Bài 6.50 trang 192 SBT đại số 10

LG a

\({\sin ^2}({180^0} - \alpha )\) \( + \tan ^2({180^0} - \alpha ){\tan ^2}({270^0} - \alpha ) \) \(  + \sin ({90^0} + \alpha )\cos(\alpha  - {360^0})\)

Lời giải chi tiết:

\({\sin ^2}({180^0} - \alpha ) \) \(  + \tan ^2({180^0} - \alpha ){\tan ^2}({270^0} - \alpha ) \) \(  + \sin ({90^0} + \alpha )\cos(\alpha  - {360^0})\)

= \({\sin ^2}\alpha  + {\tan ^2}\alpha {\cot ^2}\alpha  + {\cos ^2}\alpha  = 2\)

LG b

\({{\cos (\alpha  - {{90}^0})} \over {\sin ({{180}^0} - \alpha )}} + {{\tan (\alpha  - {{180}^0})c{\rm{os(18}}{{\rm{0}}^0} + \alpha )\sin ({{270}^0} + \alpha )} \over {\tan ({{270}^0} + \alpha )}}\)

Lời giải chi tiết:

\({{\cos (\alpha  - {{90}^0})} \over {\sin ({{180}^0} - \alpha )}} + {{\tan (\alpha  - {{180}^0})c{\rm{os(18}}{{\rm{0}}^0} + \alpha )\sin ({{270}^0} + \alpha )} \over {\tan ({{270}^0} + \alpha )}}\)

\(\begin{array}{l}
= \frac{{\cos \left( {{{90}^0} - \alpha } \right)}}{{\sin \alpha }} + \frac{{\tan \alpha .\left( { - \cos \alpha } \right)\sin \left( { - {{90}^0} + \alpha } \right)}}{{\tan \left( {{{90}^0} + \alpha } \right)}}\\
= \frac{{\sin \alpha }}{{\sin \alpha }} + \frac{{\tan \alpha .\left( { - \cos \alpha } \right)\left[ { - \sin \left( {{{90}^0} - \alpha } \right)} \right]}}{{\tan \left[ {{{90}^0} - \left( { - \alpha } \right)} \right]}}\\
= 1 + \frac{{\tan \alpha .\left( { - \cos \alpha } \right)\left( { - \cos \alpha } \right)}}{{\cot \left( { - \alpha } \right)}}\\
= 1 + \frac{{\frac{{\sin \alpha }}{{\cos \alpha }}.{{\cos }^2}\alpha }}{{ - \cot \alpha }}\\
= 1 - \frac{{\sin \alpha \cos \alpha }}{{\frac{{\cos \alpha }}{{\sin \alpha }}}}\\
= 1 - {\sin ^2}\alpha \\
= {\cos ^2}\alpha
\end{array}\)

LG c

\({{\cos ( - {{288}^0})cot{{72}^0}} \over {tan( - {{162}^0})\sin {{108}^0}}} + \tan {18^0}\)

Lời giải chi tiết:

\({{\cos ( - {{288}^0})cot{{72}^0}} \over {tan( - {{162}^0})\sin {{108}^0}}} + \tan {18^0}\)

\( = {{\cos ({{72}^0} - {{360}^0})\cot {{72}^0}} \over {\tan ({{18}^0} - {{180}^0})\sin ({{180}^0} - {{72}^0})}} - \tan {18^0}\)

= \({{{\rm{cos7}}{{\rm{2}}^0}\cot {{72}^0}} \over {\tan {{18}^0}\sin {{72}^0}}} - \tan {18^0}\)

= \({{{{\cot }^2}{{72}^0}} \over {\tan {{18}^0}}} - \tan {18^0} \) \(= {{{{\tan }^2}{{18}^0}} \over {\tan {{18}^0}}} - \tan {18^0} = 0\)

LG d

\({{\sin {{20}^0}\sin {\rm{3}}{{\rm{0}}^0}\sin {{40}^0}\sin {{50}^0}\sin {{60}^0}\sin {{70}^0}} \over {cos{{10}^0}{\rm{cos5}}{{\rm{0}}^0}}}\)

Lời giải chi tiết:

Ta có: \(\sin {70^0} = \cos {20^0},\sin {50^0} = \cos 4{{\rm{0}}^0};\) \(\sin {40^0} = cos{50^0}\).

Vì vậy

\({{\sin {{20}^0}\sin {\rm{3}}{{\rm{0}}^0}\sin {{40}^0}\sin {{50}^0}\sin {{60}^0}\sin {{70}^0}} \over {cos{{10}^0}{\rm{cos5}}{{\rm{0}}^0}}}\)

= \(\eqalign{
& {{{1 \over 2}.{{\sqrt 3 } \over 2}.\sin {{20}^0}\cos {\rm{2}}{{\rm{0}}^0}\cos {{50}^0}\cos {{40}^0}} \over {cos{{10}^0}{\rm{cos5}}{{\rm{0}}^0}}} \cr 
& = {{{1 \over 2}.{{\sqrt 3 } \over 4}\sin {{40}^0}.cos{{40}^0}} \over {{\rm{cos1}}{{\rm{0}}^0}}} \cr} \)

= \({{{{\sqrt 3 } \over {16}}\sin {{80}^0}} \over {cos{{10}^0}}} = {{\sqrt 3 } \over {16}}\)

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