「数学」三角函数公式以及部分证明
定义
在\(Rt\triangle ABC\)中，如下有六个三角函数的定义：
正弦：
\[\sin A = \frac{a}{c}
\]
级数表示：\(\sin (x)==\sum_{k=0}^{\infty} \frac{(-1)^{k} x^{1+2k}}{(1+2k)!}\)
余弦：
\[\cos A = \frac{b}{c}
\]
级数表示：\(\cos (x)=\sum_{k=0}^{\infty} \frac{(-1)^{k} x^{2 k}}{(2 k) !}\)
正切：
\[\tan A = \frac{a}{b}
\]
级数表示：\(\tan (x)=i+2 i \sum_{k=1}^{\infty}(-1)^{k} q^{2 k} \color{gray}\textrm{ for } q=e^{i x}\)
余切：
\[\cot A = \frac{b}{a}
\]
级数表示：\(\cot (x)=-i-2 i \sum_{k=1}^{\infty} q^{2 k} \color{gray}\text { for } q=e^{i x}\)
正割：
\[\sec A = \frac{c}{b}
\]
级数表示：\(\sec (x)=-2 \sum_{k=1}^{\infty}(-1)^{k} q^{-1+2 k} \color{gray}\text { for } q=e^{i x}\)
余割：
\[\csc A = \frac{c}{a}
\]
级数表示：\(\csc (x)=-2 i \sum_{k=1}^{\infty} q^{-1+2 k} \color{gray}\text { for } q=e^{i x}\)
诱导公式
关系 & 定理 & 公式
倒数关系
\[\cos \alpha \cdot \sec \alpha = 1
\]
\[\sin \alpha \cdot \csc \alpha = 1
\]
\[\tan \alpha \cdot \cot \alpha = 1
\]
平方关系
\[1 + \tan ^ 2 \alpha = \sec ^ 2 \alpha
\]
\[1 + \cot ^ 2 \alpha = \csc ^ 2 \alpha
\]
\[\sin^2 \alpha + cos ^ 2 \alpha = 1
\]
商的关系
\[\frac{\sin \alpha}{\cos \alpha} = \frac{\sec \alpha}{\csc \alpha} = \tan \alpha
\]
\[\frac{\cos \alpha}{\sin \alpha} = \frac{\csc \alpha}{\sec \alpha} = \cot \alpha
\]
正弦定理
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R = D
\]
\(R\) 为三角形外切圆半径，\(D\) 为三角形外切圆直径。
证明：
如图在 \(\triangle ABC\) 中可得 \(\sin A = \frac{h}{b}\) 和 \(\sin B = \frac{h}{a}\) 。
\[\therefore h = \sin A \times b, h = \sin B \times a \\\\ \therefore \sin A \times b = \sin B \times a \\\\ \therefore \frac{\sin A}{a} = \frac{\sin B}{b} \\\\ \therefore \frac{a}{\sin A} = \frac{b}{\sin B} \\\\ \textrm{同理：} \frac{a}{\sin A} = \frac{c}{\sin C} \\\\ \therefore \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
如图， \(\triangle CDB\) 中线段 \(CD\) 经过圆心，所以 \(\angle CBD = 90 ^ \circ\) ， \(CD = 2R\)。
\[\therefore \sin A = \sin D = \frac{CB}{CD} = \frac{a}{2R} \\\\ \therefore \frac{a}{\sin A} = 2R \\\\ \textrm{同理：} \frac{b}{\sin B} = 2R, \frac{c}{\sin C} = 2R \\\\ \therefore \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R = D
\]
余弦定理
\[a ^ 2 = b ^ 2 + c ^ 2 - 2bc\cos A,\ b ^ 2 = a ^ 2 + c ^ 2 - 2ac\cos B,\ c ^ 2 = a ^ 2 + b ^ 2 - 2ab\cos C \\\\ \rm{或} \\\\ \cos A = \frac{b ^ 2 + c ^ 2 - a ^ 2}{2bc},\ \cos B = \frac{a ^ 2 + c ^ 2 - b ^ 2}{2ac},\ \cos C = \frac{a ^ 2 + b ^ 2 - c ^ 2}{2ab}
\]
证明：
如图，在 \(\triangle ABC\) 中，令\(\vec{AB} = \vec{c}, \vec{CB} = \vec{a}, \vec{CA} = \vec{b}\)。
\[\therefore \vec{c} = \vec{AB} = \vec{CB} - \vec{CA} = \vec{a} - \vec{b} \\\\ \therefore (\vec{c}) ^ 2 = (\vec{a} - \vec{b}) ^ 2 = \vec{a} ^ 2 + \vec{b} ^ 2 - 2 \vec{a} \cdot \vec{b} \\\\ \therefore |\vec{c}| ^ 2 = |\vec{a}| ^ 2 + |\vec{b}| ^ 2 - 2 |\vec{a}| \cdot |\vec{b}| \cdot \cos C \\\\ \therefore c ^ 2 = a ^ 2 + b ^ 2 - 2ab\cos C \\\\ 同理：\cos A = \frac{b ^ 2 + c ^ 2 - a ^ 2}{2bc},\ \cos B = \frac{a ^ 2 + c ^ 2 - b ^ 2}{2ac}
\]
和角公式
\[\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta
\]
\[\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta
\]
\[\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}
\]
差角公式
\[\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta
