Giải bài 3.1 trang 32 sách bài tập toán 10 - Kết nối tri thức với cuộc sống

Đề bài

Tính giá trị của các biểu thức:

a) $A = \sin {45^ \circ } + 2\sin {60^ \circ } + \tan {120^ \circ } + \cos {135^ \circ }.$

b) $B = \tan {45^ \circ }.\cot {135^ \circ } - \sin {30^ \circ }.\cos {120^ \circ } - \sin {60^ \circ }.\cos {150^ \circ }.$

c) $C = {\cos ^2}{5^ \circ } + {\cos ^2}{25^ \circ } + {\cos ^2}{45^ \circ } + {\cos ^2}{65^ \circ } + {\cos ^2}{85^ \circ }.$

d) $D = \frac{{12}}{{1 + {{\tan }^2}{{73}^ \circ }}} - 4\tan {75^ \circ }.\cot {105^ \circ } + 12{\sin ^2}{107^ \circ } - 2\tan {40^ \circ }.\cos {60^ \circ }.\tan {50^ \circ }.$

e) $E = 4\tan {32^ \circ }.\cos {60^ \circ }.\cot {148^ \circ } + \frac{{5{{\cot }^2}108}}{{1 + {{\tan }^2}{{18}^ \circ }}} + 5{\sin ^2}{72^ \circ }.$

Lời giải chi tiết

a) $A = \sin {45^ \circ } + 2\sin {60^ \circ } + \tan {120^ \circ } + \cos {135^ \circ }$

$\begin{array}{l}A = \frac{{\sqrt 2 }}{2} + 2.\frac{{\sqrt 3 }}{2} + \left( { - \sqrt 3 } \right) + \left( {\frac{{ - \sqrt 2 }}{2}} \right)\\A = \frac{{\sqrt 2 }}{2} + \sqrt 3  + \left( { - \sqrt 3 } \right) + \left( {\frac{{ - \sqrt 2 }}{2}} \right)\\A = 0\end{array}$

b) $B = \tan {45^ \circ }.\cot {135^ \circ } - \sin {30^ \circ }.\cos {120^ \circ } - \sin {60^ \circ }.\cos {150^ \circ }$

$\begin{array}{l}B = 1.\left( { - 1} \right) - \frac{1}{2}.\left( {\frac{{ - 1}}{2}} \right) - \frac{{\sqrt 3 }}{2}.\left( {\frac{{ - \sqrt 3 }}{2}} \right)\\B =  - 1 + \frac{1}{4} + \frac{3}{4} = 0\end{array}$

c) $C = {\cos ^2}{5^ \circ } + {\cos ^2}{25^ \circ } + {\cos ^2}{45^ \circ } + {\cos ^2}{65^ \circ } + {\cos ^2}{85^ \circ }$

$\begin{array}{l}C = {\cos ^2}\left( {{{90}^ \circ } - {{85}^ \circ }} \right) + {\cos ^2}\left( {{{90}^ \circ } - {{65}^ \circ }} \right) + {\cos ^2}{45^ \circ } + {\cos ^2}{65^ \circ } + {\cos ^2}{85^ \circ }\\C = {\sin ^2}{85^ \circ } + {\cos ^2}{85^ \circ } + {\sin ^2}{65^ \circ } + {\cos ^2}{85^ \circ } + {\sin ^2}{45^ \circ }\\C = 1 + 1 + {\left( {\frac{{\sqrt 2 }}{2}} \right)^2} = 2 + \frac{1}{2} = \frac{5}{2}\end{array}$

d) $D = \frac{{12}}{{1 + {{\tan }^2}{{73}^ \circ }}} - 4\tan {75^ \circ }.\cot {105^ \circ } + 12{\sin ^2}{107^ \circ } - 2\tan {40^ \circ }.\cos {60^ \circ }.\tan {50^ \circ }$

$\begin{array}{l}D = 12{\cos ^2}{73^ \circ } + 12{\sin ^2}\left( {{{180}^ \circ } - {{73}^ \circ }} \right) - 4\tan {75^ \circ }.\cot \left( {{{180}^ \circ } - {{75}^ \circ }} \right) - 2\tan {40^ \circ }.\tan \left( {{{90}^ \circ } - {{40}^ \circ }} \right)\cos {60^ \circ }\\D = 12{\cos ^2}{73^ \circ } + 12{\sin ^2}{73^ \circ } - 4\tan {75^ \circ }\left( { - \cot {{75}^ \circ }} \right) - 2\tan {40^ \circ }.\cot {40^ \circ }.\frac{1}{2}\\D = 12 + 4 - 1 = 15\end{array}$

e) $E = 4\tan {32^ \circ }.\cos {60^ \circ }.\cot {148^ \circ } + \frac{{5{{\cot }^2}108}}{{1 + {{\tan }^2}{{18}^ \circ }}} + 5{\sin ^2}{72^ \circ }.$

$\begin{array}{l}E = 4\tan {32^ \circ }.\cot \left( {{{180}^ \circ } - {{32}^ \circ }} \right).\cos {60^ \circ } + \frac{{5{{\cot }^2}\left( {{{180}^ \circ } - {{72}^ \circ }} \right)}}{{1 + {{\tan }^2}{{18}^ \circ }}} + 5{\sin ^2}\left( {{{90}^ \circ } - {{18}^ \circ }} \right)\\E =  - 4\tan {32^ \circ }.\cot {32^ \circ }.\cos {60^ \circ } + 5{\cot ^2}{72^ \circ }.{\cos ^2}{18^ \circ } + 5{\cos ^2}{18^ \circ }\\E =  - 4\cos {60^ \circ } + 5{\cot ^2}\left( {{{90}^ \circ } - {{18}^ \circ }} \right).{\cos ^2}{18^ \circ } + 5{\cos ^2}{18^ \circ }\\E =  - 4\cos {60^ \circ } + 5{\tan ^2}{18^ \circ }.{\cos ^2}{18^ \circ } + 5{\cos ^2}{18^ \circ }\\E =  - 4\cos {60^ \circ } + 5{\sin ^2}{18^ \circ } + 5{\cos ^2}{18^ \circ }\\E =  - 4.\frac{1}{2} + 5 =  - 2 + 5 = 3\end{array}$