Bài 103039

Question

Tính giá trị các biểu thức :
$A = \sin ^21^o + \sin ^22^o+ ... + \sin ^288^o + \sin ^289^o+ \sin ^290^o$;
$B = \cos ^31^o + \cos ^32^o + ... + \cos ^3179^o+ \cos ^3180^o$;
$C = \tan1^o.\tan2^o...\tan88^o.\tan89^o$

hoàng anh thọ · Answer 1 · 5/5/2012 9:42:28 AM

$\begin{array}{l}
A = \left( {{{\sin }^2}{1^\circ} + {{\sin }^2}{{89}^\circ}} \right) + \left( {{{\sin }^2}{2^\circ} + {{\sin }^2}{{88}^\circ}} \right) + ... + \left( {{{\sin }^2}{{44}^\circ} + {{\sin }^2}{{46}^\circ}} \right)\\ + {\sin ^2}{45^\circ} + {\sin ^2}{90^\circ}\\
    = \left( {{{\sin }^2}{1^\circ} + c{\rm{o}}{{\rm{s}}^2}{1^\circ}} \right) + \left( {{{\sin }^2}{2^\circ} + c{\rm{o}}{{\rm{s}}^2}{2^\circ}} \right) + ... + \left( {{{\sin }^2}{{44}^\circ} + c{\rm{o}}{{\rm{s}}^2}{{44}^\circ}} \right) + \frac{1}{2} + 1\\
    = 44 + \frac{1}{2} + 1 = \frac{{91}}{2}
\end{array}$
$\begin{array}{l}
B = \left( {c{\rm{o}}{{\rm{s}}^3}{1^\circ} + c{\rm{o}}{{\rm{s}}^3}{{179}^\circ}} \right) + \left( {{{\cos }^3}{2^\circ} + {{\cos }^3}{{178}^\circ}} \right) + ... + \left( {{{\cos }^3}{{89}^\circ} + {{\cos }^3}{{91}^\circ}} \right)\\ + {\cos ^3}{90^\circ} + {\cos ^3}{180^\circ}\\
    = \left( {{{\cos }^3}{1^\circ} - {{\cos }^3}{1^\circ}} \right) + \left( {{{\cos }^3}{2^\circ} - {{\cos }^3}{2^\circ}} \right) + ... + \left( {{{\cos }^3}{{89}^\circ} - {{\cos }^3}{{89}^\circ}} \right) + 0 - 1 = - 1
\end{array}$
$\begin{array}{l}
C = \left( {\tan{1^\circ}.\tan{{89}^\circ}} \right)\left( {\tan{2^\circ}.\tan{{88}^\circ}} \right)...\left( {\tan{{44}^\circ}.\tan{{46}^\circ}} \right).\tan{45^\circ}\\
    = \left( {\tan{1^\circ}.\cot {1^\circ}} \right)\left( {\tan{2^\circ}.\cot {2^\circ}} \right)...\left( {\tan{{44}^\circ}.\cot {{44}^\circ}} \right).\tan{45^\circ}\\
    = 1
\end{array}$

Bài 103039

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Bài 103039

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