三角函数

三角函数基础知识

一、定义:

正弦: \(\sin A = \frac{a}{c} = \frac{对边}{斜边}\)

余弦: \(\cos A = \frac{b}{c} = \frac{邻边}{斜边}\)

正切: \(\tan A = \frac{a}{b} = \frac{对边}{邻边}\)

余切: \(\cot A = \frac{b}{a} = \frac{邻边}{对边}\)

特殊性质: |\(\sin \alpha\)| \(\leq 1\) , |\(\cos \alpha\)| \(\leq 1\)

二、特殊角三角函数

	\(0°\)	\(15°\)	\(30°\)	\(45°\)	\(60°\)	\(75°\)	\(90°\)
\(\sin\)	\(0\)	\(\frac{\sqrt{6}-\sqrt{2}}{4}\)	\(\frac{1}{2}\)	\(\frac{\sqrt{2}}{2}\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{\sqrt{6}+\sqrt{2}}{4}\)	\(1\)
\(\cos\)	\(0\)	\(\frac{\sqrt{6}+\sqrt{2}}{4}\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{\sqrt{2}}{2}\)	\(\frac{1}{2}\)	\(\frac{\sqrt{6}-\sqrt{2}}{4}\)	\(0\)
\(\tan\)	\(0\)	\(2-\sqrt{3}\)	\(\frac{\sqrt{3}}{3}\)	\(1\)	\(\sqrt{3}\)	\(2+\sqrt{3}\)	/
\(\cot\)	/	\(2+\sqrt{3}\)	\(\sqrt{3}\)	\(1\)	\(\frac{\sqrt{3}}{3}\)	\(2-\sqrt{3}\)	\(0\)

三、基本公式

1. 若 \(\angle A + \angle B = 90°\) ,则 \(\sin A = \cos B\) , \(\tan A = \cot B\)

2. \(\tan A \cdot \cot A = 1\)

3. \(\tan A = \frac{\sin A}{\cos A}\)

4. \(\sin^2 A + \cos^2 A = 1\)

四、三角形面积公式

五、两角和差公式

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

\(\cos (\alpha \pm \beta) = \cos \alpha \cdot \cos \beta \mp \sin \alpha \cdot \sin \beta\)

\(\tan (\alpha \pm \beta) = \frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \cdot \tan \beta}\)

六、倍角公式

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

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

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

七、直线斜率

1. |\(k\)| \(= \tan \theta\) , \(\theta\) 为该直线与 \(x\) 轴相交所形成的最小夹角

（以上都是初中内容，现在来点高中的）

八、半角公式

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

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

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

九、万能公式

\[\begin{cases} \cos \theta = \frac{1 - \tan ^ 2\frac{\theta}{2}}{1 + \tan ^ 2\frac{\theta}{2}} \\ \sin \theta = \frac{2\tan\frac{\theta}{2}}{1 + \tan ^ 2\frac{\theta}{2}} \end{cases} \]

十、和差化积，积化和差

\[\sin\alpha + \sin\beta = 2\sin\frac{\alpha +\beta}{2}\cos\frac{\alpha - \beta}{2} \]

\[\sin\alpha -\sin\beta = 2\sin\frac{\alpha -\beta}{2}\cos\frac{\alpha + \beta}{2} \]

\[\cos \alpha + \cos\beta = 2\cos\frac{\alpha + \beta}{2}\cos\frac{\alpha -\beta}{2} \]

\[\cos \alpha -\cos\beta = -2\sin\frac{\alpha + \beta}{2}\sin\frac{\alpha -\beta}{2} \]

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

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

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

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

十一、正余弦定理

\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

\[\cos A = \frac{b ^ 2 + c ^ 2 - a ^ 2}{2bc} \]

\[a ^ 2 = b ^ 2 + c ^ 2 - 2bc\cos A \]

十二、神秘公式（两次和差化积）

\[\sin^2x-\sin^2y =(\sin x + \sin y)(\sin x - \sin y)=\sin(x-y)\sin(x+y) \]

\[\cos^2x-\cos^2y=(\cos x+\cos y)(\cos x - \cos y) = -\sin(x-y)\sin(x+y) \]