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[tex] \cos( \alpha ) = \frac{3}{5} \\ [/tex]
[tex] {sin}^{2}( \alpha ) = 1 - {cos}^{2} ( \alpha )[/tex]
[tex] {sin}^{2}( \alpha ) = 1 - ({ \frac{3}{5} })^{2} \\ [/tex]
[tex] {sin}^{2}( \alpha ) = 1 - \frac{9}{25} \\ [/tex]
[tex] {sin}^{2}( \alpha ) = \frac{25}{25} - \frac{9}{25} \\ [/tex]
[tex] {sin}^{2}( \alpha ) = \frac{16}{25} \\ [/tex]
[tex] sin( \alpha )= ± \sqrt{ \frac{16}{25} } \\ [/tex]
In Quadrant IV , sin is negative .
Thus ;
[tex]sin( \alpha ) = - \sqrt{ \frac{16}{25} } \\ [/tex]
[tex] \sin( \alpha ) = - \frac{4}{5} \\ [/tex]
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[tex] \tan(2 \alpha ) = \frac{ \sin(2 \alpha ) }{ \cos(2 \alpha ) } \\ [/tex]
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[tex] \sin(2 \alpha ) = 2. \sin( \alpha ). \cos( \alpha ) [/tex]
[tex] \sin(2 \alpha ) = 2 \times ( - \frac{4}{5} ) \times ( \frac{3}{5} ) \\ [/tex]
[tex] \sin(2 \alpha ) = - \frac{24}{25} \\ [/tex]
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[tex] \cos(2 \alpha ) = {cos}^{2}( \alpha ) - {sin}^{2}( \alpha ) [/tex]
[tex] \cos(2 \alpha ) = ({ \frac{3}{5} })^{2} - ({ - \frac{4}{5} })^{2} \\ [/tex]
[tex] \cos(2 \alpha ) = \frac{9}{25} - \frac{16}{25} \\ [/tex]
[tex] \cos(2 \alpha ) = - \frac{7}{25} \\ [/tex]
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[tex] \tan(2 \alpha ) = \frac{ \sin(2 \alpha ) }{ \cos(2 \alpha ) } \\ [/tex]
[tex] \tan(2 \alpha ) = \frac{ - \frac{24}{25} }{ - \frac{7}{25} } \\ [/tex]
[tex] \tan(2 \alpha ) = - \frac{24}{25} \div - \frac{7}{25} \\ [/tex]
[tex] \tan(2 \alpha ) = - \frac{24}{25} \times - \frac{25}{7} \\ [/tex]
[tex] \tan(2 \alpha ) = \frac{24}{7} \\ [/tex]
Thus the correct answer is (( C )) .
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