函数与极限

基础知识

常见函数定义域

函数 定义域 说明
\(y = \frac{Q(x)}{P(x)}\) \(D = \{x\vert P(x) \ne 0\}\) 分母不为0
\(y = \sqrt[2n]{x}\) \(D = [\ 0, +\infty)\) 偶次方,被开方数必须大于等于零
\(y = \log_ax\) \(D = (\ 0, +\infty)\) 真数必须大于0
\(y = \tan x\) \(D = \{x\vert x \ne (k + \frac{1}{2})\pi, k \in Z\}\) \(\tan u = \frac{\sin u}{\cos u}, \cos u \ne 0 \)
\(y = \cot x\) \(D = \{x\vert x \ne k\pi, k \in Z\}\) \(\cot u = \frac{\cos u}{\sin u}, \sin u \ne 0\)
\(y = \arcsin x\) \(D = [-1, 1]\) \(R_{\sin x} = [-1, 1]\)
\(y = \arccos x\) \(D = [-1,1]\) \(R_{\cos x} = [-1, 1]\)

三角函数值表

弧度 \(\sin x\) \(\cos x\) \(\tan x\) \(\cot x\)
0 0 1 0 不存在
\(\frac{\pi}{6}\) \(\frac{1}{2}\) \(\frac{\sqrt{3}}{2}\) \(\frac{\sqrt{3}}{3}\) \(\sqrt{3}\)
\(\frac{\pi}{4}\) \(\frac{\sqrt{2}}{2}\) \(\frac{\sqrt{2}}{2}\) 1 1
\(\frac{\pi}{3}\) \(\frac{\sqrt{3}}{2}\) \(\frac{1}{2}\) \(\sqrt{3}\) \(\frac{\sqrt{3}}{3}\)
\(\frac{\pi}{2}\) 1 0 不存在 0

对数函数运算法则

\(\log_a(1) = 0\)

\(\log_a(a) = 1\)

\(a^{\log_ab} = b\)

\(\log_a(MN) = \log_aM + \log_aN\)

\(\log_a(\frac{M}{N}) = \log_aM - \log_aN\)

\(\log_a(M^n) = n\log_a(M)\)

换底公式: \(log_bN = \frac{\log_aN}{\log_ab}\)

三角函数变换

  • 函数关系

    1. 倒数关系

      \(\sin\alpha \cdot \csc\alpha = 1\)

      \(\cos\alpha \cdot \sec\alpha = 1\)

      \(\tan\alpha \cdot \cot\alpha = 1\)

    2. 商数关系

      \(\tan\alpha = \frac{\sin\alpha}{\cos\alpha}\)

      \(\cot\alpha = \frac{\cos\alpha}{\sin\alpha}\)

    3. 平方关系

      \(\sin^2\alpha + \cos^2\alpha = 1\)

      \(1 + \tan^2\alpha = \cot^2\alpha\)

      \(1 + \cot^2\alpha = \csc^2\alpha\)

  • 诱导公式

    \(f(\frac{k}{2}\pi \pm \alpha) = g(\alpha), k \in Z\) 奇变偶不变,符号看象限

    1. k 为偶数,函数名不变;k 为奇数,函数名余变正,正变余。
    2. 将 \(\alpha\) 视为锐角,观察原函数的函数值符号,将其赋给变换后的函数。

    若 k 为 0,符号为负:

    \(\sin(-\alpha) = -\sin(\alpha)\)

    \(\cos(-\alpha) = \cos(\alpha)\)

    \(\tan(-\alpha) = -\tan(\alpha)\)

    \(\cot(-\alpha) = -\cot(\alpha)\)

    若 k 为 1,符号为负:

    \(\sin(\frac{1}{2}\pi - \alpha) = \cos(\alpha)\)

    \(\cos(\frac{1}{2}\pi - \alpha) = \sin(\alpha)\)

    \(\tan(\frac{1}{2}\pi - \alpha) = \cot(\alpha)\)

