三角函数、反三角函数、双曲函数和反双曲函数的定义、极限、微分、积分和恒等式。
三角函数
\( \sin (\theta ) = \frac{对边}{斜边} \) 奇函数 周期\(2\pi\)
\( \cos (\theta ) = \frac{邻边}{斜边} \) 偶函数 周期\(2\pi\)
\( \tan (\theta ) = \frac{对边}{邻边} \) 奇函数 周期\(\pi\)
\( \csc (x) = \frac{1}{\sin (x)} \) 奇函数 周期\(2\pi\)
\( \sec (x) = \frac{1}{\cos (x)} \) 偶函数 周期\(2\pi\)
\( \cot (x) = \frac{1}{\tan (x)} \) 奇函数 周期\(2\pi\)
点的横坐标为\(\cos (\theta)\),纵坐标为\(\sin (\theta)\)
| 弧度 | 0 | \(\frac{\pi }{6} \) | \(\frac{\pi }{4} \) | \(\frac{\pi }{3} \) | \(\frac{\pi }{2} \) |
|---|---|---|---|---|---|
| sin | 0 | \(\frac{1}{2}\) | \(\frac{\sqrt{2} }{2}\) | \(\frac{\sqrt{3} }{2}\) | 1 |
| cos | 1 | \(\frac{\sqrt{3} }{2}\) | \(\frac{\sqrt{2} }{2}\) | \(\frac{1}{2}\) | 0 |
| tan | 0 | \(\frac{\sqrt{3} }{3}\) | 1 | \(\sqrt{3}\) | NaN |
\(\tan (x) = \frac{\sin (x)}{\cos (x)}\)
\(\cot (x) = \frac{\cos (x)}{\sin (x)}\)
\(\cos ^2(x)+\sin ^2(x)=1\)
\(1+\tan ^2(x)=\sec ^2(x)\)
\(\cot ^2(x)+1=\csc ^2(x)\)
\(\sin (x)=\cos (\frac{\pi }{2} - x)\)
\(\cos (x)=\sin (\frac{\pi }{2} - x)\)
\(\tan (x)=\cot (\frac{\pi }{2} - x)\)
\(\cot (x)=\tan (\frac{\pi }{2} - x)\)
\(\sec (x)=\csc (\frac{\pi }{2} - x)\)
\(\csc (x)=\sec (\frac{\pi }{2} - x)\)
\(\sin (A+B) = \sin (A)\cos (B)+\cos (A)\sin (B)\)
\(\cos (A+B) = \cos (A)\cos (B)-\sin (A)\sin (B)\)
\(\sin (A-B) = \sin (A)\cos (B)-\cos (A)\sin (B)\)
\(\cos (A-B) = \cos (A)\cos (B)+\sin (A)\sin (B)\)
\(\tan (A+B)=\frac{\tan A+\tan B}{1-\tan {A}\tan {B}} \)
\(\tan (A-B)=\frac{\tan A-\tan B}{1+\tan {A}\tan {B}} \)
\(\sin (2x)=2\sin (x)\cos (x)\)
\(\cos (2x) = 2\cos ^2(x)-1=1-2\sin ^2(x)\)
\(\cos ^2(\theta) = \frac{1+\cos 2\theta}{2} \)
\(\sin ^2(\theta) = \frac{1-\cos 2\theta}{2} \)
\(\sin (A-\frac{\pi }{2} )=-\cos A\)
\(\cos (A-\frac{\pi }{2} )=\sin A\)
\(\sin (A+\frac{\pi }{2} )=\cos A\)
\(\cos (A+\frac{\pi }{2} )=-\sin A\)
\(\sin {A}\sin {B}=\frac{1}{2}\cos (A-B)-\frac{1}{2} \cos (A+B)\)
\(\cos {A}\cos {B}=\frac{1}{2}\cos (A-B)+\frac{1}{2} \cos (A+B)\)
\(\sin {A}\cos {B}=\frac{1}{2}\sin (A-B)+\frac{1}{2} \sin (A+B)\)
\(\sin A+\sin B=2\sin \frac{1}{2} (A+B)\cos \frac{1}{2} (A-B)\)
\(\sin A-\sin B=2\cos \frac{1}{2} (A+B)\sin \frac{1}{2} (A-B)\)
\(\cos A+\cos B=2\cos \frac{1}{2} (A+B)\cos \frac{1}{2} (A-B)\)
