2 \sin \frac{α + β}{2} \sin \frac{α - β}{2}\]