三角函数公式
文章目录
- 诱导公式
- 公式一
- 公式二
- 公式三
- 公式四
- 公式五
- 公式六
- 两角和公式
- 倍角公式
- 半角公式
- 和差化积
- 积化和差
- 万能公式
- 降幂公式
- 正弦定理
- 余弦定理
诱导公式
记忆规律：奇变偶不变，符号看向限
奇偶是指 π 2 \frac{\large\pi}{\large2} 2π的奇数倍或偶数倍，变是指函数名。
符号看向限是指，将 α \alpha α看成锐角时，原函数值的符号。
应用技巧：
任意负角 → 3 或 1 用 公 式 任意正角 → 1 用 公 式 0-2 π → 2 或 4 用 公 式 锐角 \begin{aligned} \fcolorbox{red}{aqua}{任意负角}\xrightarrow[3或1]{用公式}&\fcolorbox{red}{aqua}{任意正角}\xrightarrow[1]{用公式}\fcolorbox{red}{aqua}{0-2}\pi\xrightarrow[2或4]{用公式}\fcolorbox{red}{aqua}{锐角} \end{aligned} 任意负角用公式3或1任意正角用公式10-2π用公式2或4锐角
公式一
sin ( α + k ⋅ 2 π ) = sin α ( k ∈ Z ) cos ( α + k ⋅ 2 π ) = cos α ( k ∈ Z ) tan ( α + k ⋅ 2 π ) = tan α ( k ∈ Z ) cot ( α + k ⋅ 2 π ) = cot α ( k ∈ Z ) sec ( α + k ⋅ 2 π ) = sec α ( k ∈ Z ) csc ( α + k ⋅ 2 π ) = csc α ( k ∈ Z ) \begin{aligned} \sin(\alpha+k\cdot2\pi)&=\sin\alpha (k\in\Z)\\ \cos(\alpha+k\cdot2\pi)&=\cos\alpha(k\in\Z)\\ \tan(\alpha+k\cdot2\pi)&=\tan\alpha(k\in\Z)\\ \cot(\alpha+k\cdot2\pi)&=\cot\alpha(k\in\Z)\\ \sec(\alpha+k\cdot2\pi)&=\sec\alpha(k\in\Z)\\ \csc(\alpha+k\cdot2\pi)&=\csc\alpha(k\in\Z)\\ \end{aligned} sin(α+k⋅2π)cos(α+k⋅2π)tan(α+k⋅2π)cot(α+k⋅2π)sec(α+k⋅2π)csc(α+k⋅2π)=sinα(k∈Z)=cosα(k∈Z)=tanα(k∈Z)=cotα(k∈Z)=secα(k∈Z)=cscα(k∈Z)
公式二
sin ( π + α ) = − sin ( α ) cos ( π + α ) = − cos ( α ) tan ( π + α ) = tan ( α ) cot ( π + α ) = cot ( α ) sec ( π + α ) = − sec ( α ) csc ( π + α ) = − csc ( α ) \begin{aligned} \sin(\pi+\alpha)&=-\sin(\alpha)\\ \cos(\pi+\alpha)&=-\cos(\alpha)\\ \tan(\pi+\alpha)&=\tan(\alpha)\\ \cot(\pi+\alpha)&=\cot(\alpha)\\ \sec(\pi+\alpha)&=-\sec(\alpha)\\ \csc(\pi+\alpha)&=-\csc(\alpha)\\ \end{aligned} sin(π+α)cos(π+α)tan(π+α)cot(π+α)sec(π+α)csc(π+α)=−sin(α)=−cos(α)=tan(α)=cot(α)=−sec(α)=−csc(α)
公式三
sin ( − α ) = − sin α cos ( − α ) = cos α tan ( − α ) = − tan α cot ( − α ) = − cot α sec ( − α ) = sec α csc ( − α ) = − csc α \begin{aligned} \sin(-\alpha)&=-\sin\alpha\\ \cos(-\alpha)&=\cos\alpha\\ \tan(-\alpha)&=-\tan\alpha\\ \cot(-\alpha)&=-\cot\alpha\\ \sec(-\alpha)&=\sec\alpha\\ \csc(-\alpha)&=-\csc\alpha\\ \end{aligned} sin(−α)cos(−α)tan(−α)cot(−α)sec(−α)csc(−α)=−sinα=cosα=−tanα=−cotα=secα=−cscα
公式四