    \(\cot(\frac{1}{2}\pi - \alpha) = \tan(\alpha)\)

  • 两角和差公式

    \(\sin(\alpha + \beta) = \sin\alpha \cdot \cos\beta + \cos\alpha \cdot \sin\beta\)

    \(\sin(\alpha - \beta) = \sin\alpha \cdot \cos\beta - \cos\alpha \cdot \sin\beta\)

    \(\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta\)

    \(\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta\)

    \(\tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}\)

    \(\tan(\alpha - \beta) = \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha\tan\beta}\)

  • 积化和差

    \(\sin\alpha \cdot \sin\beta = \frac{1}{2}[\cos(\alpha - \beta) - \cos(\alpha + \beta)]\)

    \(\cos\alpha \cdot \cos\beta = \frac{1}{2}[\cos(\alpha - \beta) + \cos(\alpha + \beta)]\)

    \(\sin\alpha \cdot \cos\beta = \frac{1}{2}[\sin(\alpha + \beta) + \sin(\alpha - \beta)]\)

    \(\cos\alpha \cdot \sin\beta = \frac{1}{2}[\sin(\alpha + \beta) - \sin(\alpha - \beta)]\)

  • 和差化积

    \(\sin\alpha + \sin\beta = 2\sin\frac{\alpha + \beta}{2}\cos\frac{\alpha - \beta}{2}\)

    \(\cos\alpha + \cos\beta = 2\cos\frac{\alpha + \beta}{2}\cos\frac{\alpha - \beta}{2}\)

    \(\sin\alpha - \sin\beta = 2\cos\frac{\alpha + \beta}{2}\sin\frac{\alpha - \beta}{2}\)

    \(\cos\alpha - \cos\beta = -2\sin\frac{\alpha + \beta}{2}\sin\frac{\alpha - \beta}{2}\)

    \(\tan\alpha + \tan\beta = \frac{\sin(\alpha + \beta)}{\cos\alpha \cdot \cos\beta}\)

  • 二倍角公式

    \(\sin2\alpha = 2\sin\alpha\cos\alpha\)

    \(\cos2\alpha = \cos^2\alpha - \sin^2\alpha = 2\cos^2\alpha - 1 = 1 - 2\sin^2\alpha\)

    \(\tan2\alpha = \frac{2\tan\alpha}{1 - tan^2\alpha}\)

  • 半角公式

    \(\sin\frac{\alpha}{2} = \pm\sqrt{\frac{1 - \cos\alpha}{2}}\)

    \(\cos\frac{\alpha}{2} = \pm\sqrt{\frac{1 + \cos\alpha}{2}}\)

    \(\tan\frac{\alpha}{2} = \pm\sqrt{\frac{1 - \cos\alpha}{1+ \cos\alpha}}\)

    \(\tan\frac{\alpha}{2} = \frac{1 - \cos\alpha}{\sin\alpha}\)

    \(\tan\frac{\alpha}{2} = \frac{\sin\alpha}{1 + \cos\alpha}\)

  • 万能公式

    \(\sin\alpha = \frac{2\tan\frac{\alpha}{2}}{1 + \tan^2\frac{\alpha}{2}}\)

    \(\cos\alpha = \frac{1- \tan^2\frac{\alpha}{2}}{1 + \tan^2\frac{\alpha}{2}}\)

    \(\tan\alpha = \frac{2\tan\frac{\alpha}{2}}{1 - \tan^2\frac{\alpha}{2}}\)

  • 正弦定理,余弦定理

    三角形 ABC 的角 A, B, C 对应的三边分别为 a, b, c 。其外接圆半径为 R ,则有

    正弦定理:\(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R\)

    余弦定理:

    \(a^2 = b^2 + c^2 - 2bc\cos A\)

    \(b^2 = a^2 + c^2 - 2ac\cos B\)

    \(c^2 = a^2 + b^2 - 2ab\cos C\)

  • 常用反三角公式

    \(\arcsin\alpha + \arccos\alpha = \frac{\pi}{2}\)