本文详细介绍了三角函数的基本关系、诱导公式、倍角公式、半角公式、和差公式、积化和差公式，以及泰勒公式等内容，并配以常见不等式和极限表达。此外，还涵盖了求导的基本规则和一些特殊函数的导数计算。是初学者和进阶者必备的参考资料。
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基础预备知识
三角函数
csc α = 1 sin α \csc \ \alpha = \cfrac{1}{\sin \ \alpha} csc α=sin α1
sec α = 1 cos α \sec \ \alpha = \cfrac{1}{\cos \ \alpha} sec α=cos α1
cot α = 1 tan α \cot \ \alpha = \cfrac{1}{\tan \ \alpha} cot α=tan α1
sin 2 α + cos 2 α = 1 \sin^2 \ \alpha + \cos^2 \ \alpha = 1 sin2 α+cos2 α=1
1 + tan 2 α = sec 2 α 1+\tan^2 \ \alpha = \sec^2 \ \alpha 1+tan2 α=sec2 α
1 + cot 2 α = csc 2 α 1+\cot^2 \ \alpha = \csc ^2 \ \alpha 1+cot2 α=csc2 α
诱导公式
θ = k π 2 + α \theta = \cfrac{k \pi}{2}+\alpha θ=2kπ+α 奇变偶不变，符号看象限
倍角公式
sin 2 α = 2 sin α cos α \sin 2\alpha=2\sin \alpha \cos \alpha sin2α=2sinαcosα
cos 2 α = cos 2 α − sin 2 α = 1 − 2 sin 2 α = 2 cos 2 α − 1 \cos 2\alpha=\cos^2 \alpha-\sin^2 \alpha = 1-2\sin^2 \alpha = 2\cos^2 \alpha-1 cos2α=cos2α−sin2α=1−2sin2α=2cos2α−1
sin 3 α = − 4 sin 3 α + 3 sin α \sin 3\alpha = -4\sin^3 \alpha+3\sin\alpha sin3α=−4sin3α+3sinα
cos 3 α = 4 cos 3 α − 3 cos α \cos3\alpha=4\cos^3\alpha-3\cos\alpha cos3α=4cos3α−3cosα
tan 2 α = 2 tan α 1 − tan 2 α \tan2\alpha = \cfrac {2\tan\alpha}{1-\tan^2\alpha} tan2α=1−tan2α2tanα
cot 2 α = cot 2 α − 1 2 cot α \cot2\alpha=\cfrac{\cot^2\alpha-1}{2\cot\alpha} cot2α=2cotαcot2α−1
半角
sin 2 α 2 = 1 2 ( 1 − cos α ) \sin^2 \cfrac \alpha 2 = \cfrac 12(1-\cos \alpha) sin22α=21(1−cosα)
cos 2 α 2 = 1 2 ( 1 + cos α ) \cos^2 \cfrac \alpha 2= \cfrac 12(1+\cos \alpha) cos22α=21(1+cosα)
sin α 2 = ± 1 − cos α 2 \sin \cfrac \alpha 2 = \pm \sqrt{\cfrac{1-\cos \alpha}{2}} sin2α=±21−cosα
cos α 2 = ± 1 + cos α 2 \cos \cfrac \alpha 2 = \pm \sqrt{\cfrac{1+\cos \alpha}{2}} cos2α=±21+cosα
tan α 2 = 1 − cos α sin α = sin α 1 + cos α = ± 1 − cos α 1 + cos α \tan \cfrac \alpha 2 =\cfrac{1-\cos\alpha}{\sin \alpha}=\cfrac{\sin\alpha}{1+\cos \alpha}=\pm \sqrt{\cfrac{1-\cos \alpha}{1+\cos\alpha}} tan2α=sinα1−cosα=1+cosαsinα=±1+cosα1−cosα
cot α 2 = sin α 1 − cos α = 1 + cos α sin α = ± 1 + cos α 1 − cos α \cot \cfrac \alpha 2 =\cfrac{\sin\alpha}{1-\cos\alpha}=\cfrac{1+\cos \alpha}{\sin\alpha}=\pm \sqrt{\cfrac{1+\cos \alpha}{1-\cos\alpha}} cot2α=1−cosαsinα=sinα1+cosα=±1−cosα1+cosα
和差
