Pembahasan Buku Sukino BAB 2 LKS 5 Matematika Peminatan Kelas XI Kurikulum 2013

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Sobat hudamath, di postingan kali ini akan kita bahas BAB LKS 5 Buku Sukino Matematika Peminatan Kelas XI Kurikulum 2013 khusus no 8. Untuk pembahasan no yang lainnya bisa sobat download di bagian bawah postingan ini. 

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LKS 5

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8.   Jika A=580° , maka berlaku...
      A.  \(\sin \frac{1}{2}A = \frac{1}{2}\left[ {\sqrt {1 + \sin A}  - \sqrt {1 - \sin A} } \right]\)
      B.  \(\sin \frac{1}{2}A =  - \left[ {\sqrt {1 + \sin A}  - \sqrt {1 - \sin A} } \right]\)
      C.  \(\sin \frac{1}{2}A =  - \frac{1}{2}\left[ {\sqrt {1 + \sin A}  + \sqrt {1 - \sin A} } \right]\)
      D.  \(\sin \frac{1}{2}A =  - \frac{1}{2}\left[ {\sqrt {1 + \sin A}  - \sqrt {1 - \sin A} } \right]\)
      E.  \(\cos \frac{1}{2}A = \left[ {\sqrt {1 + \sin A}  - \sqrt {1 - \sin A} } \right]\)

    Jawab           : C
    Pembahasan :
\(A = 580^\circ  = \left( {360^\circ  + 220^\circ } \right) = 220^\circ \left( {{\rm{Kuadran}}3} \right)\)
\(\cos 580^\circ  = \cos \left( {360^\circ  + 220^\circ } \right) = \cos 220^\circ  = \cos \left( {180^\circ  + 40^\circ } \right) =  - \cos 40^\circ \)
\(\sin 580^\circ  = \sin \left( {360^\circ  + 220^\circ } \right) = \sin 220^\circ  = \sin \left( {180^\circ  + 40^\circ } \right) =  - \sin 40^\circ \)

