Đề kiểm tra Bài tập cuối chương III (có lời giải) - Đề 3

Chọn A

Ta có \(\sin {0^{\rm{o}}} + \cos {0^{\rm{o}}} = 1\).

Chọn A

\(E = \sin {36^{\rm{o}}}\cos {6^{\rm{o}}}\sin \left( {{{90}^{\rm{o}}} + {{36}^{\rm{o}}}} \right)\cos \left( {{{90}^{\rm{o}}} - {6^{\rm{o}}}} \right) = \sin {36^{\rm{o}}}\cos {6^{\rm{o}}} - \cos {36^{\rm{o}}}\sin {6^{\rm{o}}} = \sin {30^{\rm{o}}} = \frac{1}{2}\)

Chọn B

Ta có: \(\cos A = \frac{{{b^2} + {c^2} - {a^2}}}{{2bc}} = \frac{{{{13}^2} + {{15}^2} - {{24}^2}}}{{2.13.15}} =  - \frac{7}{{15}} \Rightarrow A \simeq {117^0}49'\,.\)

Chọn C

Ta có \(A{M^2} = \frac{{A{B^2} + A{C^2}}}{2} - \frac{{B{C^2}}}{4}\)\( = \frac{{{9^2} + {{12}^2}}}{2} - \frac{{{{15}^2}}}{4} = \frac{{225}}{4}\)\( \Rightarrow AM = \frac{{15}}{2}\).

Chọn C

\(S = {\sin ^2}{2^{\rm{o}}} + {\sin ^2}{4^{\rm{o}}} + {\sin ^2}{6^{\rm{o}}} + ... + {\sin ^2}{84^{\rm{o}}} + {\sin ^2}{86^{\rm{o}}} + {\sin ^2}{88^{\rm{o}}}\)

\[ = \left( {{{\sin }^2}{2^{\rm{o}}} + {{\sin }^2}{{88}^{\rm{o}}}} \right) + \left( {{{\sin }^2}{4^{\rm{o}}} + {{\sin }^2}{{86}^{\rm{o}}}} \right) + ... + \left( {{{\sin }^2}{{44}^{\rm{o}}} + {{\sin }^2}{{46}^{\rm{o}}}} \right)\]

\( = \left( {{{\sin }^2}{2^{\rm{o}}} + {{\cos }^2}{2^{\rm{o}}}} \right) + \left( {{{\sin }^2}{4^{\rm{o}}} + {{\cos }^2}{4^{\rm{o}}}} \right) + ... + \left( {{{\sin }^2}{{44}^{\rm{o}}} + {{\cos }^2}{{44}^{\rm{o}}}} \right) = 22\).

Chọn B

\(\tan \alpha  - \cot \alpha  = 3 \Leftrightarrow {\left( {\tan \alpha  - \cot \alpha } \right)^2} = 9 \Leftrightarrow {\tan ^2}\alpha  + {\cot ^2}\alpha  - 2\tan \alpha .\cot \alpha  = 9\)

\( \Leftrightarrow {\tan ^2}\alpha  + {\cot ^2}\alpha  - 2 = 9 \Leftrightarrow {\tan ^2}\alpha  + {\cot ^2}\alpha  = 11\).

Chọn B

\({\tan ^2}x{\sin ^2}x - {\tan ^2}x + {\sin ^2}x = {\tan ^2}x\left( {{{\sin }^2}x - 1} \right) + {\sin ^2}x = \frac{{{{\sin }^2}x}}{{{{\cos }^2}x}}\left( { - {{\cos }^2}x} \right) + {\sin ^2}x = 0\).

Chọn A

Ta có: \(S = m_a^2 + m_b^2 + m_c^2 = \frac{{{b^2} + {c^2}}}{2} - \frac{{{a^2}}}{4} + \frac{{{a^2} + {c^2}}}{2} - \frac{{{b^2}}}{4} + \frac{{{a^2} + {b^2}}}{2} - \frac{{{c^2}}}{4} = \frac{3}{4}({a^2} + {b^2} + {c^2}).\)