三角公式

Posted by Zcat233 on October 26, 2024

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:

\[\sin α + \sin β = 2 \sin \frac{α + β}{2} \cos \frac{α - β}{2}\] \[\sin α - \sin β = 2 \cos \frac{α + β}{2} \sin \frac{α - β}{2}\] \[\cos α + \cos β = 2 \cos \frac{α + β}{2} \cos \frac{α - β}{2}\] \[\cos α - \cos β = -2 \sin \frac{α + β}{2} \sin \frac{α - β}{2}\]