sin ( π − α ) = sin α cos ( π − α ) = − cos α tan ( π − α ) = − tan α cot ( π − α ) = − cot α sec ( π − α ) = − sec α csc ( π − α ) = csc α \begin{aligned} \sin(\pi-\alpha)&=\sin\alpha\\ \cos(\pi-\alpha)&=-\cos\alpha\\ \tan(\pi-\alpha)&=-\tan\alpha\\ \cot(\pi-\alpha)&=-\cot\alpha\\ \sec(\pi-\alpha)&=-\sec\alpha\\ \csc(\pi-\alpha)&=\csc\alpha\\ \end{aligned} sin(π−α)cos(π−α)tan(π−α)cot(π−α)sec(π−α)csc(π−α)=sinα=−cosα=−tanα=−cotα=−secα=cscα
公式五
sin ( π 2 − α ) = cos α cos ( π 2 − α ) = sin α tan ( π 2 − α ) = cot α cot ( π 2 − α ) = tan α sec ( π 2 − α ) = csc α csc ( π 2 − α ) = sec α \begin{aligned} \sin(\frac{\pi}{2}-\alpha)&=\cos\alpha\\ \cos(\frac{\pi}{2}-\alpha)&=\sin\alpha\\ \tan(\frac{\pi}{2}-\alpha)&=\cot\alpha\\ \cot(\frac{\pi}{2}-\alpha)&=\tan\alpha\\ \sec(\frac{\pi}{2}-\alpha)&=\csc\alpha\\ \csc(\frac{\pi}{2}-\alpha)&=\sec\alpha\\ \end{aligned} sin(2π−α)cos(2π−α)tan(2π−α)cot(2π−α)sec(2π−α)csc(2π−α)=cosα=sinα=cotα=tanα=cscα=secα
公式六
sin ( π 2 + α ) = cos α cos ( π 2 + α ) = − sin α tan ( π 2 + α ) = − cot α cot ( π 2 + α ) = − tan α sec ( π 2 + α ) = − csc α csc ( π 2 + α ) = sec α \begin{aligned} \sin(\frac{\pi}{2}+\alpha)&=\cos\alpha\\ \cos(\frac{\pi}{2}+\alpha)&=-\sin\alpha\\ \tan(\frac{\pi}{2}+\alpha)&=-\cot\alpha\\ \cot(\frac{\pi}{2}+\alpha)&=-\tan\alpha\\ \sec(\frac{\pi}{2}+\alpha)&=-\csc\alpha\\ \csc(\frac{\pi}{2}+\alpha)&=\sec\alpha\\ \end{aligned} sin(2π+α)cos(2π+α)tan(2π+α)cot(2π+α)sec(2π+α)csc(2π+α)=cosα=−sinα=−cotα=−tanα=−cscα=secα
两角和公式
sin ( α + β ) = sin α cos β + cos α sin β sin ( α − β ) = sin α cos β − cos α sin β cos ( α + β ) = cos α cos β − sin α sin β cos ( α − β ) = cos α cos β + sin α sin β tan ( α + β ) = tan α + tan β 1 − tan α tan β tan ( α − β ) = tan α − tan β 1 + tan α tan β cot ( α + β ) = cot α cot β − 1 cot β + cot α cot ( α − β ) = cot α cot β + 1 cot β − cot α \begin{aligned} \sin(\alpha+\beta)&=\sin\alpha\cos\beta+\cos\alpha\sin\beta\\ \sin(\alpha-\beta)&=\sin\alpha\cos\beta-\cos\alpha\sin\beta\\ \cos(\alpha+\beta)&=\cos\alpha\cos\beta-\sin\alpha\sin\beta\\ \cos(\alpha-\beta)&=\cos\alpha\cos\beta+\sin\alpha\sin\beta\\\\ \tan(\alpha+\beta)&=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}\\\\ \tan(\alpha-\beta)&=\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}\\\\ \cot(\alpha+\beta)&=\frac{\cot\alpha\cot\beta-1}{\cot\beta+\cot\alpha}\\\\ \cot(\alpha-\beta)&=\frac{\cot\alpha\cot\beta+1}{\cot\beta-\cot\alpha} \end{aligned} sin(α+β)sin(α−β)cos(α+β)cos(α−β)tan(α+β)tan(α−β)cot(α+β)cot(α−β)=sinαcosβ+cosαsinβ=sinαcosβ−cosαsinβ=cosαcosβ−sinαsinβ=cosαcosβ+sinαsinβ=1−tanαtanβtanα+tanβ=1+tanαtanβtanα−tanβ=cotβ+cotαcotαcotβ−1=cotβ−cotαcotαcotβ+1
