三角函数公式

周期公式

基本周期

\(\sin x, \cos x:周期2\pi\)
\(\tan x, \cot x:周期\pi\)

缩放周期

\(\sin (ax + b), \cos (ax + b) \implies T = \frac{2\pi}{|a|}\)
\(\tan (ax + b), \cot (ax + b) \implies T = \frac{\pi}{|a|}\)

平方立方后的周期

\(\sin^2 x, \cos^2 x:周期\pi\)
\(\sin^3 x, \cos^3 x:周期2\pi\)
\(\sin ^n x, \cos ^n x:n为偶数T \implies \pi,\quad n为奇数T \implies 2\pi\)

和函数的周期

\(f(x)周期T_1，g(x)周期T_2，则f(x)+g(x)的周期是T_1,T_2的最小公倍数\)

乘积的周期

与和函数一样是两个周期的最小公倍数

基础公式

\(\sin^2x+\cos^2x = 1\)
\(\tan x = \frac{\sin x}{\cos x}, \cot x = \frac{\cos x}{\sin x}\)
\(\sec = \frac{1}{\cos x}, \csc = \frac{1}{\sin x}\)
\(1 + \tan^2x = \sec^2x, 1 + \cot^2x = \csc^2x\)

诱导公式（奇偶 / 周期转换，口诀：奇变偶不变，符号看象限）

1.\(\sin(-x) = -\sin x,\cos(-x) = \cos x,\tan(-x) = -\tan x\)
2.\(\sin(\pi-x) = \sin x,\cos(\pi-x) = -\cos x,\tan(\pi-x) = -\tan x\)
3.\(\sin(\frac{\pi}{2}-x) = \cos x,\cos(\frac{\pi}{2}-x) = \sin x\)
4.\(\sin(\frac{\pi}{2}+x) = \cos x,\cos(\frac{\pi}{2}+x) = -\sin x\)
5.\(\sin(\pi + x) = -\sin x,\cos(\pi + x) = -\cos x,\tan(\pi + x) = \tan x\)

和差角公式

1.\(\sin(A\pm B) = \sin A\cos B\pm \cos A\sin B\)
2.\(\cos(A\pm B) = \cos A\cos B\mp \sin A\sin B\)
3.\(\tan(A\pm B) = \frac{\tan A\pm \tan B}{1\mp \tan A\tan B}\)

倍角公式

1.\(\sin 2x = 2\sin x\cos x\)
2.\(\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x\)
3.\(\tan 2x = \frac{2\tan x}{1 - \tan^2 x}\)

半角公式（降幂/升幂）

1.\(\sin^2 \frac{x}{2} = \frac{1 - \cos x}{2}\)
2.\(\cos^2 \frac{x}{2} = \frac{1 + \cos x}{2}\)
3.\(\tan \frac{x}{2} = \frac{\sin x}{1 + \cos x} = \frac{1 - \cos x}{\sin x}\)

和差化积/积化和差（积分/极限）

和差化积

1.\(\sin A + \sin B = 2\sin \frac{A + B}{2} \cos \frac{A - B}{2}\)
2.\(\sin A - \sin B = 2\cos \frac{A + B}{2} \sin \frac{A - B}{2}\)
3.\(\cos A + \cos B = 2\cos \frac{A + B}{2} \cos \frac{A - B}{2}\)
4.\(\cos A - \cos B = -2\sin \frac{A + B}{2} \sin \frac{A - B}{2}\)

积化和差

1.\(\sin A\cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]\)
2.\(\cos A\sin B = \frac{1}{2}[\sin(A+B) - \sin(A-B)]\)
3.\(\cos A\cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)]\)
4.\(\sin A\sin B = -\frac{1}{2}[\cos(A+B) - \cos(A-B)]\)

辅助角公式（求最值/化简）

1.\(a\sin x + b\cos x = \sqrt{a^2 + b^2}\sin(x + \phi),其中\tan\phi = \frac{b}{a}\)
2.\(a\sin x + b\cos x = \sqrt{a^2 + b^2}\cos(x - \theta),其中\tan\theta = \frac{a}{b}\)

不等式

% ========== 核心高频（x→0时的不等式，求极限/夹逼准则必考） ==========
\(\sin x < x,\quad x>0\)
\(x < \tan x,\quad 0<x<\frac{\pi}{2}\)
\(\sin x < x < \tan x,\quad 0<x<\frac{\pi}{2}\)
\(|\sin x| \le |x|,\quad \forall x\in\mathbb{R}\)

% ========== 函数有界性（全题型通用） ==========
\(|\sin x| \le 1\)
\(|\cos x| \le 1\)

% ========== 高阶近似不等式（泰勒展开相关，积分/极限进阶用） ==========
\(x - \frac{x^3}{6} < \sin x < x,\quad x>0\)
\(\cos x > 1 - \frac{x^2}{2},\quad x\neq0\)

% ========== 数列极限夹逼专用 ==========
\(0 < \sin\frac{1}{n} < \frac{1}{n},\quad n\in\mathbb{N}^*\)
\(0 < \cos\frac{1}{n} < 1,\quad n\in\mathbb{N}^*\)

% ========== 区间内单调性/不等式 ==========
\(\sin x > \frac{2x}{\pi},\quad 0<x<\frac{\pi}{2}\)
\(\cos x > 1 - \frac{x^2}{2!} + \frac{x^4}{4!},\quad x\in\mathbb{R}\)