20 câu trắc nghiệm Toán 11 Chân trời sáng tạo Các công thức lượng giác có đáp án
20 câu hỏi
Chọn đẳng thức sai trong các đẳng thức sau:
\[{\rm{sin}}\left( {{\rm{a + b}}} \right){\rm{ = sin}}\left( {\rm{a}} \right){\rm{cos}}\left( {\rm{b}} \right){\rm{ + cos}}\left( {\rm{a}} \right){\rm{sin}}\left( {\rm{b}} \right)\]
\[{\rm{sin}}\left( {{\rm{a + b}}} \right){\rm{ = sin}}\left( {\rm{a}} \right){\rm{cos}}\left( {\rm{a}} \right){\rm{ + cos}}\left( {\rm{b}} \right){\rm{sin}}\left( {\rm{b}} \right)\]
\[{\rm{sin}}\left( {{\rm{a}} - {\rm{b}}} \right){\rm{ = sin}}\left( {\rm{a}} \right){\rm{cos}}\left( {\rm{a}} \right) - {\rm{cos}}\left( {\rm{b}} \right){\rm{sin}}\left( {\rm{b}} \right)\]
\[{\rm{sin}}\left( {{\rm{a}} - {\rm{b}}} \right){\rm{ = sin}}\left( {\rm{b}} \right){\rm{cos}}\left( {\rm{a}} \right) - {\rm{cos}}\left( {\rm{a}} \right){\rm{sin}}\left( {\rm{b}} \right)\]
Cho tam giác nhọn ABC. Đẳng thức sai trong các đẳng thức sau là:
\[{\rm{sin}}\left( {{\rm{B + C}}} \right){\rm{ = }} - {\rm{sin}}\left( {\rm{A}} \right)\]
\[{\rm{cos}}\left( {{\rm{B + C}}} \right){\rm{ = }} - {\rm{cos}}\left( {\rm{A}} \right)\]
\[{\rm{tan}}\left( {{\rm{B + C}}} \right){\rm{ = }} - {\rm{tan}}\left( {\rm{A}} \right)\]
\[{\rm{cot}}\left( {{\rm{B + C}}} \right){\rm{ = }} - {\rm{cot}}\left( {\rm{A}} \right)\]
Trong các mệnh đề sau, mệnh đề nào là mệnh đề sai:
\[{\rm{sin}}\left( {{\rm{2a}}} \right){\rm{ = 2sin}}\left( {\rm{a}} \right){\rm{cos}}\left( {\rm{a}} \right)\]
\[{\rm{sin}}\left( {{\rm{2a}}} \right){\rm{ = sin}}\left( {\rm{a}} \right){\rm{cos}}\left( {\rm{a}} \right)\]
\[{\rm{cos}}\left( {{\rm{2a}}} \right){\rm{ = co}}{{\rm{s}}^{\rm{2}}}\left( {\rm{a}} \right) - {\rm{si}}{{\rm{n}}^{\rm{2}}}\left( {\rm{a}} \right)\]
\[{\rm{cos}}\left( {{\rm{2a}}} \right){\rm{ = 1}} - {\rm{2si}}{{\rm{n}}^{\rm{2}}}\left( {\rm{a}} \right)\]
Trong các khẳng định sau, khẳng định nào là đúng ?
\[{\rm{cos}}\left( {{\rm{2a}}} \right){\rm{ = si}}{{\rm{n}}^{\rm{2}}}\left( {\rm{a}} \right) - {\rm{co}}{{\rm{s}}^{\rm{2}}}\left( {\rm{a}} \right)\]
\[{\rm{cos}}\left( {{\rm{2a}}} \right){\rm{ = 1}} - {\rm{2co}}{{\rm{s}}^{\rm{2}}}\left( {\rm{a}} \right)\]
\[{\rm{co}}{{\rm{s}}^{\rm{2}}}\left( {\rm{a}} \right){\rm{ + si}}{{\rm{n}}^{\rm{2}}}\left( {\rm{a}} \right){\rm{ = 1}}\]
\[{\rm{co}}{{\rm{s}}^{\rm{2}}}\left( {\rm{a}} \right) - {\rm{si}}{{\rm{n}}^{\rm{2}}}\left( {\rm{a}} \right){\rm{ = 1}}\]
Chọn khẳng định sai:
\[{\rm{cos}}\left( {\rm{a}} \right){\rm{ + cos}}\left( {\rm{b}} \right){\rm{ = 2cos}}\left( {\frac{{{\rm{a + b}}}}{{\rm{2}}}} \right){\rm{cos}}\left( {\frac{{{\rm{a}} - {\rm{b}}}}{{\rm{2}}}} \right)\]
