高中数学必备三角函数公式大全，轻松掌握三角变换，秒杀难题！

掌握高中三角函数公式是解决相关问题的关键。以下是一些核心公式，助你轻松应对三角变换，秒杀难题：

1. 基本定义：

- \(\sin \theta = \frac{\text{对边}}{\text{斜边}}\)

- \(\cos \theta = \frac{\text{邻边}}{\text{斜边}}\)

- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

2. 同角三角函数基本关系式：

- \(\sin^2 \theta + \cos^2 \theta = 1\)

- \(1 + \tan^2 \theta = \sec^2 \theta\)

- \(1 + \cot^2 \theta = \csc^2 \theta\)

3. 诱导公式（即三角函数的符号变换）：

- \(\sin(-\theta) = -\sin \theta\)

- \(\cos(-\theta) = \cos \theta\)

- \(\tan(-\theta) = -\tan \theta\)

- \(\sin(\pi - \theta) = \sin \theta\)

- \(\cos(\pi - \theta) = -\cos \theta\)

- \(\tan(\pi - \theta) = -\tan \theta\)

- \(\sin(\pi + \theta) = -\sin \theta\)

- \(\cos(\pi + \theta) = -\cos \theta\)

- \(\tan(\pi + \theta) = \tan \theta\)

4. 和差角公式：

- \(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\)

- \(\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta\)

- \(\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\)

- \(\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\)

- \(\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\)

- \(\tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}\)

5. 倍角公式：

- \(\sin 2\theta = 2\sin \theta \cos \theta\)

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

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

6. 半角公式：

- \(\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}\)

- \(\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}\)

- \(\tan \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} = \frac{1 - \cos \theta}{\sin \theta}\)

7. 积化和差公式：

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

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

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

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

8. 和差化积公式：

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

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

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

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

通过熟练掌握这些公式，你可以轻松应对各种三角变换问题，从而在考试中取得优异成绩。不断练习和应用这些公式，你会发现解题变得更加简单和快速。