sin ( π 2 ± α ) = cos α sin ( π ± α ) = ∓ sin α cos ( π 2 ± α ) = ∓ sin α sin ( π ± α ) = − cos α \sin(\frac{\pi}{2}\pm\alpha) = \cos\alpha\\ \sin(\pi\pm\alpha) = \mp\sin\alpha\\ \cos(\frac{\pi}{2}\pm\alpha) = \mp\sin\alpha\\ \sin(\pi\pm\alpha) = -\cos\alpha\\ sin(2π±α)=cosαsin(π±α)=∓sinαcos(2π±α)=∓sinαsin(π±α)=−cosα
sin 3 α = − 4 sin 3 α + 3 sin α cos 3 α = 4 cos 3 α − 3 cos α \sin 3\alpha = -4\sin^3\alpha+3\sin\alpha\\ \cos 3\alpha=4\cos^3\alpha-3\cos\alpha sin3α=−4sin3α+3sinαcos3α=4cos3α−3cosα
sin 2 α 2 = 1 2 ( 1 − cos α ) cos 2 α 2 = 1 2 ( 1 + cos α ) tan α 2 = 1 − cos α sin α = sin α 1 + cos α = ± 1 − cos α 1 + cos α cot α 2 = sin α 1 − cos α = 1 + cos α sin α = ± 1 + cos α 1 − cos α \sin^2\frac{\alpha}{2} = \frac{1}{2}(1-\cos\alpha)\\ \cos^2\frac{\alpha}{2} = \frac{1}{2}(1+\cos\alpha)\\ \tan\frac{\alpha}{2} = \frac{1-\cos\alpha}{\sin\alpha} = \frac{\sin\alpha}{1+\cos\alpha} = \pm\sqrt{\frac{1-\cos\alpha}{1+\cos\alpha}}\\ \cot\frac{\alpha}{2} = \frac{\sin\alpha}{1-\cos\alpha} = \frac{1+\cos\alpha}{\sin\alpha} = \pm\sqrt{\frac{1+\cos\alpha}{1-\cos\alpha}}\\ sin22α=21(1−cosα)cos22α=21(1+cosα)tan2α=sinα1−cosα=1+cosαsinα=±1+cosα1−cosαcot2α=1−cosαsinα=sinα1+cosα=±1−cosα1+cosα
cot ( α + β ) = cot α cot β − 1 cot β + cot α \cot(\alpha+\beta) = \frac{\cot\alpha\cot\beta-1}{\cot\beta+\cot\alpha} cot(α+β)=cotβ+cotαcotαcotβ−1
cot ( α − β ) = cot α cot β + 1 cot β − cot α \cot(\alpha-\beta) = \frac{\cot\alpha\cot\beta+1}{\cot\beta-\cot\alpha} cot(α−β)=cotβ−cotαcotαcotβ+1
sin α cos β = sin ( α + β ) + sin ( α − β ) 2 cos α sin β = sin ( α + β ) − sin ( α − β ) 2 cos α cos β = cos ( α + β ) + c o s ( α − β ) 2 sin α sin β = − cos ( α + β ) − cos ( α − β ) 2 \sin\alpha\cos\beta=\frac{\sin(\alpha+\beta)+\sin(\alpha-\beta)}{2}\\ \cos\alpha\sin\beta=\frac{\sin(\alpha+\beta)-\sin(\alpha-\beta)}{2}\\ \cos\alpha\cos\beta=\frac{\cos(\alpha+\beta)+cos(\alpha-\beta)}{2}\\ \sin\alpha\sin\beta=-\frac{\cos(\alpha+\beta)-\cos(\alpha-\beta)}{2} sinαcosβ=2sin(α+β)+sin(α−β)cosαsinβ=2sin(α+β)−sin(α−β)cosαcosβ=2cos(α+β)+cos(α−β)sinαsinβ=−2cos(α+β)−cos(α−β)
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 \sin\alpha+\sin\beta = 2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}\\ \sin\alpha-\sin\beta = 2\cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}\\ \cos\alpha+\cos\beta = 2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}\\ \cos\alpha-\cos\beta = -2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2} sinα+sinβ=2sin2α+βcos2α−βsinα−sinβ=2cos2α+βsin2α−βcosα+cosβ=2cos2α+βcos2α−βcosα−cosβ=−2sin2α+βsin2α−β
