sin⁡α 2 =±1−cos⁡α 2\sin \frac{\alpha}{2}=\pm \sqrt{\frac{1-\cos \alpha}{2}}sin2α​=±21−cosα​​

cos⁡α 2 =±1+cos⁡α 2\cos \frac{\alpha}{2}=\pm \sqrt{\frac{1+\cos \alpha}{2}}cos2α​=±21+cosα​​

cos⁡α=2 cos ⁡2 α 2 −1=1−2 sin ⁡2 α 2 \cos \alpha=2 \cos ^{2} \frac{\alpha}{2}-1=1-2 \sin ^{2} \frac{\alpha}{2}cosα=2cos22α​−1=1−2sin22α​

tan⁡α 2 =±1−cos⁡α1+cos⁡α= sin ⁡ α 1 + cos ⁡ α= 1 − cos ⁡ α sin ⁡ α\tan \frac{\alpha}{2}=\pm \sqrt{\frac{1-\cos \alpha}{1+\cos \alpha}}=\frac{\sin \alpha}{1+\cos \alpha}=\frac{1-\cos \alpha}{\sin \alpha}tan2α​=±1+cosα1−cosα​​=1+cosαsinα​=sinα1−cosα​

cot⁡α 2 = 1 + cos ⁡ α sin ⁡ α= sin ⁡ α 1 − cos ⁡ α\cot \frac{\alpha}{2}=\frac{1+\cos \alpha}{\sin \alpha}=\frac{\sin \alpha}{1-\cos \alpha}cot2α​=sinα1+cosα​=1−cosαsinα​

sec⁡α 2 = ±sec ⁡ α + 1 2 sec ⁡ α2 sec ⁡ α sec ⁡ α + 1= ±4 sec⁡ 3α + sec⁡ 2α 2 sec ⁡ αsec ⁡ α + 1\sec \frac{\alpha}{2}=\frac{\pm \sqrt{\frac{\sec \alpha+1}{2 \sec \alpha}} 2 \sec \alpha}{\sec \alpha+1}=\frac{\pm \sqrt{\frac{4 \sec ^{3} \alpha+\sec ^{2} \alpha}{2 \sec \alpha}}}{\sec \alpha+1}sec2α​=secα+1±2secαsecα+1​​2secα​=secα+1±2secα4sec3α+sec2α​​​

csc⁡α 2 = ±sec ⁡ α − 1 2 sec ⁡ α2 sec ⁡ α sec ⁡ α − 1= ±4 sec⁡ 3α − sec⁡ 2α 2 sec ⁡ αsec ⁡ α − 1\csc \frac{\alpha}{2}=\frac{\pm \sqrt{\frac{\sec \alpha-1}{2 \sec \alpha}} 2 \sec \alpha}{\sec \alpha-1}=\frac{\pm \sqrt{\frac{4 \sec ^{3} \alpha-\sec ^{2} \alpha}{2 \sec \alpha}}}{\sec \alpha-1}csc2α​=secα−1±2secαsecα−1​​2secα​=secα−1±2secα4sec3α−sec2α​​​

sin⁡2α=2sin⁡αcos⁡α\sin 2 \alpha=2 \sin \alpha \cos \alphasin2α=2sinαcosα

cos⁡2α= cos ⁡2 α− sin ⁡2 α=2 cos ⁡2 α−1=1−2 sin ⁡2 α\cos 2 \alpha=\cos ^{2} \alpha-\sin ^{2} \alpha=2 \cos ^{2} \alpha-1=1-2 \sin ^{2} \alphacos2α=cos2α−sin2α=2cos2α−1=1−2sin2α

tan⁡2α= 2 tan ⁡ α 1 −tan⁡2 α\tan 2 \alpha=\frac{2 \tan \alpha}{1-\tan ^{2} \alpha}tan2α=1−tan2α2tanα​

cot⁡2α=cot⁡2 α − 1 2 cot ⁡ α\cot 2 \alpha=\frac{\cot ^{2} \alpha-1}{2 \cot \alpha}cot2α=2cotαcot2α−1​

