Explicație pas cu pas:
[tex] \tan(\alpha ) = \frac{ \sin( \alpha ) }{ \cos( \alpha ) }[/tex]
[tex] \tan( \alpha ) = 2 = > \frac{ \sin( \alpha ) }{ \cos( \alpha ) } = 2 \\ = > \sin( \alpha ) = 2 \cos( \alpha ) \\ < = > { \sin^{2}( \alpha ) } = 4 \cos^{2}( \alpha )[/tex]
[tex]\sin^{2}( \alpha ) + \cos^{2}( \alpha ) = 1[/tex]
[tex]= > \sin^{2}( \alpha ) = 1 - \cos^{2}( \alpha )[/tex]
[tex]1 - \cos^{2}( \alpha ) = 4\cos^{2}( \alpha ) \\ 5\cos^{2}( \alpha ) = 1 = > \cos^{2}( \alpha ) = \frac{1}{5} [/tex]
[tex]\tan( 2\alpha ) = \frac{2 \tan( \alpha ) }{1 - \tan^{2} ( \alpha ) }[/tex]
[tex]\tan( 2\alpha ) = \frac{2 \times 2}{1 - {2}^{2} } = \frac{4}{ - 3}[/tex]
[tex]= > \tan( 2\alpha ) = - \frac{4}{3}[/tex]
[tex] \sin( 2\alpha ) = 2 \sin( \alpha ) \cos( \alpha )[/tex]
[tex]\sin( 2\alpha ) = 2(2 \cos( \alpha ) ) \cos( \alpha ) = 4 \cos^{2} ( \alpha ) = 4 \times \frac{1}{5} = \frac{4}{5}[/tex]
[tex]= > \sin( 2\alpha ) = \frac{4}{5} [/tex]
[tex]\cos(2 \alpha ) = 2 \cos^{2} ( \alpha ) - 1[/tex]
[tex]\cos(2 \alpha ) = 2 \times \frac{1}{5} - 1 = \frac{2}{5} - 1 = - \frac{3}{5} [/tex]
[tex] = > \cos(2 \alpha ) = - \frac{3}{5} [/tex]
