正切半角公式維基百科,自由的 encyclopedia 正切半角公式又稱萬能公式,這一組公式有四個功能: 將角統一為 α 2 {\displaystyle {\frac {\alpha }{2}}} [1]; 將函數名稱統一為 tan {\displaystyle \tan } ; 任意實數都可以 tan α 2 {\displaystyle \tan {\frac {\alpha }{2}}} 的形式表達,可用正切函數換元。 在某些積分中,可以將含有三角函數的積分變為有理分式的積分。 因此,這組公式被稱為以切表弦公式,簡稱以切表弦。它們是由二倍角公式求得的。 sin α = 2 tan α 2 1 + tan 2 α 2 {\displaystyle \sin \alpha ={\frac {2\tan {\frac {\alpha }{2}}}{1+\tan ^{2}{\frac {\alpha }{2}}}}} cos α = 1 − tan 2 α 2 1 + tan 2 α 2 {\displaystyle \cos \alpha ={\frac {1-\tan ^{2}{\frac {\alpha }{2}}}{1+\tan ^{2}{\frac {\alpha }{2}}}}} tan α = 2 tan α 2 1 − tan 2 α 2 {\displaystyle \tan \alpha ={\frac {2\tan {\frac {\alpha }{2}}}{1-\tan ^{2}{\frac {\alpha }{2}}}}} cot α = 1 − tan 2 α 2 2 tan α 2 {\displaystyle \cot \alpha ={\frac {1-\tan ^{2}{\frac {\alpha }{2}}}{2\tan {\frac {\alpha }{2}}}}} sec α = 1 + tan 2 α 2 1 − tan 2 α 2 {\displaystyle \sec \alpha ={\frac {1+\tan ^{2}{\frac {\alpha }{2}}}{1-\tan ^{2}{\frac {\alpha }{2}}}}} csc α = 1 + tan 2 α 2 2 tan α 2 {\displaystyle \csc \alpha ={\frac {1+\tan ^{2}{\frac {\alpha }{2}}}{2\tan {\frac {\alpha }{2}}}}} 而被稱為萬能公式的原因是利用 tan α 2 {\displaystyle \tan {\frac {\alpha }{2}}} 的代換可以解決一些有關三角函數的積分。參見三角換元法。 tan ( η 2 ± θ 2 ) = sin η ± sin θ cos η + cos θ = − cos η − cos θ sin η ∓ sin θ , tan ( ± θ 2 ) = ± sin θ 1 + cos θ = ± tan θ sec θ + 1 = ± 1 csc θ + cot θ , ( η = 0 ) tan ( ± θ 2 ) = 1 − cos θ ± sin θ = sec θ − 1 ± tan θ = ± ( csc θ − cot θ ) , ( η = 0 ) tan ( π 4 ± θ 2 ) = 1 ± sin θ cos θ = sec θ ± tan θ = csc θ ± 1 cot θ , ( η = π 2 ) tan ( π 4 ± θ 2 ) = cos θ 1 ∓ sin θ = 1 sec θ ∓ tan θ = cot θ csc θ ∓ 1 , ( η = π 2 ) 1 − tan θ 2 1 + tan θ 2 = 1 − sin θ 1 + sin θ . {\displaystyle {\begin{aligned}\tan \left({\frac {\eta }{2}}\pm {\frac {\theta }{2}}\right)&={\frac {\sin \eta \pm \sin \theta }{\cos \eta +\cos \theta }}=-{\frac {\cos \eta -\cos \theta }{\sin \eta \mp \sin \theta }},\\[10pt]\tan \left(\pm {\frac {\theta }{2}}\right)&={\frac {\pm \sin \theta }{1+\cos \theta }}={\frac {\pm \tan \theta }{\sec \theta +1}}={\frac {\pm 1}{\csc \theta +\cot \theta }},~~~~(\eta =0)\\[10pt]\tan \left(\pm {\frac {\theta }{2}}\right)&={\frac {1-\cos \theta }{\pm \sin \theta }}={\frac {\sec \theta -1}{\pm \tan \theta }}=\pm (\csc \theta -\cot \theta ),~~~~(\eta =0)\\[10pt]\tan \left({\frac {\pi }{4}}\pm {\frac {\theta }{2}}\right)&={\frac {1\pm \sin \theta }{\cos \theta }}=\sec \theta \pm \tan \theta ={\frac {\csc \theta \pm 1}{\cot \theta }},~~~~(\eta ={\frac {\pi }{2}})\\[10pt]\tan \left({\frac {\pi }{4}}\pm {\frac {\theta }{2}}\right)&={\frac {\cos \theta }{1\mp \sin \theta }}={\frac {1}{\sec \theta \mp \tan \theta }}={\frac {\cot \theta }{\csc \theta \mp 1}},~~~~(\eta ={\frac {\pi }{2}})\\[10pt]{\frac {1-\tan {\frac {\theta }{2}}}{1+\tan {\frac {\theta }{2}}}}&={\sqrt {\frac {1-\sin \theta }{1+\sin \theta }}}.\end{aligned}}}