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∑ k = 1 n k 2 = 1 2 + 2 2 + 3 2 + ⋯ + n 2 = n ( n + 1 ) ( 2 n + 1 ) 6 \sum_{k=1}^nk^2 = 1^2+2^2+3^2+\dots+n^2 = \frac{n(n+1)(2n+1)}{6} k=1∑nk2=12+22+32+⋯+n2=6n(n+1)(2n+1)
x 1 , 2 = − b ± b 2 − 4 a c 2 a x_{1,2} = \frac{-b\pm\sqrt{b^2-4ac}}{2a} x1,2=2a−b±b2−4ac
x 1 + x 2 = − b a x 1 x 2 = c a x_1+x_2=-\frac{b}{a}\\ x_1x_2 = \frac{c}{a} x1+x2=−abx1x2=ac
( a + b ) 3 = a 3 + 3 a 2 b + 3 a b 2 + b 3 ( a − b ) 3 = a 3 − 3 a 2 b + 3 a b 2 − b 3 a 3 − b 3 = ( a − b ) ( a 2 + a b + b 2 ) a 3 + b 3 = ( a + b ) ( a 2 − a b + b 2 ) a n − b n = ( a − b ) ∑ i = 0 n − 1 a n − 1 − i b i a n + b n = ( a − b ) ∑ i = 0 n − 1 ( − 1 ) i a n − 1 − i b i ( n 为 正 奇 数 ) (a+b)^3=a^3+3a^2b+3ab^2+b^3\\ (a-b)^3=a^3-3a^2b+3ab^2-b^3\\ a^3-b^3 = (a-b)(a^2+ab+b^2)\\ a^3+b^3 = (a+b)(a^2-ab+b^2)\\ a^n-b^n = (a-b)\sum_{i=0}^{n-1}a^{n-1-i}\ b^i\\ a^n+b^n = (a-b)\sum_{i=0}^{n-1}(-1)^{i}a^{n-1-i}\ b^i(n 为正奇数) (a+b)3=a3+3a2b+3ab2+b3(a−b)3=a3−3a2b+3ab2−b3a3−b3=(a−b)(a2+ab+b2)a3+b3=(a+b)(a2−ab+b2)an−bn=(a−b)i=0∑n−1an−1−i bian+bn=(a−b)i=0∑n−1(−1)ian−1−i bi(n为正奇数)
( 2 n ) ! ! = 2 ∗ 4 ∗ 6 ∗ 8 ∗ 10 ∗ ⋯ ∗ ( 2 n ) = 2 n ∗ n ! ( 2 n − 1 ) ! ! = 1 ∗ 3 ∗ 5 ∗ 7 ∗ ⋯ ∗ ( 2 n − 1 ) (2n)!! = 2*4*6*8*10*\dots*(2n)=2^n*n!\\ (2n-1)!! = 1 * 3* 5*7*\dots*(2n-1) (2n)!!=2∗4∗6∗8∗10∗⋯∗(2n)=2n∗n!(2n−1)!!=1∗3∗5∗7∗⋯∗(2n−1)
∣ a ± b ∣ ≤ ∣ a ∣ + ∣ b ∣ ∣ ∣ a ∣ − ∣ b ∣ ∣ ≤ ∣ a − b ∣ ∣ a 1 ± a 2 ± ⋯ ± a n ∣ ≤ ∣ a 1 ∣ + ∣ a 2 ∣ + ⋯ + ∣ a n ∣ |a\pm b|\leq|a|+|b|\\ ||a|-|b||\leq|a-b|\\ |a_1\pm a_2\pm \dots\pm a_n|\leq|a_1|+|a_2|+\dots+|a_n|\\ ∣a±b∣≤∣a∣+∣b∣∣∣a∣−∣b∣∣≤∣a−b∣∣a1±a2±⋯±an∣≤∣a1∣+∣a2∣+⋯+∣an∣
a b ≤ a + b 2 ≤ a 2 + b 2 2 ( a , b > 0 ) a b c 3 ≤ a + b + c 3 ≤ a 2 + b 2 + c 2 3 ( a , b , c > 0 ) \sqrt{ab}\leq \frac{a+b}{2}\leq\sqrt{\frac{a^2+b^2}{2}}\ \ \ \ \ (a,b>0)\\ \sqrt[3]{abc}\leq\frac{a+b+c}{3}\leq\sqrt{\frac{a^2+b^2+c^2}{3}}\ \ \ \ (a,b,c>0)\\ ab≤2a+b≤2a2+b2 (a,b>0)3abc≤3a+b+c≤3a2+b2+c2 (a,b,c>0)
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1 x + 1 < ln ( 1 + 1 x ) < 1 x ( x > 0 ) \frac{1}{x+1}<\ln(1+\frac{1}{x})<\frac{1}{x}\ \ \ \ \ (x>0) x+11<ln(1+x1)<x1 (x>0)