「三角函数公式」三角函数公式总结

三角函数公式

求导运算

$(\sin x)^{'} = \cos x (\sin x)'=\cos x$ (sinx)′=cosx $(\cos x)^{'} = \sin x (\cos x)'=\sin x$ (cosx)′=sinx $(\tan x)^{'} = \frac{1}{\cos^{2} x} = \sec^{2} x (\tan x)'=\frac{1}{\cos^2 x }=\sec^2 x$ (tanx)′=cos2x1=sec2x $(\cot x)^{'} = - \csc^{2} x = - \frac{1}{\sin^{2} x} (\cot x)'=-\csc ^2 x=-\frac{1}{\sin ^2 x}$ (cotx)′=−csc2x=−sin2x1 $(\sec x)^{'} = \frac{\sin x}{\cos^{2} x} = \sec x \tan x (\sec x)'=\frac{\sin x}{\cos^ 2 x}=\sec x\tan x$ (secx)′=cos2xsinx=secxtanx $(\arcsin x)^{'} = \frac{1}{\sqrt{1 - x^{2}}} (\arcsin x)'=\frac{1}{\sqrt{1-x^2}}$ (arcsinx)′=1−x21 $(\arccos x)^{'} = - \frac{1}{\sqrt{1 - x^{2}}} (\arccos x )'=-\frac{1}{\sqrt{1-x^2}}$ (arccosx)′=−1−x21 $(\arctan x)^{'} = \frac{1}{1 + x^{2}} (\arctan x)'=\frac{1}{1+x^2}$ (arctanx)′=1+x21 $(a r c c o t x)^{'} = - \frac{1}{1 + x^{2}} (\mathrm{arccot}\,x)'=-\frac{1}{1+x^2}$ (arccotx)′=−1+x21 $(a r c s e c x)^{'} = \frac{1}{∣ x ∣ \sqrt{x^{2} - 1}} (\mathrm{arcsec}\, x)'=\frac{1}{|x|\sqrt{x^2-1}}$ (arcsecx)′=∣x∣x2−11 $(a r c c s c x)^{'} = - \frac{1}{∣ x ∣ \sqrt{x^{2} - 1}} (\mathrm{arccsc}\, x)'=-\frac{1}{|x|\sqrt{x^2-1}}$ (arccscx)′=−∣x∣x2−11

和差角公式

$\sin (α \pm β) = \sin α \cos β \pm \cos α \sin β \sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta$ sin(α±β)=sinαcosβ±cosαsinβ $\cos (α \pm β) = \cos α \cos β \mp \sin α \sin β \cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta$ cos(α±β)=cosαcosβ∓sinαsinβ $\tan (α \pm β) = \frac{\tan α \pm \tan β}{1 \mp \tan α \tan β} \tan(\alpha\pm\beta)=\frac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\tan\beta}$ tan(α±β)=1∓tanαtanβtanα±tanβ $\cot (α \pm β) = \frac{\cot α \cot β \mp 1}{\cot β \pm \cot α} \cot(\alpha\pm\beta)=\frac{\cot\alpha\cot\beta\mp1}{\cot\beta\pm\cot\alpha}$ cot(α±β)=cotβ±cotαcotαcotβ∓1

和差化积

$\sin α \pm \sin β = 2 \sin \frac{α \pm β}{2} \cos \frac{α \mp β}{2} \sin\alpha\pm\sin\beta=2\sin\frac{\alpha\pm\beta}{2}\cos\frac{\alpha\mp\beta}{2}$ sinα±sinβ=2sin2α±βcos2α∓β $\cot α + \cos β = 2 \cos \frac{α + β}{2} \cos \frac{α - β}{2} \cot\alpha+\cos\beta=2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}$ cotα+cosβ=2cos2α+βcos2α−β $\cot α - \cos β = - 2 \sin \frac{α + β}{2} \sin \frac{α - β}{2} \cot\alpha-\cos\beta=-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}$ cotα−cosβ=−2sin2α+βsin2α−β $\tan α + \tan β = \frac{\sin (α + β)}{\cos α \cos β} \tan \alpha+\tan\beta=\frac{\sin(\alpha+\beta)}{\cos\alpha\cos\beta}$ tanα+tanβ=cosαcosβsin(α+β)

