初识三角函数

三角函数六边形记忆法则

规律1. 六边形对角线互为倒数（倒数关系）

$\cos x=\dfrac{1}{\sec x}$ 或者 $\sec x=\dfrac{1}{\cos x}$

$\sin x=\dfrac{1}{c s c x}$ 或者 $\csc x=\dfrac{1}{\sin x}$

$\tan x=\dfrac{1}{\cot x}$ 或者 $\cot x=\dfrac{1}{\tan x}$

规律2. 灰色三角形上端的平方之和等于下端的平方 (平方关系)

sin²x + cos²x = 1 tan²x + 1 = sec²x 1 + cot²x = csc²x

规律3. 任意一点的值等于这一点顺时针的第一个值与第二个值的比值

$$\begin{aligned} &\tan x=\dfrac{\sin x}{\cos x} \\\\ &\sin x=\dfrac{\cos x}{\cot x} \\\\ &\cos x=\dfrac{\cot x}{\csc x} \\\\ &\cot x=\dfrac{\csc x}{\sec x} \\\\ &\csc x=\dfrac{\sec x}{\tan x} \\\\ &\sec x=\dfrac{\tan x}{\sin x} \end{aligned}$$

规律4. 任意一点的值等于紧挨着这一点的两个端点的值的积

tan x = sin x × sec x sin x = cos x × tan x

sin 函数的各常数

y = Asin (ωx + φ), x ∈ [0, +∞),其中A > 0, ω > 0.

A叫做振幅，表示做简谐运动物体离开平衡位置的最大距离。
这个简谐运动的频率$f=\dfrac{1}{T}=\dfrac{\omega}{2\pi}$
ωx + φ称为相位
x = 0时的相位φ称为初相

基本性质与图像变换

	y = sin x	y = cos x	y = tan x
奇偶性	奇函数	偶函数	奇函数
单调性	$\left[ {-\dfrac{\pi}{2} + 2k\pi,\dfrac{\pi }{2} + 2k\pi}\right]$上递增 $\left[ {\dfrac{\pi}{2} + 2k\pi,\dfrac{3\pi}{2} + 2k\pi}\right]$上递碱	[−π + 2kπ, 2kπ]上递增 [2kπ, π + 2kπ]上递减	$\left( { - \dfrac{\pi }{2} + k\pi,\dfrac{\pi }{2}+k\pi} \right)$上递增
周期性	y = Asin (ωx + φ)(ω ≠ 0) $T=\dfrac{2\pi }{\mid \omega \mid }$	y = Acos (ωx + φ)(ω ≠ 0) $T=\dfrac{2\pi }{\mid \omega \mid }$	y = Atan (ωx + φ)(ω ≠ 0) $T=\dfrac{\pi }{\mid \omega \mid }$

图像变换

平移变换：左加右减，上加下减
周期变换：y = sin x → y = sin ωx，横坐标伸长/缩短为原来的$\dfrac{1}{\omega}$倍
振幅变换：y = sin x → y = Asin x，纵坐标伸长/缩短为原来的A倍

三角函数恒等变换

诱导公式

口诀

奇变偶不变，符号看象限。

“符号看象限”的含义是：把角α看做锐角，不考虑α角所在象限，看$n·(\dfrac{\pi}{2})±α$是第几象限角，从而得到等式右边是正号还是负号。可结合单位圆判断。

$$\left\{ \begin{array}{l} \sin (2k\pi + \alpha ) = \sin \alpha \\ \cos (2k\pi + \alpha ) = \cos \alpha \\ \tan (2k\pi + \alpha ) = \tan \alpha \\ \cot (2k\pi + \alpha ) = \cot \alpha \end{array} \right.$$

$$\left\{ \begin{array}{l} \sin (\pi + \alpha ) = - \sin \alpha \\ \cos (\pi + \alpha ) = - \cos \alpha \\ \tan (\pi + \alpha ) = \tan \alpha \\ \cot (\pi + \alpha ) = \cot \alpha \end{array} \right.$$

$$\left\{ \begin{array}{l} \sin (\pi - \alpha ) = \sin \alpha \\ \cos (\pi - \alpha ) = - \cos \alpha \\ \tan (\pi - \alpha ) = - \tan \alpha \\ \cot (\pi - \alpha ) = - \cot \alpha \end{array} \right.$$

$$\left\{ \begin{array}{l} \sin \left( {\dfrac{\pi }{2} + \alpha } \right) = \cos \alpha \\ \sin \left( {\dfrac{\pi }{2} - \alpha } \right) = \cos \alpha \\ \cos \left( {\dfrac{\pi }{2} + \alpha } \right) = - \sin \alpha \\ \cos \left( {\dfrac{\pi }{2} - \alpha } \right) = \sin \alpha \\ \tan \left( {\dfrac{\pi }{2} + \alpha } \right) = - \cot \alpha \\ \tan \left( {\dfrac{\pi }{2} - \alpha } \right) = \cot \alpha \\ \cot \left( {\dfrac{\pi }{2} + \alpha } \right) = - \tan \alpha \\ \cot \left( {\dfrac{\pi }{2} - \alpha } \right) = \tan \alpha \end{array} \right.$$