sec⁡2α=sec⁡2 α +csc⁡2 αcsc⁡2 α −sec⁡2 α=sec⁡2 αcsc⁡2 αcsc⁡2 α −sec⁡2 α\sec 2 \alpha=\frac{\sec ^{2} \alpha+\csc ^{2} \alpha}{\csc ^{2} \alpha-\sec ^{2} \alpha}=\frac{\sec ^{2} \alpha \csc ^{2} \alpha}{\csc ^{2} \alpha-\sec ^{2} \alpha}sec2α=csc2α−sec2αsec2α+csc2α​=csc2α−sec2αsec2αcsc2α​

csc⁡2α=sec⁡2 α +csc⁡2 α 2 sec ⁡ α csc ⁡ α=sec⁡2 α csc ⁡ α2 \csc 2 \alpha=\frac{\sec ^{2} \alpha+\csc ^{2} \alpha}{2 \sec \alpha \csc \alpha}=\frac{\sec ^{2} \alpha \csc \alpha}{2}csc2α=2secαcscαsec2α+csc2α​=2sec2αcscα​

曲率K= ∣y′′ ∣(1+y′ 2 )32曲率 K=\frac{\left|y^{\prime \prime}\right|}{\left(1+y^{\prime 2}\right)^{\frac{3}{2}}}曲率K=(1+y′2)23​∣y′′∣​

曲率半径ρ=1 K =(1+y′ 2 )32 ∣y′′ ∣曲率半径 \rho=\frac{1}{K}=\frac{\left(1+y^{\prime 2}\right)^{\frac{3}{2}}}{\left|y^{\prime \prime}\right|}曲率半径ρ=K1​=∣y′′∣(1+y′2)23​​

设直线L\mathrm{L}L 的方程为A x +B y +C=0\mathrm{Ax}+\mathrm{By}+\mathrm{C}=0Ax+By+C=0 ，点P\mathrm{P}P 的坐标为(x0,y0)(x 0, y 0)(x0,y0) ，则点P\mathrm{P}P 到直线L\mathrm{L}L 的距离为: ∣ Ax0 + By0 + C ∣A2+B2\frac{\left|A x_{0}+B y_{0}+C\right|}{\sqrt{A^{2}+B^{2}}}A2+B2​∣Ax0​+By0​+C∣​

(xα )′ =αxα − 1, (ax )′ =a x ln⁡a, (ex )′ =e x , (log⁡a x )′ =1x ln ⁡ a,(ln⁡x) ′ =1 x \left(x^{\alpha}\right)^{\prime}=\alpha x^{\alpha-1}, \quad\left(a^{x}\right)^{\prime}=a^{x} \ln a, \quad\left(e^{x}\right)^{\prime}=e^{x}, \quad\left(\log _{a} x\right)^{\prime}=\frac{1}{x \ln a}, \quad(\ln x)^{\prime}=\frac{1}{x}(xα)′=αxα−1,(ax)′=axlna,(ex)′=ex,(loga​x)′=xlna1​,(lnx)′=x1​

(sin⁡x) ′ =cos⁡x,(cos⁡x) ′ =−sin⁡x,(arcsin⁡x) ′ =1 1−x2,(arccos⁡x) ′ =−1 1−x2(\sin x)^{\prime}=\cos x, \quad(\cos x)^{\prime}=-\sin x, \quad(\arcsin x)^{\prime}=\frac{1}{\sqrt{1-x^{2}}}, \quad(\arccos x)^{\prime}=-\frac{1}{\sqrt{1-x^{2}}}(sinx)′=cosx,(cosx)′=−sinx,(arcsinx)′=1−x2​1​,(arccosx)′=−1−x2​1​

(tan⁡x) ′ = sec ⁡2 x,(cot⁡x) ′ =− csc ⁡2 x,(arctan⁡x) ′ =11 +x2,(arccot⁡x) ′ =−11 +x2(\tan x)^{\prime}=\sec ^{2} x, \quad(\cot x)^{\prime}=-\csc ^{2} x, \quad(\arctan x)^{\prime}=\frac{1}{1+x^{2}}, \quad(\operatorname{arccot} x)^{\prime}=-\frac{1}{1+x^{2}}(tanx)′=sec2x,(cotx)′=−csc2x,(arctanx)′=1+x21​,(arccotx)′=−1+x21​

