数学基础知识总结 —— 3. 常用三角函数公式表
文章目录积化和差公式和差化积公式归一化公式倍（半）角公式、降（升）幂公式万能公式积化和差公式sinαcosβ=12[sin(α+β)+sin(α−β)]\sin \alpha \cos \beta = \frac{1}{2} \left [\sin(\alpha + \beta)+ \sin (\alpha - \beta) \right ]sinαcosβ=21[sin(α+β)+si
文章目录
积化和差公式
sin α cos β = 1 2 [ sin ( α + β ) + sin ( α − β ) ] \sin \alpha \cos \beta = \frac{1}{2} \left [ \sin(\alpha + \beta) + \sin (\alpha - \beta) \right ] sinαcosβ=21[sin(α+β)+sin(α−β)]
cos α sin β = 1 2 [ sin ( α + β ) − sin ( α − β ) ] \cos \alpha \sin \beta = \frac{1}{2} \left [ \sin(\alpha + \beta) - \sin (\alpha - \beta) \right ] cosαsinβ=21[sin(α+β)−sin(α−β)]
cos α cos β = 1 2 [ cos ( α + β ) + cos ( α − β ) ] \cos \alpha \cos \beta = \frac{1}{2} \left [ \cos(\alpha + \beta) + \cos (\alpha - \beta) \right ] cosαcosβ=21[cos(α+β)+cos(α−β)]
sin α sin β = − 1 2 [ cos ( α + β ) − cos ( α − β ) ] \sin \alpha \sin \beta = - \frac{1}{2} \left [ \cos(\alpha + \beta) - \cos (\alpha - \beta) \right ] sinαsinβ=−21[cos(α+β)−cos(α−β)]
和差化积公式
sin α + sin β = 2 sin α + β 2 cos α − β 2 \sin \alpha + \sin \beta = 2 \sin \frac{\alpha + \beta}{2} \cos \frac{\alpha - \beta}{2} sinα+sinβ=2sin2α+βcos2α−β
sin α − sin β = 2 cos α + β 2 sin α − β 2 \sin \alpha - \sin \beta = 2 \cos \frac{\alpha + \beta}{2} \sin \frac{\alpha - \beta}{2} sinα−sinβ=2cos2α+βsin2α−β
cos α + cos β = 2 cos α + β 2 cos α − β 2 \cos \alpha + \cos \beta = 2 \cos \frac{\alpha + \beta}{2} \cos \frac{\alpha - \beta}{2} cosα+cosβ=2cos2α+βcos2α−β
cos α − cos β = − 2 sin α + β 2 sin α − β 2 \cos \alpha - \cos \beta = -2 \sin \frac{\alpha + \beta}{2} \sin \frac{\alpha - \beta}{2} cosα−cosβ=−2sin2α+βsin2α−β
tan α + tan β = sin ( α + β ) cos α ⋅ cos β \tan \alpha + \tan \beta = \frac{\sin( \alpha + \beta )}{\cos \alpha \cdot \cos \beta} tanα+tanβ=cosα⋅cosβsin(α+β)
归一化公式
sin 2 x + c o s 2 x = 1 \sin^2 x + cos^2 x = 1 sin2x+cos2x=1
sec 2 x − tan 2 x = 1 \sec^2 x - \tan^2 x = 1 sec2x−tan2x=1
cosh 2 x − sinh 2 x = 1 \cosh^2 x - \sinh^2 x= 1 cosh2x−sinh2x=1
倍（半）角公式、降（升）幂公式
sin 2 x = 1 2 ( 1 − cos 2 x ) \sin^2 x = \frac{1}{2} (1 - \cos 2x) sin2x=21(1−cos2x)
cos 2 x = 1 2 ( 1 + cos 2 x ) \cos^2 x = \frac{1}{2} (1 + \cos 2x) cos2x=21(1+cos2x)
tan 2 x = 1 − cos 2 x 1 + cos 2 x \tan^2 x = \frac{1 - \cos 2x}{1 + \cos 2x} tan2x=1+cos2x1−cos2x
sin x = 2 sin x 2 cos x 2 \sin x = 2 \sin \frac{x}{2} \cos \frac{x}{2} sinx=2sin2xcos2x
cos x = 2 cos 2 x 2 − 1 = 1 − 2 sin 2 x 2 = cos 2 x 2 − sin 2 x 2 \cos x = 2 \cos^2 \frac{x}{2} - 1 = 1 - 2 \sin^2 \frac{x}{2} = \cos^2 \frac{x}{2} - \sin^2 \frac{x}{2} cosx=2cos22x−1=1−2sin22x=cos22x−sin22x
tan x = 2 tan ( x / 2 ) 1 − tan 2 ( x / 2 ) \tan x = \frac{2 \tan(x / 2)}{1 - \tan^2 (x / 2)} tanx=1−tan2(x/2)2tan(x/2)
万能公式
令 u = tan x 2 u = \tan \frac{x}{2} u=tan2x，于是有
sin x = 2 u 1 + u 2 \sin x = \frac{2u}{1 + u^2} sinx=1+u22u
cos x = 1 − u 2 1 + u 2 \cos x = \frac{1 - u^2}{1 + u^2} cosx=1+u21−u2
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