三角函数常用公式集锦
欢迎来到三角函数世界
两角和(差)公式
$$
sin ( \alpha + \beta ) = sin ( \alpha ) cos ( \beta ) + cos ( \alpha ) sin( \beta )
$$
$$
sin ( \alpha - \beta ) = sin ( \alpha ) cos ( \beta ) - cos ( \alpha ) sin ( \beta )
$$
$$
cos ( \alpha + \beta ) = cos ( \alpha ) cos ( \beta ) - sin ( \alpha ) sin ( \beta )
$$
$$
cos ( \alpha - \beta ) = cos ( \alpha ) cos ( \beta ) + sin ( \alpha ) sin ( \beta )
$$
$$
tan ( \alpha + \beta ) = \frac{tan \alpha + tan \beta }{1 - tan \alpha tan \beta }
$$
$$
tan ( \alpha - \beta ) = \frac{tan \alpha - tan \beta }{1 + tan \alpha tan \beta }
$$
倍角公式
$$
sin ( 2 \theta ) = 2sin ( \theta ) cos ( \theta )
$$
$$
\begin{align}
cos ( 2 \theta ) &= cos^2 ( \theta ) - sin^2 ( \theta ) \\
&= 2cos^2 ( \theta - 1 ) \\
&= 1 - 2 sin^2 ( \theta )
\end{align}
$$
$$
tan ( 2\theta ) = \frac{ 2 tan ( \theta ) }{ 1 - tan^2 ( \theta ) }
$$
半角公式
$$
sin ( \frac{ \theta }{ 2 } ) = \pm \sqrt {\frac{1 - cos ( \theta )}{2}}
$$
$$
cos ( \frac{\theta}{2} ) = \pm \sqrt {\frac{1 + cos ( \theta )}{2}}
$$
$$
\begin {align}
tan ( \frac{\theta}{2} ) &= \pm \sqrt {\frac{1 - cos ( \theta )}{1 + cos ( \theta )}} \\
&= \frac {sin ( \theta )} {1 + cos ( \theta )} \\
&= \frac {1 - cos ( \theta )} {sin ( \theta )}
\end {align}
$$
积化和差
$$
sin ( \alpha ) cos ( \beta ) = \frac{1}{2}[sin (\alpha + \beta) + sin (\alpha - \beta)]
$$
$$
cos ( \alpha ) sin ( \beta ) = \frac{1}{2}[sin (\alpha + \beta) - sin (\alpha - \beta)]
$$
$$
cos ( \alpha ) cos ( \beta ) = \frac{1}{2}[cos (\alpha + \beta) + cos (\alpha - \beta)]
$$
$$
sin ( \alpha ) sin ( \beta ) = -\frac{1}{2}[cos (\alpha + \beta) - cos (\alpha - \beta)]
$$
和差化积
$$
sin ( \alpha ) + sin ( \beta ) = 2 sin ( \frac{\alpha + \beta}{2} ) cos ( \frac{\alpha - \beta}{2} )
$$
$$
sin ( \alpha ) - sin ( \beta ) = 2 cos ( \frac{\alpha + \beta}{2} ) sin ( \frac{\alpha - \beta}{2} )
$$
$$
cos ( \alpha ) + cos ( \beta ) = 2 cos ( \frac{\alpha + \beta}{2} ) cos ( \frac{\alpha - \beta}{2} )
$$
$$
cos ( \alpha ) - cos ( \beta ) = -2 sin ( \frac{\alpha + \beta}{2} ) sin ( \frac{\alpha - \beta}{2} )
$$
Test
$$
\frac{\partial u}{\partial t} = h^2 \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right)
$$
$$
\begin{align}
\sqrt{37} & = \sqrt{\frac{73^2-1}{12^2}} \\
& = \sqrt{\frac{73^2}{12^2}\cdot\frac{73^2-1}{73^2}} \\
& = \sqrt{\frac{73^2}{12^2}}\sqrt{\frac{73^2-1}{73^2}} \\
& = \frac{73}{12}\sqrt{1 - \frac{1}{73^2}} \\
& \approx \frac{73}{12}\left(1 - \frac{1}{2\cdot73^2}\right)
\end{align}
$$