(a ± b)2 = a2 ± 2ab + b2
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc
a2 - b2 = (a - b)(a + b)
(a ± b)3 = a3 ± 3a2b + 3ab2 ± b3
a3 ± b3 = (a ± b)(a2 ∓ ab + b2)
an - bn = (a - b)(an-1 + an-2b + an-3b2 + … + abn-2+ bn-1)
合比定理 a + b b \frac{a + b}{b} ba+b = c + d d \frac{c + d}{d} dc+d
分比定理 a − b b \frac{a - b}{b} ba−b = c − d d \frac{c - d}{d} dc−d
合分比定理 a + b a − b \frac{a + b}{a - b} a−ba+b = c + d c − d \frac{c + d}{c - d} c−dc+d
根 = − b ± b 2 − 4 a c 2 a \frac{-b ± \sqrt{b^2 - 4ac}}{2a} 2a−b±b2−4ac
韦达定理:x1 + x2 = - b a \frac{b}{a} ab,x1x2 = c a \frac{c}{a} ac
判别式△ = b2 - 4ac
△ > 0,方程有两个不等实根
△ = 0,方程有两个相等实根
△ < 0,方程有两个共轭虚根
换底公式logaM = logbM / logba
loga1 = 0
logaa = 1
等差数列,设a1为首项,an为通项,d为公差,Sn为n项和,则:
an = a1 + (n - 1)d
Sn = a 1 + a n 2 \frac{a~1~ + a~n~}{2} 2a 1 +a n n
设a, b, c成等差数列,则等差中项b = 1 2 \frac{1}{2} 21(a + c)
等比数列,设a1为首项,q为公比,an为通项,则:
通项an = a1qn-1
前n项和Sn = a 1 ( 1 − q n ) 1 − q \frac{a1~(1 - q^n)}{1 - q} 1−qa1 (1−qn) = a 1 − a n q 1 − q \frac{a1~ - an~q}{1 - q} 1−qa1 −an q
常用的几种数列的和
1 + 2 + 3 + … + n = 1 2 \frac{1}{2} 21n(n - 1)
12 + 22 + 32 + … + n2 = 1 6 \frac{1}{6} 61n(n + 1)(2n + 1)
13 + 23 + 33 + … + n3 = [ 1 2 \frac{1}{2} 21n(n + 1)]2
1 * 2 + 2 * 3 + … + n(n+1) = 1 3 \frac{1}{3} 31n(n + 1)(n + 2)
1 * 2 * 3 + 2 * 3 * 4 + … + n(n + 1)(n + 2) = 1 4 \frac{1}{4} 41n(n + 1)(n + 2)(n + 3)
排列:Pnm = n(n - 1)(n - 2)…[n - (m - 1)]
全排列:Pnn = n(n - 1)…3 * 2 * 1 = n!
组合:Cnm = n ( n − 1 ) . . . ( n − m + 1 ) m ! \frac{n(n - 1)...(n - m + 1)}{m!} m!n(n−1)...(n−m+1) = n ! m ! ( n − m ) ! \frac{n!}{m!(n - m)!} m!(n−m)!n!
组合的性质:
Cnm = Cnn-m
Cnm = Cn-1m + Cn-1m-1
二项式定理:
(a + b)n = an + nan-1b + n ( n − 1 ) 2 ! \frac{n(n - 1)}{2!} 2!n(n−1)an-2b2 + … + n ( n − 1 ) . . . ( n − ( k − 1 ) ) k ! \frac{n(n - 1)...(n - (k - 1))}{k!} k!n(n−1)...(n−(k−1))an-kbk + … + bn
三角函数间的关系:
sinα cscα = 1
cosα secα = 1
tanα cotα = 1
sin2α + cos2α = 1
1 + tan2α = sec2α
1 + cot2α = csc2α
tanα = s i n α c o s α \frac{sinα}{cosα} cosαsinα
cotα = c o s α s i n α \frac{cosα}{sinα} sinαcosα
倍角三角函数:
sin2α = 2 sinα cosα
cos2α = cos2α - sin2α = 1 - 2sin2α = 2cos2α - 1
tan2α = 2 t a n α 1 − t a n 2 α \frac{2tanα}{1 - tan^2α} 1−tan2α2tanα
cot2α = 1 − c o t 2 α 2 c o t α \frac{1 - cot^2α}{2cotα} 2cotα1−cot2α
sin2α = 1 − c o s 2 α 2 \frac{1 - cos2α}{2} 21−cos2α
cos2α = 1 + c o s 2 α 2 \frac{1 + cos2α}{2} 21+cos2α
三角函数的和差化积与积化和差公式:
sinα + sinβ = 2sin α + β 2 \frac{α + β}{2} 2α+βcos α − β 2 \frac{α - β}{2} 2α−β
sinα - sinβ = 2cos α + β 2 \frac{α + β}{2} 2α+βsin α − β 2 \frac{α - β}{2} 2α−β
cosα + cosβ = 2cos α + β 2 \frac{α + β}{2} 2α+βcos α − β 2 \frac{α - β}{2} 2α−β
cosα - cosβ = -2sin α + β 2 \frac{α + β}{2} 2α+βsin α − β 2 \frac{α - β}{2} 2α−β
sinα cosβ = 1 2 \frac{1}{2} 21[sin(α + β) + sin(α - β)]
cosα cosβ = 1 2 \frac{1}{2} 21[cos(α + β) + cos(α - β)]
cosα sinβ = 1 2 \frac{1}{2} 21[sin(α + β) - sin(α - β)]
sinα sinβ = 1 2 \frac{1}{2} 21[cos(α + β) - cos(α - β)]
正弦定理:
a s i n A \frac{a}{sinA} sinAa = b s i n B \frac{b}{sinB} sinBb = c s i n C \frac{c}{sinC} sinCc = 2R,R为外接圆半径
余弦定理:
a2 = b2 + c2 - 2bc cosA
b2 = c2 + a2 - 2ca cosB
c2 = a2 + b2 - 2ab cosC
反三角函数的恒等式:
arcsinx + arccosx = π 2 \frac{π}{2} 2π
arctanx + arccotx = π 2 \frac{π}{2} 2π