\]
\[\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta
\]
\[\tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}
\]
和差化积
\[\sin \alpha+\sin \beta=2 \sin \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)
\]
\[\sin \alpha-\sin \beta=2 \sin \left(\frac{\alpha-\beta}{2}\right) \cos \left(\frac{\alpha+\beta}{2}\right)
\]
\[\cos \alpha+\cos \beta=2 \cos \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)
\]
\[\cos \alpha-\cos \beta=-2 \sin \left(\frac{\alpha+\beta}{2}\right) \sin \left(\frac{\alpha-\beta}{2}\right)
\]
积化和差
\[\cos \alpha \sin \beta=\frac{1}{2}[\sin (\alpha+\beta)-\sin (\alpha-\beta)]
\]
\[\sin \alpha \cos \beta=\frac{1}{2}[\sin (\alpha+\beta)+\sin (\alpha-\beta)]
\]
\[\cos \alpha \cos \beta=\frac{1}{2}[\cos (\alpha+\beta)+\cos (\alpha-\beta)]
\]
\[\sin \alpha \sin \beta=-\frac{1}{2}[\cos (\alpha+\beta)-\cos (\alpha-\beta)]
\]
倍角公式
\[\sin 2 \alpha = 2 \sin \alpha \cos \alpha
\]
\[\cos 2 \alpha = \cos ^ 2 \alpha - \sin ^ 2 \alpha
\]
\[\tan 2 \alpha = \frac{2 \tan \alpha}{1 - \tan ^ 2 \alpha}
\]
\[\cot 2 \alpha=\frac{\cot ^{2} \alpha-1}{2 \cot \alpha}
\]
\[\sec 2 \alpha=\frac{\sec ^{2} \alpha}{1-\tan ^{2} \alpha}
\]
\[\csc 2 \alpha=\frac{1}{2} \sec \alpha \csc \alpha
\]
半角公式
\[\sin \left(\frac{\alpha}{2}\right) = \sqrt{\frac{1-\cos \alpha}{2}}
\]
\[\cos \left(\frac{\alpha}{2}\right) = \sqrt{\frac{1+\cos \alpha}{2}}
\]
\[\tan \left(\frac{\alpha}{2}\right) = \csc \alpha-\cot \alpha
\]
\[\cot \left(\frac{\alpha}{2}\right) = \csc \alpha+\cot \alpha
\]
\[\sec \left(\frac{\alpha}{2}\right) = \sqrt{\frac{2 \sec \alpha}{\sec \alpha+1}}
\]
\[\csc \left(\frac{\alpha}{2}\right) = \sqrt{\frac{2 \sec \alpha}{\sec \alpha-1}}
\]
Attachment
常用三角函数值对照表：
| 角\(\alpha\) | 弧度 | \(\sin\)值 | \(\cos\)值 | \(\tan\)值 |
|---|---|---|---|---|
| \(0^\circ\) | \(0\) | \(0\) | \(1\) | \(0\) |
| \(15^\circ\) | \(\frac{\pi}{12}\) | \(\frac{\sqrt{6} - \sqrt{2}}{4}\) | \(\frac{\sqrt{6} + \sqrt{2}}{4}\) | \(2 - \sqrt{3}\) |
| \(22.5^\circ\) | \(\frac{\pi}{8}\) | \(\frac{\sqrt{2 - \sqrt{2}}}{2}\) | \(\frac{\sqrt{2 + \sqrt{2}}}{2}\) | \(-1 + \sqrt{2}\) |
| \(30^\circ\) | \(\frac{\pi}{6}\) | \(\frac{1}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{\sqrt{3}}{3}\) |
| \(45^\circ\) | \(\frac{\pi}{4}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(1\) |
| \(60^\circ\) | \(\frac{\pi}{3}\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{2}\) | \(\sqrt{3}\) |
| \(75^\circ\) | \(\frac{5\pi}{12}\) | \(\frac{\sqrt{6} + \sqrt{2}}{4}\) | \(\frac{\sqrt{6} - \sqrt{2}}{4}\) | \(2 + \sqrt{3}\) |
| \(90^\circ\) | \(\frac{\pi}{2}\) | \(1\) | \(0\) | \(\rm{无}\) |
| \(120^\circ\) | \(\frac{2\pi}{3}\) | \(\frac{\sqrt{3}}{2}\) | \(-\frac{1}{2}\) | \(-\sqrt{3}\) |
| \(135^\circ\) | \(\frac{3\pi}{4}\) | \(\frac{\sqrt{2}}{2}\) | \(-\frac{\sqrt{2}}{2}\) | \(-1\) |
| \(150^\circ\) | \(\frac{5\pi}{6}\) | \(\frac{1}{2}\) | \(-\frac{\sqrt{3}}{2}\) | \(\frac{\sqrt{3}}{3}\) |
| \(180^\circ\) | \(\pi\) | \(0\) | \(-1\) | \(0\) |
| \(270^\circ\) | \(\frac{3\pi}{2}\) | \(-1\) | \(0\) | \(\rm{无}\) |
| \(360^\circ\) | \(2\pi\) | \(0\) | \(1\) | \(0\) |
