三角函数诱导公式

三角函数诱导公式

\[\sin ( \alpha \pm \beta ) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \]

\[\cos( \alpha \pm \beta ) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta \]

\[\tan ( \alpha \pm \beta ) = \frac{ \tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta } \]

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

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

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

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

\[a \sin \alpha + b \cos \alpha = \sqrt[ 2 ]{ a^2 + b^2 } \sin (\alpha + \gamma ) \]

\[\sin \gamma = \frac{ a }{ \sqrt[ 2 ]{ a^2 + b^2 } },\cos \gamma = \frac{ b }{ \sqrt[ 2 ]{ a^2 + b^2 } } \]

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

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

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

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

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

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

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

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

Tags : 三角函数 数学