Trigonometry

1 定义

\(\sin\theta=\dfrac{y}{r}\)，正弦：sine / saɪn /

\(\cos\theta=\dfrac{x}{r}\)，余弦：cosine / ˈkoʊsaɪn /

\(\tan\theta=\dfrac{y}{x}\)，正切：tangent / ˈtændʒənt /

\(\cot\theta=\dfrac{x}{y}\)，余切：cotangent / koʊˈtændʒənt /

\(\sec\theta=\dfrac{r}{x}\)，正割：secant / ˈsiːkənt /

\(\csc\theta=\dfrac{r}{y}\)，余割：cosecant / koʊˈsiːkənt /

2 特殊值

	$\sin$	$\cos$	$\tan$	$\cot$	$\sec$	$\csc$
$0(0^\circ)$	$0$	$1$	$0$	——	$1$	——
$\dfrac{\pi}{6}(30^\circ)$	$\dfrac{1}{2}$	$\dfrac{\sqrt 3}{2}$	$\dfrac{\sqrt 3}{3}$	$\sqrt 3$	$\dfrac{2 \sqrt 3}{3}$	$2$
$\dfrac{\pi}{4}(45^\circ)$	$\dfrac{\sqrt 2}{2}$	$\dfrac{\sqrt 2}{2}$	$1$	$1$	$\sqrt 2$	$\sqrt 2$
$\dfrac{\pi}{3}(60^\circ)$	$\dfrac{\sqrt 3}{2}$	$\dfrac{1}{2}$	$\sqrt 3$	$\dfrac{\sqrt 3}{3}$	$2$	$\dfrac{2 \sqrt 3}{3}$
$\dfrac{\pi}{2}(90^\circ)$	$1$	$0$	——	$0$	——	$0$

3 负角公式

\[ \begin{aligned} \sin(-\theta) &= -\sin\theta & \cos(-\theta) &= \cos\theta \\ \tan(-\theta) &= -\tan\theta & \cot(-\theta) &= -\cot\theta \\ \sec(-\theta) &= \sec\theta & \csc(-\theta) &= -\csc\theta \end{aligned} \]

4 倒数公式

根据定义可以得到如下关系式：

\[ \sin\theta\csc\theta = 1 \qquad \cos\theta\sec\theta = 1 \qquad \tan\theta\cot\theta = 1 \]

5 乘积公式

同样根据定义，可以用另外两个三角函数表示其中一个：

\[ \begin{aligned} \sin\theta &= \cos\theta \, \tan\theta \qquad &\cos\theta = \sin\theta \, \cot\theta \\ \tan\theta &= \sin\theta \, \sec\theta \qquad &\cot\theta = \cos\theta \, \csc\theta \\ \sec\theta &= \tan\theta \, \csc\theta \qquad &\csc\theta = \cot\theta \, \sec\theta \end{aligned} \]

6 商公式

\[ \tan\theta = \dfrac{\sin\theta}{\cos\theta} \qquad \cot\theta = \dfrac{\cos\theta}{\sin\theta} \]

7 平方和公式

\[ \sin^2\theta + \cos^2\theta = 1 \qquad \tan^2\theta + 1 = \sec^2\theta \qquad \cot^2\theta +1 = \csc^2\theta \]

8 和差角公式

余弦和角公式：\(\cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta\)

余弦差角公式：\(\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta\)

正弦和角公式：\(\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta\)

正弦差角公式：\(\sin(\alpha - \beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta\)

正切和角公式：\(\tan(\alpha+\beta) = \dfrac{\tan\alpha + \tan\beta}{1-\tan\alpha\tan\beta}\)

正切差角公式：\(\tan(\alpha - \beta) = \dfrac{\tan\alpha - \tan\beta}{1+\tan\alpha\tan\beta}\)

9 诱导公式

奇变偶不变，符号看象限

\[ \begin{aligned} \cos\left(\alpha - \frac{\pi}{2}\right) &= \sin\alpha \qquad &\sin\left(\alpha - \frac{\pi}{2}\right) &= -\cos\alpha \\ \ \\ \cos\left(\frac{\pi}{2} - \alpha\right) &= \sin\alpha \qquad &\sin\left(\frac{\pi}{2} - \alpha\right) &= \cos\alpha \\ \ \\ \cos\left(\alpha + \frac{\pi}{2}\right) &= -\sin\alpha \qquad &\sin\left(\alpha + \frac{\pi}{2}\right) &= \cos\alpha \end{aligned} \]

10 二倍角公式

10.1 二倍\(\rightarrow\)平方

\[ \begin{aligned} \sin 2\alpha &= 2\sin\alpha\cos\alpha \\ \ \\ \cos 2\alpha &= \cos^2\alpha-\sin^2\alpha \\ \ \\ &= 2\cos^2\alpha - 1 \\ \ \\ &= 1 - 2\sin^2\alpha \\ \ \\ \tan 2\alpha &= \frac{2\tan\alpha}{1-\tan^2\alpha} \end{aligned} \]

10.2 平方\(\rightarrow\)二倍

\[ \sin^2\alpha = \frac{1-\cos2\alpha}{2} \qquad \cos^2\alpha = \frac{1+\cos2\alpha}{2} \qquad \tan^2\alpha = \frac{1-\cos2\alpha}{1+\cos2\alpha} \]

10.3 半角公式

\[ \tan^2\alpha = \frac{\sin\alpha}{1+\cos\alpha} = \frac{1-\cos\alpha}{\sin\alpha} \]

11 万能代换公式

\[ \sin\alpha = \frac{2\tan\frac{\alpha}{2}}{1+\tan^2\frac{\alpha}{2}} \qquad \cos\alpha = \frac{1-\tan\frac{\alpha}{2}}{1+\tan^2\frac{\alpha}{2}} \qquad \tan\alpha = \frac{2\tan\frac{\alpha}{2}}{1-\tan^2\frac{\alpha}{2}} \]

12 积化和差

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

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

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

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

13 和差化积

\(\sin\alpha + \sin\beta = 2\sin\dfrac{\alpha+\beta}{2} \cos\dfrac{\alpha-\beta}{2}\)

\(\sin\alpha - \sin\beta = 2\cos\dfrac{\alpha+\beta}{2} \sin\dfrac{\alpha-\beta}{2}\)

\(\cos\alpha + \cos\beta = 2\cos\dfrac{\alpha+\beta}{2} \cos\dfrac{\alpha-\beta}{2}\)

\(\cos\alpha - \cos\beta = -2\sin\dfrac{\alpha+\beta}{2} \sin\dfrac{\alpha-\beta}{2}\)

14 辅助角公式

\(a\sin x + b\cos x = \sqrt{a^2 + b^2} \, \sin\left( x + \arctan\frac{b}{a} \right)\)

\(a\sin x - b\cos x = \sqrt{a^2 + b^2} \, \sin\left( x - \arctan\frac{b}{a} \right)\)