倍角公式
tan 2 α = 2 tan α 1 − tan 2 α sin 2 α = 2 sin α cos α cos 2 α = cos 2 α − sin 2 α = 2 cos 2 α − 1 = 1 − 2 sin 2 α \begin{aligned} \tan2\alpha&=\frac{2\tan\alpha}{1-\tan^2\alpha}\\\\ \sin2\alpha&=2\sin\alpha\cos\alpha\\\\ \cos2\alpha&=\cos^2\alpha-\sin^2\alpha\\ &=2\cos^2\alpha-1\\ &=1-2\sin^2\alpha \end{aligned} tan2αsin2αcos2α=1−tan2α2tanα=2sinαcosα=cos2α−sin2α=2cos2α−1=1−2sin2α
半角公式
sin α 2 = ± 1 − cos α 2 cos α 2 = ± 1 + cos α 2 tan α 2 = sin α 1 + cos α = 1 − cos α sin α = ± 1 − cos α 1 + cos α cot α 2 = 1 + cos α sin α = sin α 1 − cos α = ± 1 + cos α 1 − cos α \begin{aligned} \sin\frac{\alpha}{2}&=\pm\sqrt{\frac{1-\cos\alpha}{2}}\\\\ \cos\frac{\alpha}{2}&=\pm\sqrt{\frac{1+\cos\alpha}{2}}\\\\ \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}}\\\\ \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}}\\\\ \end{aligned} sin2αcos2αtan2α=1+cosαsinαcot2α=sinα1+cosα=±21−cosα=±21+cosα=sinα1−cosα=±1+cosα1−cosα=1−cosαsinα=±1−cosα1+cosα
和差化积
sin α + sin β = 2 sin ( α + β 2 ) cos ( α − β 2 ) sin α − sin β = 2 cos ( α + β 2 ) sin ( α − β 2 ) cos α + cos β = 2 cos ( α + β 2 ) cos ( α − β 2 ) cos α − cos β = − 2 sin ( α + β 2 ) sin ( α − β 2 ) tan α + tan β = sin ( α + β ) cos α + cos β \begin{aligned} \sin\alpha+\sin\beta&=2\sin\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)\\\\ \sin\alpha-\sin\beta&=2\cos\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right)\\\\ \cos\alpha+\cos\beta&=2\cos\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)\\\\ \cos\alpha-\cos\beta&=-2\sin\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right)\\\\ \tan\alpha+\tan\beta&=\frac{\sin(\alpha+\beta)}{\cos\alpha+\cos\beta} \end{aligned} sinα+sinβsinα−sinβcosα+cosβcosα−cosβtanα+tanβ=2sin(2α+β)cos(2α−β)=2cos(2α+β)sin(2α−β)=2cos(2α+β)cos(2α−β)=−2sin(2α+β)sin(2α−β)=cosα+cosβsin(α+β)
积化和差