\(\sin \frac{1}{2}A =  - \sqrt {\frac{{1 - \cos A}}{2}} \)
\(\sin 290^\circ  =  - \sqrt {\frac{{1 - \cos 580^\circ }}{2}} \)
\( =  - \sqrt {\frac{{1 + \cos 40^\circ }}{2}} \)
\( =  - \sqrt {\frac{{\left( {1 + \cos 40^\circ } \right){{\cos }^2}40^\circ }}{{2{{\cos }^2}40^\circ }}} \)
\( =  - \sqrt {\frac{{{{\cos }^2}40^\circ  + {{\cos }^3}40^\circ }}{{2{{\cos }^2}40^\circ }}} \)
\( =  - \sqrt {\frac{{1 - {{\sin }^2}40^\circ  + {{\cos }^2}40^\circ \cos 40^\circ }}{{2{{\cos }^2}40^\circ }}} \)
\(= - \sqrt {\frac{{\left( {1 - {{\sin }^2}40^\circ } \right) + \left( {1 - {{\sin }^2}40^\circ } \right)\sqrt {\left( {1 - {{\sin }^2}40^\circ } \right)} }}{{2{{\cos }^2}40^\circ }}} \)
\( =  - \sqrt {\frac{{\left( {1 - {{\sin }^2}40^\circ } \right) + \left( {1 - {{\sin }^2}40^\circ } \right)\sqrt {\left( {1 - {{\sin }^2}40^\circ } \right)} }}{{2{{\cos }^2}40^\circ }}.\frac{2}{2}} \)
\( =  - \sqrt {\frac{{2\left( {1 - {{\sin }^2}40^\circ } \right) + 2\left( {1 - {{\sin }^2}40^\circ } \right)\sqrt {\left( {1 - {{\sin }^2}40^\circ } \right)} }}{{4{{\cos }^2}40^\circ }}} \)
\( =  - \sqrt {\frac{{2\left( {1 - \sin 40^\circ } \right)\left( {1 + \sin 40^\circ } \right) + 2\left( {1 - {{\sin }^2}40^\circ } \right)\sqrt {\left( {1 - {{\sin }^2}40^\circ } \right)} }}{{4{{\cos }^2}40^\circ }}} \)
\( =  - \sqrt {\frac{{\left( {1 - \sin 40^\circ } \right)\left( {1 + \sin 40^\circ } \right)\left( {\left( {1 + \sin 40^\circ } \right) + \left( {1 - \sin 40^\circ } \right)} \right) + 2\left( {1 - {{\sin }^2}40^\circ } \right)\sqrt {\left( {1 - {{\sin }^2}40^\circ } \right)} }}{{4{{\cos }^2}40^\circ }}} \)
\( =  - \sqrt {\frac{{\left( {1 - \sin 40^\circ } \right)\left( {1 + \sin 40^\circ } \right)\left( {1 + \sin 40^\circ } \right) + \left( {1 - \sin 40^\circ } \right)\left( {1 + \sin 40^\circ } \right)\left( {1 - \sin 40^\circ } \right) + 2\left( {1 - {{\sin }^2}40^\circ } \right)\sqrt {\left( {1 - {{\sin }^2}40^\circ } \right)} }}{{4{{\cos }^2}40^\circ }}} \)
\( =  - \sqrt {\frac{{{{\left( {1 + \sin 40^\circ } \right)}^2}\left( {1 - \sin 40^\circ } \right) + {{\left( {1 - \sin 40^\circ } \right)}^2}\left( {1 + \sin 40^\circ } \right) + 2\left( {1 - {{\sin }^2}40^\circ } \right)\sqrt {1 - {{\sin }^2}40^\circ } }}{{4{{\cos }^2}40^\circ }}} \)
\( =  - \sqrt {\left[ {\frac{{{{\left( {\left( {1 + \sin 40^\circ } \right)\sqrt {1 - \sin 40^\circ }  + \left( {1 - \sin 40^\circ } \right)\sqrt {1 + \sin 40^\circ } } \right)}^2}}}{{{2^2}{{\cos }^2}40^\circ }}} \right]} \)
\( =  - \sqrt {{{\left\{ {\left[ {\frac{{\left( {1 + \sin 40^\circ } \right)\sqrt {1 - \sin 40^\circ }  + \left( {1 - \sin 40^\circ } \right)\sqrt {1 + \sin 40^\circ } }}{{2\cos 40^\circ }}} \right]} \right\}}^2}} \)
\( =  - \left[ {\frac{{\left( {1 + \sin 40^\circ } \right)\sqrt {1 - \sin 40^\circ }  + \left( {1 - \sin 40^\circ } \right)\sqrt {1 + \sin 40^\circ } }}{{2\cos 40^\circ }}} \right]\)
\( =  - \left[ {\frac{{\left( {1 + \sin 40^\circ } \right)\sqrt {1 - \sin 40^\circ } }}{{2\cos 40^\circ }} + \frac{{\left( {1 - \sin 40^\circ } \right)\sqrt {1 + \sin 40^\circ } }}{{2\cos 40^\circ }}} \right]\)
\( =  - \left[ {\frac{{\left( {1 + \sin 40^\circ } \right)\sqrt {1 - \sin 40^\circ } }}{{2\sqrt {{{\cos }^2}40^\circ } }} + \frac{{\left( {1 - \sin 40^\circ } \right)\sqrt {1 + \sin 40^\circ } }}{{2\sqrt {{{\cos }^2}40^\circ } }}} \right]\)
\( =  - \left[ {\frac{{\left( {1 + \sin 40^\circ } \right)\sqrt {1 - \sin 40^\circ } }}{{2\sqrt {1 - {{\sin }^2}40^\circ } }} + \frac{{\left( {1 - \sin 40^\circ } \right)\sqrt {1 + \sin 40^\circ } }}{{2\sqrt {1 - {{\sin }^2}40^\circ } }}} \right]\)
\( =  - \left[ {\frac{{\left( {1 + \sin 40^\circ } \right)\sqrt {1 - \sin 40^\circ } }}{{2\sqrt {1 + \sin 40^\circ } \sqrt {1 - \sin 40^\circ } }} + \frac{{\left( {1 - \sin 40^\circ } \right)\sqrt {1 + \sin 40^\circ } }}{{2\sqrt {1 - \sin 40^\circ } \sqrt {1 + \sin 40^\circ } }}} \right]\)
\( =  - \left[ {\frac{{1 + \sin 40^\circ }}{{2\sqrt {1 + \sin 40^\circ } }} + \frac{{1 - \sin 40^\circ }}{{2\sqrt {1 - \sin 40^\circ } }}} \right]\)
\( =  - \left[ {\frac{{\sqrt {{{\left( {1 + \sin 40^\circ } \right)}^2}} }}{{2\sqrt {1 + \sin 40^\circ } }} + \frac{{\sqrt {{{\left( {1 - \sin 40^\circ } \right)}^2}} }}{{2\sqrt {1 - \sin 40^\circ } }}} \right]\)
\( =  - \left[ {\frac{{\sqrt {1 + \sin 40^\circ } }}{2}.\frac{{\sqrt {1 + \sin 40^\circ } }}{{\sqrt {1 + \sin 40^\circ } }} + \frac{{\sqrt {1 - \sin 40^\circ } }}{2}.\frac{{\sqrt {1 - \sin 40^\circ } }}{{\sqrt {1 - \sin 40^\circ } }}} \right]\)
\( =  - \left[ {\frac{{\sqrt {1 + \sin 40^\circ } }}{2} + \frac{{\sqrt {1 - \sin 40^\circ } }}{2}} \right]\)
\( =  - \left[ {\frac{{\sqrt {1 + \sin 40^\circ }  + \sqrt {1 - \sin 40^\circ } }}{2}} \right]\)
\( =  - \frac{1}{2}\left[ {\sqrt {1 + \sin 40^\circ }  + \sqrt {1 - \sin 40^\circ } } \right]\)
\( =  - \frac{1}{2}\left[ {\sqrt {1 - \sin 580^\circ }  + \sqrt {1 + \sin 580^\circ } } \right]\)
\( =  - \frac{1}{2}\left[ {\sqrt {1 - \sin A}  + \sqrt {1 + \sin A} } \right]\)
\( =  - \frac{1}{2}\left[ {\sqrt {1 + \sin A}  + \sqrt {1 - \sin A} } \right]\)

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