三角函数运算公式非常多,涵盖了基本恒等式、和差公式、倍角公式、半角公式、积化和差以及和差化积等。以下是一些基本和重要的三角函数公式:
基本恒等式
- $\sin^2 x + \cos^2 x = 1$
- $1 + \tan^2 x = \sec^2 x$
- $1 + \cot^2 x = \csc^2 x$
和角公式
- $\sin(x + y) = \sin x \cos y + \cos x \sin y$
- $\sin(x - y) = \sin x \cos y - \cos x \sin y$
- $\cos(x + y) = \cos x \cos y - \sin x \sin y$
- $\cos(x - y) = \cos x \cos y + \sin x \sin y$
- $\tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}$
- $\tan(x - y) = \frac{\tan x - \tan y}{1 + \tan x \tan y}$
倍角公式
- $\sin 2x = 2\sin x \cos x$
- $\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x$
- $\tan 2x = \frac{2\tan x}{1 - \tan^2 x}$
半角公式
- $\sin^2 x = \frac{1 - \cos 2x}{2}$
- $\cos^2 x = \frac{1 + \cos 2x}{2}$
- $\tan^2 x = \frac{1 - \cos 2x}{1 + \cos 2x}$
积化和差
- $\sin x \sin y = \frac{1}{2}[\cos(x - y) - \cos(x + y)]$
- $\cos x \cos y = \frac{1}{2}[\cos(x - y) + \cos(x + y)]$
- $\sin x \cos y = \frac{1}{2}[\sin(x + y) + \sin(x - y)]$
和差化积
- $\sin x + \sin y = 2 \sin\left(\frac{x + y}{2}\right) \cos\left(\frac{x - y}{2}\right)$
- $\sin x - \sin y = 2 \cos\left(\frac{x + y}{2}\right) \sin\left(\frac{x - y}{2}\right)$
- $\cos x + \cos y = 2 \cos\left(\frac{x + y}{2}\right) \cos\left(\frac{x - y}{2}\right)$
- $\cos x - \cos y = -2 \sin\left(\frac{x + y}{2}\right) \sin\left(\frac{x - y}{2}\right)$
特殊角的三角函数值
- $\sin 0 = 0, \cos 0 = 1, \tan 0 = 0$
- $\sin \frac{\pi}{6} = \frac{1}{2}, \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}, \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}}$
- $\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}, \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}, \tan \frac{\pi}{4} = 1$
- $\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}, \cos \frac{\pi}{3} = \frac{1}{2}, \tan \frac{\pi}{3} = \sqrt{3}$
- $\sin \frac{\pi}{2} = 1, \cos \frac{\pi}{2} = 0, \tan \frac{\pi}{2}$ is undefined
这些是三角函数中最常用的一些公式。在解决具体问题时,根据需要可能还会用到其他更复杂的公式。
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