和差角公式

sin (α ± β) = sin α ⋅ cos β ± cos α ⋅ sin β cos (α ± β) = cos α ⋅ cos β ∓ sin α ⋅ sin β

二倍角公式

$$\sin 2\alpha = 2\sin \alpha \cos \alpha = \dfrac{{2\tan \alpha }}{{1 + {{\tan }^2}\alpha }}\\\cos 2\alpha = {\cos ^2}\alpha - {\sin ^2}\alpha = 1 - 2{\sin ^2}\alpha = 2{\cos ^2}\alpha - 1\\\tan 2\alpha = \dfrac{{2\tan \alpha }}{{1 - {{\tan }^2}\alpha}}$$

半角公式

$$ \sin \dfrac{\alpha }{2} = \pm \sqrt {\dfrac{{1 - \cos \alpha }}{2}} \\ \cos \dfrac{\alpha }{2} = \pm \sqrt {\dfrac{{1 + \cos \alpha }}{2}} \\ \tan \dfrac{\alpha }{2} = \pm \sqrt {\dfrac{{1 - \cos \alpha }}{{1 + \cos \alpha }}} = \dfrac{{\sin \alpha }}{{1 + \cos \alpha }} = \dfrac{{1 - \cos \alpha }}{{\sin \alpha }} $$

二倍角、半角公式可用于降次。

万能公式

$$\sin \alpha = \dfrac{2\tan \dfrac{\alpha}{2}}{1 + \tan ^2 \dfrac{\alpha }{2}}$$

$$\cos \alpha = \dfrac{1 - \tan ^2 \dfrac{\alpha }{2}}{1 + \tan ^2 \dfrac{\alpha }{2}}$$

$$\tan \alpha = \dfrac{2\tan \dfrac{\alpha}{2}}{1 - \tan ^2 \dfrac{\alpha }{2}}$$

积化和差

$$\sin \alpha \cdot \cos \beta = \dfrac{1}{2}\left[ {\sin \left( {\alpha + \beta } \right) + \sin \left( {\alpha - \beta } \right)} \right]$$

$$\cos \alpha \cdot \sin \beta = \dfrac{1}{2}\left[ {\sin \left( {\alpha + \beta } \right) - \sin \left( {\alpha - \beta } \right)} \right]$$

$$\cos \alpha \cdot \cos \beta = \dfrac{1}{2}\left[ {\cos \left( {\alpha + \beta } \right) + \cos \left( {\alpha - \beta } \right)} \right]$$

$$\sin \alpha \cdot \sin \beta = - \dfrac{1}{2}\left[ {\cos \left( {\alpha + \beta } \right) - \cos \left( {\alpha - \beta } \right)} \right]$$

口诀

正和正在先，正差正后迁；余和一色余，余差翻了天。

和差化积

$$\sin \alpha + \sin \beta = 2\sin \dfrac{\alpha + \beta }{2} \cdot \cos \dfrac{\alpha - \beta }{2}$$

$$\sin \alpha - \sin \beta = 2\cos \dfrac{\alpha + \beta }{2} \cdot \sin \dfrac{\alpha - \beta }{2}$$

$$\cos \alpha + \cos \beta = 2\cos \dfrac{\alpha + \beta }{2} \cdot \cos \dfrac{\alpha - \beta }{2}$$

$$\cos \alpha - \cos \beta = - 2\sin \dfrac{\alpha + \beta }{2} \cdot \sin \dfrac{\alpha - \beta }{2}$$

口诀

正加正，正在前；正减正，余在前；
余加余，余并肩；余减余，负正弦。

$ A (t+u)+B (w t+z)$$= (t-- ) $

其他公式

$$\sin 3 \alpha = 3 \sin \alpha - 4{\sin ^3}\alpha = 4 \cdot \sin (60^\circ - \alpha ) \cdot \sin \alpha \cdot \sin (60^\circ + \alpha)$$ $$\cos 3\alpha = 4{\cos ^3}\alpha - 3\cos \alpha = 4 \cdot \cos (60^\circ - \alpha ) \cdot \cos \alpha \cdot \cos (60^\circ + \alpha )$$ $$\tan 3\alpha = \tan (60^\circ - \alpha ) \cdot \tan \alpha \cdot \tan (60^\circ + \alpha )$$ $${\sin ^2}\alpha - {\sin ^2}\beta = \sin (\alpha + \beta ) \cdot \sin(\alpha - \beta )$$ $${\cos ^2}\alpha - {\cos ^2}\beta = - \sin (\alpha + \beta ) \cdot \sin(\alpha - \beta )$$ 【辅助角公式】$a\sin x + b\cos x = \sqrt {{a^2} + {b^2}} \sin (x + \theta )$,其中$\tan\theta=\dfrac{b}{a}$.