sin α cos β = 1 2 [ sin ( α + β ) + sin ( α − β ) ] cos α sin β = 1 2 [ sin ( α + β ) − sin ( α − β ) ] sin α sin β = 1 2 [ cos ( α − β ) − cos ( α + β ) ] cos α cos β = 1 2 [ cos ( α − β ) + cos ( α + β ) ] \begin{aligned} \sin\alpha\cos\beta&=\frac{1}{2}[\sin(\alpha+\beta)+\sin(\alpha-\beta)]\\\\ \cos\alpha\sin\beta&=\frac{1}{2}[\sin(\alpha+\beta)-\sin(\alpha-\beta)]\\\\ \sin\alpha\sin\beta&=\frac{1}{2}[\cos(\alpha-\beta)-\cos(\alpha+\beta)]\\\\ \cos\alpha\cos\beta&=\frac{1}{2}[\cos(\alpha-\beta)+\cos(\alpha+\beta)]\\\\ \end{aligned} sinαcosβcosαsinβsinαsinβcosαcosβ=21[sin(α+β)+sin(α−β)]=21[sin(α+β)−sin(α−β)]=21[cos(α−β)−cos(α+β)]=21[cos(α−β)+cos(α+β)]
万能公式
sin α = 2 tan α 2 1 + tan 2 α 2 cos α = 1 − tan 2 α 2 1 + tan 2 α 2 tan α = 2 tan α 2 1 − tan 2 α 2 \begin{aligned} \sin\alpha&=\frac{2\tan\frac{\alpha}{2}}{1+\tan^2\frac{\alpha}{2}}\\\\ \cos\alpha&=\frac{1-\tan^2\frac{\alpha}{2}}{1+\tan^2\frac{\alpha}{2}}\\\\ \tan\alpha&=\frac{2\tan\frac{\alpha}{2}}{1-\tan^2\frac{\alpha}{2}}\\\\ \end{aligned} sinαcosαtanα=1+tan22α2tan2α=1+tan22α1−tan22α=1−tan22α2tan2α
降幂公式
sin 2 α = 1 − cos 2 α 2 cos 2 α = 1 + cos 2 α 2 tan 2 α = 1 − cos 2 α 1 + cos 2 α \begin{aligned} \sin^2\alpha=\frac{1-\cos2\alpha}{2}\\\\ \cos^2\alpha=\frac{1+\cos2\alpha}{2}\\\\ \tan^2\alpha=\frac{1-\cos2\alpha}{1+\cos2\alpha}\\\\ \end{aligned} sin2α=21−cos2αcos2α=21+cos2αtan2α=1+cos2α1−cos2α
正弦定理
在任意 △ \triangle △ABC中，角 α , β , γ \alpha,\beta,\gamma α,β,γ所对的边长分别为a,b,c,
三角形外接圆半径为R,则有
a sin α = b sin β = c sin γ = 2 R \begin{aligned} \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R \\ \end{aligned} sinαa=sinβb=sinγc=2R
变形得
a = 2 R sin α b = 2 R sin β c = 2 R sin γ a : b : c = sin α : sin β : sin γ S = 1 2 a b sin γ = 1 2 a c sin β = 1 2 b c sin α \begin{aligned} a&=2R\sin\alpha\\b&=2R\sin\beta\\c&=2R\sin\gamma\\\\ a:b:c&=\sin\alpha:\sin\beta:\sin\gamma\\\\ S&=\frac{1}{2}ab\sin\gamma\\&=\frac{1}{2}ac\sin\beta\\&=\frac{1}{2}bc\sin\alpha \end{aligned} abca:b:cS=2Rsinα=2Rsinβ=2Rsinγ=sinα:sinβ:sinγ=21absinγ=21acsinβ=21bcsinα
余弦定理
a 2 = b 2 + c 2 − 2 b c cos α b 2 = a 2 + c 2 − 2 a c cos β c 2 = a 2 + b 2 − 2 a b cos γ \begin{aligned} a^2&=b^2+c^2-2bc\cos\alpha\\ b^2&=a^2+c^2-2ac\cos\beta\\ c^2&=a^2+b^2-2ab\cos\gamma \end{aligned} a2b2c2=b2+c2−2bccosα=a2+c2−2accosβ=a2+b2−2abcosγ
也可表示为
cos α = b 2 + c 2 − a 2 2 b c cos β = a 2 + c 2 − b 2 2 a c cos γ = a 2 + b 2 − c 2 2 a b \begin{aligned} \cos\alpha=\frac{b^2+c^2-a^2}{2bc}\\ \cos\beta=\frac{a^2+c^2-b^2}{2ac}\\ \cos\gamma=\frac{a^2+b^2-c^2}{2ab} \end{aligned} cosα=2bcb2+c2−a2cosβ=2aca2+c2−b2cosγ=2aba2+b2−c2