\(\cos A-\cos B=-2\sin \frac{1}{2} (A+B)\sin \frac{1}{2} (A-B)\)
余弦定律：若\(\theta\)是 c 的对边， \(c^2=a^2+b^2-2ab\cos (\theta)\)
反三角函数
\(\arcsin \)是\(\sin \)的反函数，值域是\(\left[-\frac{\pi}{2},\frac{\pi }{2}\right]\)，是奇函数
\(\arccos \)是\(\cos \)的反函数，值域是\([0,\pi]\)
\(\arctan \)是\(\tan \)的反函数，值域是\(\left(-\frac{\pi}{2},\frac{\pi }{2}\right)\)
| \(\arcsin x \) | \(\frac{\pi}{3} \) | \(\frac{\pi}{4} \) | \(\frac{\pi}{6} \) | \(-\frac{\pi}{6} \) | \(-\frac{\pi}{4} \) | \(-\frac{\pi}{3} \) |
|---|---|---|---|---|---|---|
| x | \(\frac{\sqrt{3} }{2} \) | \(\frac{\sqrt{2} }{2} \) | \(\frac{1}{2}\) | \(-\frac{1}{2}\) | \(-\frac{\sqrt{2} }{2} \) | \(-\frac{\sqrt{3} }{2} \) |
| \(\arccos x \) | \(\frac{\pi}{6} \) | \(\frac{\pi}{4} \) | \(\frac{\pi}{3} \) | \(\frac{2}{3} \pi\) | \(\frac{3}{4} \pi\) | \(\frac{5}{6} \pi\) |
|---|---|---|---|---|---|---|
| x | \(\frac{\sqrt{3} }{2} \) | \(\frac{\sqrt{2} }{2} \) | \(\frac{1}{2}\) | \(-\frac{1}{2}\) | \(-\frac{\sqrt{2} }{2} \) | \(-\frac{\sqrt{3} }{2} \) |
| \(\arctan x \) | \(\frac{\pi}{3} \) | \(\frac{\pi}{4} \) | \(\frac{\pi}{6} \) | \(-\frac{\pi}{6} \) | \(-\frac{\pi}{4} \) | \(-\frac{\pi}{3} \) |
|---|---|---|---|---|---|---|
| x | \(\sqrt{3}\) | 1 | \(\frac{\sqrt{3} }{3} \) | \(-\frac{\sqrt{3} }{3} \) | 1 | \(-\sqrt{3}\) |
\( \arccos x+\arccos\left ( -x \right )= \pi \)
\(\arcsin x+\arccos x=\frac{\pi}{2} \)
双曲函数
双曲正弦: \(\sinh x = \frac{e^{x}-e^{-x}}{2}, x\in R\)，奇函数，单调递增
双曲余弦: \(\cosh x = \frac{e^{x}+e^{-x}}{2}, x\in R\)，偶函数，值域\(\left [ 1,\infty \right )\)
双曲正切: \(\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}, x\in R\)，奇函数，单调递增，值域\(\left [ -1,1 \right )\)
\(\sinh (x+y) = \sinh x \cosh y+\cosh x \sinh y\)
\(\sinh (x-y) = \sinh x \cosh y-\cosh x \sinh y\)
\(\cosh (x+y) = \cosh x \cosh y+\sinh x \sinh y\)
\(\cosh (x-y) = \cosh x \cosh y-\sinh x \sinh y\)
\(\cosh^2 x - \sinh^2 x = 1\)
\(\sinh 2x = 2\sinh x \cosh x\)
\(\cosh 2x = \sinh^2 x +\cosh^ x\)
反双曲函数
反双曲正弦 \(\sinh^{-1} =\ln(x+\sqrt{x^2+1}),x\in\left (-\infty , +\infty \right ) \)，奇函数，单调递增
反双曲余弦 \(\cosh^{-1} =\ln(x+\sqrt{x^2-1}),x\in\left [1 , +\infty \right ) \)，单调递增
反双曲正切 \(\tanh^{-1} = \frac{1}{2}\ln{\frac{1+x}{1-x} } ,x\in\left ( -1,1 \right ) \)，奇函数，单调递增