正切半角公式又稱萬能公式,這一組公式有四個功能: 將角統一為 α 2 {\displaystyle {\frac {\alpha }{2}}} [1]; 將函數名稱統一為 tan {\displaystyle \tan } ; 任意實數都可以 tan α 2 {\displaystyle \tan {\frac {\alpha }{2}}} 的形式表達,可用正切函數換元。 在某些積分中,可以將含有三角函數的積分變為有理分式的積分。 因此,這組公式被稱為以切表弦公式,簡稱以切表弦。它們是由二倍角公式求得的。 sin α = 2 tan α 2 1 + tan 2 α 2 {\displaystyle \sin \alpha ={\frac {2\tan {\frac {\alpha }{2}}}{1+\tan ^{2}{\frac {\alpha }{2}}}}} cos α = 1 − tan 2 α 2 1 + tan 2 α 2 {\displaystyle \cos \alpha ={\frac {1-\tan ^{2}{\frac {\alpha }{2}}}{1+\tan ^{2}{\frac {\alpha }{2}}}}} tan α = 2 tan α 2 1 − tan 2 α 2 {\displaystyle \tan \alpha ={\frac {2\tan {\frac {\alpha }{2}}}{1-\tan ^{2}{\frac {\alpha }{2}}}}} cot α = 1 − tan 2 α 2 2 tan α 2 {\displaystyle \cot \alpha ={\frac {1-\tan ^{2}{\frac {\alpha }{2}}}{2\tan {\frac {\alpha }{2}}}}} sec α = 1 + tan 2 α 2 1 − tan 2 α 2 {\displaystyle \sec \alpha ={\frac {1+\tan ^{2}{\frac {\alpha }{2}}}{1-\tan ^{2}{\frac {\alpha }{2}}}}} csc α = 1 + tan 2 α 2 2 tan α 2 {\displaystyle \csc \alpha ={\frac {1+\tan ^{2}{\frac {\alpha }{2}}}{2\tan {\frac {\alpha }{2}}}}} 而被稱為萬能公式的原因是利用 tan α 2 {\displaystyle \tan {\frac {\alpha }{2}}} 的代換可以解決一些有關三角函數的積分。參見三角換元法。 tan ( η 2 ± θ 2 ) = sin η ± sin θ cos η + cos θ = − cos η − cos θ sin η ∓ sin θ , tan ( ± θ 2 ) = ± sin θ 1 + cos θ = ± tan θ sec θ + 1 = ± 1 csc θ + cot θ , ( η = 0 ) tan ( ± θ 2 ) = 1 − cos θ ± sin θ = sec θ − 1 ± tan θ = ± ( csc θ − cot θ ) , ( η = 0 ) tan ( π 4 ± θ 2 ) = 1 ± sin θ cos θ = sec θ ± tan θ = csc θ ± 1 cot θ , ( η = π 2 ) tan ( π 4 ± θ 2 ) = cos θ 1 ∓ sin θ = 1 sec θ ∓ tan θ = cot θ csc θ ∓ 1 , ( η = π 2 ) 1 − tan θ 2 1 + tan θ 2 = 1 − sin θ 1 + sin θ . {\displaystyle {\begin{aligned}\tan \left({\frac {\eta }{2}}\pm {\frac {\theta }{2}}\right)&={\frac {\sin \eta \pm \sin \theta }{\cos \eta +\cos \theta }}=-{\frac {\cos \eta -\cos \theta }{\sin \eta \mp \sin \theta }},\\[10pt]\tan \left(\pm {\frac {\theta }{2}}\right)&={\frac {\pm \sin \theta }{1+\cos \theta }}={\frac {\pm \tan \theta }{\sec \theta +1}}={\frac {\pm 1}{\csc \theta +\cot \theta }},~~~~(\eta =0)\\[10pt]\tan \left(\pm {\frac {\theta }{2}}\right)&={\frac {1-\cos \theta }{\pm \sin \theta }}={\frac {\sec \theta -1}{\pm \tan \theta }}=\pm (\csc \theta -\cot \theta ),~~~~(\eta =0)\\[10pt]\tan \left({\frac {\pi }{4}}\pm {\frac {\theta }{2}}\right)&={\frac {1\pm \sin \theta }{\cos \theta }}=\sec \theta \pm \tan \theta ={\frac {\csc \theta \pm 1}{\cot \theta }},~~~~(\eta ={\frac {\pi }{2}})\\[10pt]\tan \left({\frac {\pi }{4}}\pm {\frac {\theta }{2}}\right)&={\frac {\cos \theta }{1\mp \sin \theta }}={\frac {1}{\sec \theta \mp \tan \theta }}={\frac {\cot \theta }{\csc \theta \mp 1}},~~~~(\eta ={\frac {\pi }{2}})\\[10pt]{\frac {1-\tan {\frac {\theta }{2}}}{1+\tan {\frac {\theta }{2}}}}&={\sqrt {\frac {1-\sin \theta }{1+\sin \theta }}}.\end{aligned}}}