积化和差

$\cos α \sin β = \frac{1}{2} [\sin (α + β) - \sin (α - β)] \cos\alpha\sin\beta=\frac{1}{2}[\sin(\alpha+\beta)-\sin(\alpha-\beta)]$ cosαsinβ=21[sin(α+β)−sin(α−β)] $\sin α \cos β = \frac{1}{2} [\sin (α + β) + \sin (α - β)] \sin\alpha\cos\beta=\frac{1}{2}[\sin(\alpha+\beta)+\sin(\alpha-\beta)]$ sinαcosβ=21[sin(α+β)+sin(α−β)] $\cos α \cos β = \frac{1}{2} [\cos (α + β) + \cos (α - β)] \cos\alpha\cos\beta=\frac{1}{2}[\cos(\alpha+\beta)+\cos(\alpha-\beta)]$ cosαcosβ=21[cos(α+β)+cos(α−β)] $\sin α \sin β = \frac{1}{2} [\cos (α - β) - c o s (α + β)] \sin\alpha\sin\beta=\frac{1}{2}[\cos(\alpha-\beta)-cos(\alpha+\beta)]$ sinαsinβ=21[cos(α−β)−cos(α+β)]

二倍角公式

$\sin 2 α = 2 \sin α \cos α = \frac{2 \tan α}{1 + \tan^{2} α} \sin2\alpha=2\sin\alpha\cos\alpha=\frac{2\tan\alpha}{1+\tan^2\alpha}$ sin2α=2sinαcosα=1+tan2α2tanα $\cos 2 α = 2 \cos^{2} α - 1 = 1 - 2 \sin^{2} α = \frac{1 - \tan^{2} α}{1 + \tan^{2} α} \cos2\alpha=2\cos^2\alpha-1=1-2\sin^2\alpha=\frac{1-\tan^2\alpha}{1+\tan^2\alpha}$ cos2α=2cos2α−1=1−2sin2α=1+tan2α1−tan2α $\tan 2 α = \frac{2 \tan α}{1 - \tan^{2} α} \tan2\alpha=\frac{2\tan\alpha}{1-\tan^2\alpha}$ tan2α=1−tan2α2tanα

半角公式

$\sin \frac{α}{2} = \pm \sqrt{\frac{1 - \cos α}{2}} \sin\frac{\alpha}{2}=\pm\sqrt{\frac{1-\cos\alpha}{2}}$ sin2α=±21−cosα $\cos \frac{α}{2} = \pm \sqrt{\frac{1 + \cos α}{2}} \cos\frac{\alpha}{2}=\pm\sqrt{\frac{1+\cos\alpha}{2}}$ cos2α=±21+cosα $\tan \frac{α}{2} = \frac{\sin α}{1 + \cos α} = \frac{1 - \cos α}{\sin α} = \pm \sqrt{\frac{1 - \cos α}{1 + \cos α}} \tan\frac{\alpha}{2}=\frac{\sin\alpha}{1+\cos\alpha}=\frac{1-\cos\alpha}{\sin\alpha}=\pm\sqrt{\frac{1-\cos\alpha}{1+\cos\alpha}}$ tan2α=1+cosαsinα=sinα1−cosα=±1+cosα1−cosα $\cot \frac{α}{2} = \frac{1 + \cos α}{\sin α} = \frac{\sin α}{1 - \cos α} = \pm \sqrt{\frac{1 + \cos α}{1 - \cos α}} \cot\frac{\alpha}{2}=\frac{1+\cos\alpha}{\sin\alpha}=\frac{\sin\alpha}{1-\cos\alpha}=\pm\sqrt{\frac{1+\cos\alpha}{1-\cos\alpha}}$ cot2α=sinα1+cosα=1−cosαsinα=±1−cosα1+cosα

万能公式

$\sin α = \frac{2 t a n \frac{α}{2}}{1 + \tan^{2} \frac{α}{2}} \sin\alpha=\frac{2tan\frac{\alpha}{2}}{1+\tan^2\frac{\alpha}{2}}$ sinα=1+tan22α2tan2α $\cos α = \frac{1 - \tan^{2} \frac{α}{2}}{1 + \tan^{2} \frac{α}{2}} \cos\alpha=\frac{1-\tan^2\frac{\alpha}{2}}{1+\tan^2\frac{\alpha}{2}}$ cosα=1+tan22α1−tan22α $\tan α = \frac{2 \tan \frac{α}{2}}{1 - \tan^{2} \frac{α}{2}} \tan \alpha=\frac{2\tan\frac{\alpha}{2}}{1-\tan^2\frac{\alpha}{2}}$ tanα=1−tan22α2tan2α

辅助角公式

$a \sin α + b \cos α = \sqrt{a^{2} + b^{2}} \sin (α + φ), \tan φ = \frac{b}{a} a\sin\alpha+b\cos\alpha=\sqrt{a^2+b^2}\sin(\alpha+\varphi),\,\tan\varphi=\frac{b}{a}$ asinα+bcosα=a2+b2sin(α+φ),tanφ=ab

三角函数公式总结