三角函数的极限和导数
\( \lim_{x \to 0} \frac{\sin{x}}{x} =1 \)
\( \lim_{x \to 0} \frac{\tan{x}}{x} =1 \)
\( \lim_{x \to 0} \frac{1-\cos{x}}{x^2} =\frac{1}{2} \)
\( \lim_{x \to 0} \frac{1-\cos{x}}{x} = 0 \)
\( \sin{x}<x<\tan{x},0<x<\frac{\pi}{2} \)
\( \cos{x}<\frac{\sin{x}}{x}<1,0<x<\frac{\pi}{2} \)
\( \frac{1}{\sin{x}}>\frac{1}{x}>\frac{\cos{x}}{\sin{x}}, 0<x<\frac{\pi}{2} \)
三角函数的导数
\(\sin’{x} = \cos{x}\)
\(\cos’{x} = -\sin{x}\)
\(\tan’{x} = \sec^2{x}\)
\(\sec’{x} = \sec{x}\tan{x}\)
\(\csc’{x} = -\csc{x}\cot{x}\)
\(\cot’{x} = -\csc^2{x}\)
反三角函数的导数
\( {\arcsin}’ x = \frac{1}{\sqrt{1-x^2} }\)
\( {\arccos}’ x = -\frac{1}{\sqrt{1-x^2} }\)
\( {\arctan}’ x = \frac{1}{1+x^2} \)
\( {\operatorname{arcsec} }’ x = \frac{1}{ \left | x \right | \sqrt{x^2-1}} \)
\( {\operatorname{arccsc} }’ x = -\frac{1}{ \left | x \right | \sqrt{x^2-1}} \)
\( {\operatorname{arccot} }’ x = -\frac{1}{x^2+1} \)
双曲函数的导数
反双曲函数的导数
\( {\cosh^{-1} }’ x = \frac{1}{\sqrt{x^2-1}} \)
\( {\sinh^{-1} }’ x = \frac{1}{\sqrt{x^2+1}} \)
\( {\tanh^{-1} }’ x = \frac{1}{1-x^2} \)
\( {\coth^{-1} }’ x = \frac{1}{1-x^2} \)
\( {\operatorname{sech}^{-1} }’ x = -\frac{1}{x\sqrt{1-x^2}} \)
\( {\operatorname{csch}^{-1} }’ x = \frac{1}{\left | x \right | \sqrt{1-x^2}} \)
三角函数的积分
\( \int \sin x \mathrm{d}x = -\cos x+C \)
\( \int \cos x \mathrm{d}x = \sin x+C \)
\( \int \tan x \mathrm{d}x = -\ln{\left | \cos x \right | }+C \)
\( \int \cot x \mathrm{d}x = \ln{\left | \sin x \right | }+C \)
\( \int \sec x \mathrm{d}x = \ln{\left | \tan{\left ( \frac{\pi}{4}+\frac{x}{2} \right ) } \right | } +C=\ln{\left | \sec x+\tan x \right | }+C \)
\( \int \csc x \mathrm{d}x = \ln{\left | \tan{ \frac{x}{2} } \right | } +C=\ln{\left | \csc x - \cot x \right | }+C \)
反三角函数的积分
\( \int \arcsin \frac{x}{a} \mathrm{d}x = x\arcsin {\frac{x}{a} }+\sqrt{a^2-x^2}+C \)
\( \int \arccos \frac{x}{a} \mathrm{d}x = x\arccos {\frac{x}{a} }-\sqrt{a^2-x^2}+C \)
\( \int \arctan \frac{x}{a} \mathrm{d}x = x\arctan {\frac{x}{a} }-\frac{a}{2} \ln(a^2+x^2)+C \)
双曲函数的积分
\( \int \sinh x\mathrm{d}x = \cosh x+C \)
\( \int \cosh x\mathrm{d}x = \sinh x+C \)
\( \int \tanh x\mathrm{d}x = \ln{\sinh x}+C \)