1 基礎
- 一元二次方程的根 \(x_{1,2} = \cfrac {-b \pm \sqrt{b^2 - 4ac}}{2a}\)，並且\(x_1 + x_2 = -\cfrac ba, \ \ x_1 x_2 = \cfrac ca\)
- \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)，\((a-b)^3 = a^3 -3a^2b+3ab^2-b^3\)
- \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)， \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
- \((a+b)^n = \displaystyle \sum_{k=0}^{n} C_n^k a^k b^{n-k}\)
2 對數
- \(\log _a (MN) = \log_a M + \log_a N\)
- \(\log_a \cfrac MN = \log_a M - \log_a N\)
- \(\log_a M^n = n \log_a M\)
3 三角函數
- \(\csc x = \cfrac 1{\sin x}, \ \sec x = \cfrac 1{\cos x}, \ \cot x = \cfrac 1{\tan x}\)
- \(\sin^2 x + \cos^2 x = 1, \ 1 + \tan^2 x = \sec^2 x, \ 1 + \cot^2 x = \csc^2 x\)
- \(\sin (\alpha \pm \beta) = \sin \alpha \cos \beta \pm \sin \beta \cos \alpha\)
- \(\cos (\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta\)
- \(\tan (\alpha \pm \beta) = \cfrac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}\)
- \(\sin 2\alpha = 2\sin \alpha \cos \alpha\)
- \(\cos 2\alpha = \cos^2 \alpha - \sin^2 \alpha\)
- \(\sin^2 \alpha = \cfrac 12 (1 - \cos 2\alpha), \cos ^2 \alpha = \cfrac 12 (1 + \cos 2\alpha)\)
- \(a\sin \alpha + b\cos \alpha = \sqrt{a^2 + b^2} \sin (\alpha + \varphi), \ \varphi = \arctan \cfrac ba\)
- 積化和差
- \(\sin \alpha \sin \beta = -\cfrac 12 [\cos (\alpha + \beta) - \cos (\alpha -\beta)]\)
- \(\sin \alpha \cos \beta = \cfrac 12 [\sin (\alpha + \beta) + \sin (\alpha - \beta)]\)
- \(\cos \alpha \cos \beta = \cfrac 12 [\cos (\alpha + \beta) + \cos (\alpha -\beta)]\)
- 和差化積
- \(\sin \alpha + \sin \beta = 2\sin \cfrac {\alpha + \beta}2 \cos \cfrac {\alpha -\beta}2\)
- \(\cos \alpha + \cos \beta = 2 \cos \cfrac {\alpha + \beta}2 \cos \cfrac {\alpha -\beta}2\)
- \(\sin \alpha - \sin \beta = 2 \sin \cfrac {\alpha -\beta}2 \cos \cfrac {\alpha + \beta}2\)
- \(\cos \alpha -\cos \beta = -2 \sin \cfrac {\alpha + \beta}2 \sin \cfrac {\alpha - \beta}2\)
4 不等式
- \(2|ab| \le a^2 + b^2\)
- \(|a \pm b| \le |a| + |b|\)
- \(|a_1 \pm a_2 \pm \cdots \pm a_n| \le |a_1| + |a_2| + \cdots + |a_n|\)
- \(|a - b| \ge |a| - |b|\)
- \(\cfrac {a_1 + a_2 + \cdots + a_n}n \ge \sqrt[n]{a_1 a_2 \cdots a_n}\)
- \(\left|\cfrac {a_1 + a_2 + \cdots + a_n}n \right| \le \sqrt{\cfrac {a_1^2 + a_2^2 + \cdots + a_n^2}n}\)
- \(x,y,p,q \gt 0, \cfrac 1p + \cfrac 1q = 1\)，則\(xy \le \cfrac {x^p}p + \cfrac {y^q}q\)
- \((a^2 + b^2)(c^2 + d^2) \ge (ac+bd)^2\)
5 數列
- 等差數列 \(a_n = a_1 + (n-1)d\)：\(S_n = \cfrac n2 (a_1 + a_n)\)
- 等比數列 \(a_n = a_1 q^{n-1}\)：\(S_n = \cfrac {a_1(1-q^n)}{1-q}\)
常用數列的和：
- \(1 + 2 + 3 + \cdots + n = \cfrac {n(n+1)}2\)
- \(1 + 3 + 5 + \cdots + (2n-1) = n^2\)
- \(1^2 + 2^2 + 3^2 + \cdots + n^2 = \cfrac {n(n+1)(2n+1)}6\)
- \(1^3 + 2^3 + 3^3 + \cdots + n^3 = \left[ \cfrac {n(n+1)}2 \right]^2\)
6 排列組合
- \(A_n^m = \cfrac {n!}{(n-m)!}, C_n^m = \cfrac {n!}{m! (n-m)!}\)
- 特別的，規定\(0! = 1\)。
- \(C_n^m = C_n^{n-m},\ \ C_{n+1}^m = C_n^m + C_n^{m-1}\)
7 曲線和曲面
- 圓錐體積\(V = \cfrac 13 sh\)，球的體積\(V = \cfrac 43 \pi R^3\)，球表面積\(S = 4 \pi R^2\)
- 橢圓 \(\cfrac {x^2}{a^2} + \cfrac {y^2}{b^2} = 1\)，雙曲線 \(\cfrac {x^2}{a^2} - \cfrac {y^2}{b^2} = 1\)，拋物線 \(y^2 = 2px\)
- 橢球面\(\cfrac {x^2}{a^2} + \cfrac {y^2}{b^2} + \cfrac {z^2}{c^2} = 1\)，二次錐面\(\cfrac {x^2}{a^2} + \cfrac {y^2}{b^2} - \cfrac {z^2}{c^2} = 0\)
- 單頁雙曲面\(\cfrac {x^2}{a^2} + \cfrac {y^2}{b^2} - \cfrac {z^2}{c^2} = 1\)，雙葉雙曲面\(\cfrac {x^2}{a^2} - \cfrac {y^2}{b^2} - \cfrac {z^2}{c^2} = 1\)
- 橢圓拋物面\(\cfrac {x^2}{2p} + \cfrac {y^2}{2q} = z\)，雙曲拋物面/馬鞍面\(\cfrac {x^2}{2p} - \cfrac {y^2}{2q} = z\)
8 極坐標
- 弧長 \(l = r \theta\)
- 扇形面積 \(S = \frac 12 r^2 \theta\)
- 極坐標換直角坐標 \(\begin{cases} x = r(\theta) \cos \theta \\ y = r(\theta) \sin \theta \end{cases}\)
- 圓心在\(x\)軸\(\cfrac a2\)處的圓 \(\rho = a \cos \theta\)，圓心在\(y\)軸\(\cfrac a2\)處的圓 \(\rho = a \sin \theta\)
- 擺線\(\begin{cases} x = a(\theta - \sin \theta) \\ y = a(1 - \cos \theta) \\ \end{cases}\)
- 阿基米德螺線 \(\rho = a \theta, \ (a \gt 0)\)