三角函数
sin(x) * csc(x) = 1
cos(x) * sec(x) = 1
tan(x) * cot(x) = 1
三角换元
奇变偶不变，符号看象限
$ sin(x + 2k\pi) = sin(x) ~~~~~ sin(-x) = -sin(x) \(
\) cos(x + 2k\pi) = cos(x) ~~~~~ cos(-x) = cos(x) \(
\) tan(x + 2k\pi) = tan(x) ~~~~~ tan(-x) = -tan(x) $
$ sin(\pi + x) = -sin(x) ~~~~~ sin(\pi - x) = sin(x) \( \) cos(\pi + x) = -cos(x) ~~~~~ cos(\pi - x) = -cos(x) \( \) tan(\pi + x) = tan(x) ~~~~~~~ tan(\pi - x) = -tan(x) $
$ sin(\frac{\pi}{2} + x) = cos(x) ~~~~~ sin(\frac{\pi}{2} - x) = cos(x) \( \) cos(\frac{\pi}{2} + x) = -sin(x) ~~~~~ cos(\frac{\pi}{2} - x) = sin(x) \( \) tan(\frac{\pi}{2} + x) = -cot(x) ~~~~~ tan(\frac{\pi}{2} - x) = cot(x) $
$ sin(\frac{3\pi}{2} + x) = -cos(x) ~~~~~ sin(\frac{3\pi}{2} - x) = -cos(x) \( \) cos(\frac{3\pi}{2} + x) = sin(x) ~~~~~~ cos(\frac{3\pi}{2} - x) = -sin(x) \( \) tan(\frac{3\pi}{2} + x) = -cot(x) ~~~~~ tan(\frac{3\pi}{2} - x) = cot(x) $
和角公式
$ sin(a + b) = sin(a)cos(b) + cos(a)sin(b) \( \) sin(a - b) = sin(a)cos(b) - cos(a)sin(b) $
$ cos(a + b) = cos(a)cos(b) - sin(a)sin(b) \( \) cos(a - b) = cos(a)cos(b) + sin(a)sin(b) $
$ tan(a + b) = \frac{tan(a) + tan(b)}{1 - tan(a)tan(b)} \( \) tan(a - b) = \frac{tan(a) - tan(b)}{1 + tan(a)tan(b)} $
二倍角公式和万能公式
$ sin(2x) = 2sin(a)cos(a) = \frac{2tan(x)}{1 + tan^{2}(x)} \( \) cos(2x) = cos^{2}(x) - sin^{2}(x) = \frac{2tan(x)}{1 - tan^{2}(x)} $
其他
$ \frac{sin(x)}{1 + cos(x)} = tan(\frac{\theta}{2}) \( \) \frac{sin(x)}{1 - cos(x)} = tan^{-1}(\frac{\theta}{2}) $
$ 1 + tan^{2}(x) = sec^{2}(x) \( \) 1 + cot^{2}(x) = csc^{2}(x) $
$ sin^{2}(x) + cos^{2}(x) = 1 $
$ \frac{1 - cos(x)}{1 + cos(x)} = tan^{2}(\frac{x}{2}) $
导数
| $ 原函数 $ | $ 导数 $ | $ 特殊情况 $ | $ 导数 $ |
|---|---|---|---|
| $ a $ | $ 0 $ | ||
| $ x^{a} $ | $ ax^{a - 1} $ | ||
| $ a^{x} $ | $ a^{x}ln(a) $ | $ e^{x} $ | $ e^{x} $ |
| $ \log_{a}{x} $ | $ \frac{1}{xln(a)} $ | $ ln(x) $ | $ \frac{1}{x} $ |
| $ sin(x) $ | $ cos(x) $ | ||
| $ cos(x) $ | $ sin(x) $ | ||
| $ tan(x) $ | $ sec^{2}(x) $ | ||
| $ csc(x) $ | $ -csc(x)tan(x) $ | ||
| $ sec(x) $ | $ sec(x)tan(x) $ | ||
| $ cot(x) $ | $ -csc^{2}(x) $ |
微分方程
$ y'' + py' + qy = 0 ~~~ y = e^{rx} \( \) 将 ~~ y = e^{rx} ~~ 代入 ~~ y'' + py' + qy = 0 ~~~ y = e^{rx} \( \) 得 ~~ (r^{2} + pr + q)e^{rx} = 0 \( \) 即 \( \) r^{2} + pr + q = 0 \( \) 解出r的两根r_{1},r_{2} \( \) 按下表即可求出微分方程的通解 $
| $ 特征方程r^{2} + pr + q = 0的两根r_{1},r_{2}$ | $微分方程y'' + py' + qy = 0的通解 $ |
|---|---|
| $ r_{1} \ne r_{2}$ | $ y = C_{1}e^{r_{1}x} + C_{2}e^{r_{2}x} $ |
| $ r_{1} = r_{2} = r$ | $ y = (C_{1} + C_{2}x)e^{rx} $ |
| $ r_{1} = \alpha + i \beta , r_{2} = \alpha-i\beta $ | $ y = e^{\alpha x}(C_{1}cos\beta x + C_{2}sin \beta x) $ |
$ \int e^{kx} ~ sinax ~ dx = \frac {1} {k^{2} + a^{2}}\left|\begin {array}{c}
(e^{kx})' & (sinax)' \
e^{kx} & sinax \
\end{array}\right| + C$