(sec⁡x) ′ =sec⁡xtan⁡x,(csc⁡x) ′ =−csc⁡xcot⁡x(\sec x)^{\prime}=\sec x \tan x, \quad(\csc x)^{\prime}=-\csc x \cot x(secx)′=secxtanx,(cscx)′=−cscxcotx

a x −1∼xln⁡aa^{x}-1 \sim x \ln aax−1∼xlna

arcsin⁡(a)x∼sin⁡(a)x∼(a)x\arcsin (a) x \sim \sin (a) x \sim(a) xarcsin(a)x∼sin(a)x∼(a)x

arctan⁡(a)x∼tan⁡(a)x∼(a)x\arctan (a) x \sim \tan (a) x \sim(a) xarctan(a)x∼tan(a)x∼(a)x

ln⁡(1+x)∼x\ln (1+x) \sim xln(1+x)∼x

1 + x− 1 − x∼x\sqrt{1+x}-\sqrt{1-x} \sim x1+x​−1−x​∼x

(1+ax) b −1∼abx(1+a x)^{b}-1 \sim a b x(1+ax)b−1∼abx

1 + a xb −1∼a b x\sqrt[b]{1+a x}-1 \sim \frac{a}{b} xb1+ax​−1∼ba​x

1−cos⁡x∼ x 22 1-\cos x \sim \frac{x^{2}}{2}1−cosx∼2x2​

x−ln⁡(1+x)∼ x 22 x-\ln (1+x) \sim \frac{x^{2}}{2}x−ln(1+x)∼2x2​

tan⁡x−sin⁡x∼ x 32 \tan x-\sin x \sim \frac{x^{3}}{2}tanx−sinx∼2x3​

tan⁡x−x∼ x 33 \tan x-x \sim \frac{x^{3}}{3}tanx−x∼3x3​

x−arctan⁡x∼ x 33 x-\arctan x \sim \frac{x^{3}}{3}x−arctanx∼3x3​

x−sin⁡x∼ x 36 x-\sin x \sim \frac{x^{3}}{6}x−sinx∼6x3​

arcsin⁡x−x∼ x 36 \arcsin x-x \sim \frac{x^{3}}{6}arcsinx−x∼6x3​

e x =1+x+ x 2 2 !+⋯+ x n n !+⋯=∑n = 0∞x n n !e^{x}=1+x+\frac{x^{2}}{2 !}+\cdots+\frac{x^{n}}{n !}+\cdots=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}ex=1+x+2!x2​+⋯+n!xn​+⋯=∑n=0∞​n!xn​

sin⁡x=x−13 !x 3 +⋯+(−1) n 1( 2 n + 1 ) !x2 n + 1+⋯=∑n = 0∞ (−1) nx2n+1 ( 2 n + 1 ) !\sin x=x-\frac{1}{3 !} x^{3}+\cdots+(-1)^{n} \frac{1}{(2 n+1) !} x^{2 n+1}+\cdots=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n+1}}{(2 n+1) !}sinx=x−3!1​x3+⋯+(−1)n(2n+1)!1​x2n+1+⋯=∑n=0∞​(−1)n(2n+1)!x2n+1​

cos⁡x=1−12 !x 2 +⋯+(−1) n 1( 2 n ) !x2 n+⋯=∑n = 0∞ (−1) nx2n ( 2 n ) !\cos x=1-\frac{1}{2 !} x^{2}+\cdots+(-1)^{n} \frac{1}{(2 n) !} x^{2 n}+\cdots=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n}}{(2 n) !}cosx=1−2!1​x2+⋯+(−1)n(2n)!1​x2n+⋯=∑n=0∞​(−1)n(2n)!x2n​

ln⁡(1+x)=x−1 2 x 2 +⋯+(−1)n − 1 x nn +⋯=∑n = 0∞ (−1)n − 1 x nn ,−1