sin ( α ± β ) = sin α cos β ± cos α sin β \sin(\alpha \pm \beta)=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta sin(α±β)=sinαcosβ±cosαsinβ
cos ( α ± β ) = cos α cos β ∓ sin α sin β \cos(\alpha \pm \beta)=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta cos(α±β)=cosαcosβ∓sinαsinβ
tan ( α ± β ) = tan α ± tan β 1 ∓ tan α tan β \tan(\alpha \pm \beta) = \cfrac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta} tan(α±β)=1∓tanαtanβtanα±tanβ
cot ( α ± β ) = cot α cot β ∓ 1 cot β ± cot α \cot(\alpha \pm \beta) = \cfrac{\cot \alpha \cot \beta \mp 1}{\cot \beta \pm \cot \alpha} cot(α±β)=cotβ±cotαcotαcotβ∓1
积化和差
记住第一个，其他用诱导公式推
sin α cos β = 1 2 [ sin ( α + β ) + sin ( α − β ) ] \sin \alpha \cos \beta = \cfrac 12[\sin(\alpha + \beta)+\sin(\alpha-\beta)] sinαcosβ=21[sin(α+β)+sin(α−β)]
cos α sin β = 1 2 [ sin ( α + β ) − sin ( α − β ) ] \cos \alpha \sin \beta = \cfrac 12[\sin(\alpha + \beta)-\sin(\alpha-\beta)] cosαsinβ=21[sin(α+β)−sin(α−β)]
cos α cos β = 1 2 [ cos ( α + β ) + cos ( α − β ) ] \cos \alpha \cos \beta = \cfrac 12[\cos(\alpha + \beta)+\cos(\alpha-\beta)] cosαcosβ=21[cos(α+β)+cos(α−β)]
sin α sin β = 1 2 [ cos ( α − β ) − cos ( α + β ) ] \sin \alpha \sin \beta = \cfrac 12[\cos(\alpha - \beta)-\cos(\alpha+\beta)] sinαsinβ=21[cos(α−β)−cos(α+β)]
和差化积
记住第一个，其他用诱导公式推
sin α + sin β = 2 sin α + β 2 cos α − β 2 \sin \alpha +\sin \beta = 2\sin\cfrac{\alpha+\beta}{2}\cos \cfrac{\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 \cfrac{\alpha+\beta}{2}\sin\cfrac{\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\cfrac{\alpha+\beta}{2}\cos \cfrac{\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\cfrac{\alpha+\beta}{2} \sin \cfrac{\alpha-\beta}{2} cosα−cosβ=−2sin2α+βsin2α−β
万能公式
u = tan x 2 ⇒ sin x = 2 u 1 + u 2 , cos x = 1 − u 2 1 + u 2 u=\tan \cfrac x2 \Rightarrow \sin x =\cfrac{2u}{1+u^2},\cos x=\cfrac{1-u^2}{1+u^2} u=tan2x⇒sinx=1+u22u,cosx=1+u21−u2
因式分解公式
( a + b ) 3 = a 3 + 3 a 2 b + 3 a b 2 + b 3 (a+b)^3=a^3+3a^2b+3ab^2+b^3 (a+b)3=a3+3a2b+3ab2+b3
( a − b ) 3 = a 3 − 3 a 2 b + 3 a b 2 − b 3 (a-b)^3=a^3-3a^2b+3ab^2-b^3 (a−b)3=a3−3a2b+3ab2−b3
a 3 − b 3 = ( a − b ) ( a 2 + a b + b 2 ) a^3-b^3=(a-b)(a^2+ab+b^2) a3−b3=(a−
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