First Chapter

同角三角函数的基本关系式

倒数关系	商的关系	平方关系
$\displaystyle{tan\alpha cot\alpha = 1}$	$\displaystyle{\frac{sin\alpha}{cos\alpha} = tan\alpha = \frac{sec\alpha}{csc\alpha}}$	$\displaystyle{sin^2\alpha + cos^2\alpha = 1}$
$\displaystyle{sin\alpha csc\alpha = 1}$	$\displaystyle{\frac{cos\alpha}{sin\alpha} = cot\alpha = \frac{csc\alpha}{sec\alpha}}$	$\displaystyle{1 + tan^2\alpha = sec^2\alpha}$
$\displaystyle{cos\alpha sec\alpha = 1}$		$\displaystyle{1 + cot^2\alpha = csc^2\alpha}$

和角公式	万能公式
$\displaystyle{sin(\alpha \pm \beta) = sin\alpha cos\beta \pm cos\alpha sin\beta}$	$\displaystyle{sin2\alpha = \frac{2tan\alpha}{1+tan^2 \alpha}}$
$\displaystyle{cos(\alpha \pm \beta) = cos\alpha cos\beta \mp sin\alpha sin\beta}$	$\displaystyle{cos2\alpha = \frac{1-tan^2\alpha}{1+tan^2 \alpha}}$
$\displaystyle{tan(\alpha \pm \beta) = \frac{tan\alpha \pm tan\beta}{1\mp tan\alpha tan\beta}}$	$\displaystyle{tan2\alpha = \frac{2tan\alpha}{1-tan^2 \alpha}}$

半角的正弦、余弦和正切公式	三角函数的降幂公式
$\displaystyle{sin\frac{\alpha}{2} = \pm\sqrt{\frac{1-cos\alpha}{2}}}$	$\displaystyle{sin^2\alpha = \frac{1-cos2\alpha}{2}}$
$\displaystyle{cos\frac{\alpha}{2} = \pm\sqrt{\frac{1+cos\alpha}{2}}}$	$\displaystyle{cos^2\alpha = \frac{1+cos2\alpha}{2}}$
$\displaystyle{tan\frac{\alpha}{2} = \pm\sqrt{\frac{1-cos\alpha}{1+cos\alpha}} = \frac{1-cos\alpha}{sin\alpha} = \frac{sin\alpha}{1+cos\alpha}}$	$\displaystyle{sin2\alpha = 2sin\alpha cos\alpha}$

二倍角的正弦、余弦和正切公式	三倍角的正弦、余弦和正切公式
$\displaystyle{sin2\alpha = 2sin\alpha cos\alpha}$	$\displaystyle{sin3\alpha = - 4sin^3\alpha + 3sin\alpha}$
$\displaystyle{cos2\alpha = cos^2\alpha - sin^2\alpha = 2cos^2\alpha -1 = 1-2sin^2\alpha}$	$\displaystyle{cos3\alpha = 4cos^3\alpha - 3sin\alpha}$
$\displaystyle{tan2\alpha = - \frac{2tan\alpha}{1-tan^2\alpha}}$	$\displaystyle{tan3\alpha = - \frac{3tan\alpha - tan3\alpha}{1-3tan2\alpha}}$

三角函数的和差化积公式	三角函数的积化和差公式
$\displaystyle{sin\alpha + sin\beta = 2sin\frac{\alpha + \beta}{2}cos\frac{\alpha - \beta}{2}}$	$\displaystyle{sin\alpha cos \beta} = \frac{1}{2}[sin(\alpha + \beta) + sin(\alpha - \beta)]$
$\displaystyle{sin\alpha - sin\beta = 2cos\frac{\alpha + \beta}{2}sin\frac{\alpha - \beta}{2}}$	$\displaystyle{cos\alpha sin \beta} = \frac{1}{2}[sin(\alpha + \beta) - sin(\alpha - \beta)]$
$\displaystyle{cos\alpha + cos\beta = 2cos\frac{\alpha + \beta}{2}cos\frac{\alpha - \beta}{2}}$	$\displaystyle{cos\alpha cos \beta} = \frac{1}{2}[cos(\alpha + \beta) + cos(\alpha - \beta)]$
$\displaystyle{cos\alpha - cos\beta = -2sin\frac{\alpha + \beta}{2}sin\frac{\alpha - \beta}{2}}$	$\displaystyle{sin\alpha sin \beta} = -\frac{1}{2}[cos(\alpha + \beta) - cos(\alpha - \beta)]$

辅助角公式

$asinx \pm bcosx = \sqrt{a^2 + b^2}sin(x\pm \phi)$