\[{\rm{cos}}\left( {\rm{a}} \right) - {\rm{cos}}\left( {\rm{b}} \right){\rm{ = }} - {\rm{2cos}}\left( {\frac{{{\rm{a + b}}}}{{\rm{2}}}} \right){\rm{cos}}\left( {\frac{{{\rm{a}} - {\rm{b}}}}{{\rm{2}}}} \right)\]
\[{\rm{sin}}\left( {\rm{a}} \right){\rm{ + sin}}\left( {\rm{b}} \right){\rm{ = 2sin}}\left( {\frac{{{\rm{a + b}}}}{{\rm{2}}}} \right){\rm{cos}}\left( {\frac{{{\rm{a}} - {\rm{b}}}}{{\rm{2}}}} \right)\]
\[{\rm{sin}}\left( {\rm{a}} \right) - {\rm{sin}}\left( {\rm{b}} \right){\rm{ = }} - {\rm{2sin}}\left( {\frac{{{\rm{a + b}}}}{{\rm{2}}}} \right){\rm{cos}}\left( {\frac{{{\rm{a}} - {\rm{b}}}}{{\rm{2}}}} \right)\]
Trong các mệnh đề sau, tìm mệnh đề đúng:
\[{\rm{cos}}\left( {{\rm{a + }}\frac{{\rm{\pi }}}{{\rm{3}}}} \right){\rm{ = cos}}\left( {\rm{a}} \right){\rm{ + }}\frac{{\rm{1}}}{{\rm{2}}}\]
\[{\rm{cos}}\left( {{\rm{a + }}\frac{{\rm{\pi }}}{{\rm{3}}}} \right){\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{sin}}\left( {\rm{a}} \right) - \frac{{\sqrt {\rm{3}} }}{{\rm{2}}}{\rm{cos}}\left( {\rm{a}} \right)\]
\[{\rm{cos}}\left( {{\rm{a + }}\frac{{\rm{\pi }}}{{\rm{3}}}} \right){\rm{ = }}\frac{{\sqrt {\rm{3}} }}{{\rm{2}}}{\rm{sin}}\left( {\rm{a}} \right) - \frac{{\rm{1}}}{{\rm{2}}}{\rm{cos}}\left( {\rm{a}} \right)\]
\[{\rm{cos}}\left( {{\rm{a + }}\frac{{\rm{\pi }}}{{\rm{3}}}} \right){\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{cos}}\left( {\rm{a}} \right) - \frac{{\sqrt {\rm{3}} }}{{\rm{2}}}{\rm{sin}}\left( {\rm{a}} \right)\]
Cho biết \[\frac{{\rm{\pi }}}{{\rm{2}}}{\rm{ < x < \pi }}\]và \[{\rm{sin}}\left( {\rm{x}} \right){\rm{ = }}\frac{{\rm{1}}}{{\rm{3}}}\]. Tính\[{\rm{cos}}\left( {\rm{x}} \right)\]A. \[{\rm{cos}}\left( {\rm{x}} \right){\rm{ = }}\frac{{\rm{2}}}{{\rm{3}}}\]
\[{\rm{cos}}\left( {\rm{x}} \right){\rm{ = }} - \frac{2}{3}\]
\[{\rm{cos}}\left( {\rm{x}} \right){\rm{ = }}\frac{{{\rm{2}}\sqrt {\rm{2}} }}{{\rm{3}}}\]
\[{\rm{cos}}\left( {\rm{x}} \right){\rm{ = }} - \frac{{2\sqrt 2 }}{3}\]
Cho \[{\rm{tan}}\left( {\rm{x}} \right){\rm{ = 5}}\]. Tính giá trị của\[{\rm{P = }}\frac{{{\rm{3sin}}\left( {\rm{x}} \right) - {\rm{4cos}}\left( {\rm{x}} \right)}}{{{\rm{cos}}\left( {\rm{x}} \right){\rm{ + 2sin}}\left( {\rm{x}} \right)}}\]
1
– 1
\[\frac{{11}}{{19}}\]
\[\frac{{19}}{{11}}\]
Cho \[{\rm{sin}}\left( {\rm{\alpha }} \right){\rm{ + cos}}\left( {\rm{\beta }} \right){\rm{ = }}\frac{{\rm{5}}}{{\rm{4}}}\], khi đó \(\sin \left( {2\alpha } \right)\)có giá trị bằng:
\[\frac{{16}}{9}\]
\[\frac{6}{9}\]
\[\frac{9}{{16}}\]
\(\frac{9}{6}\)
Cho\[\sin \left( \alpha \right) = \frac{1}{{\sqrt 3 }}\] với\(0 < \alpha < \frac{\pi }{2}\). Tính giá trị của\[\sin \left( {\alpha + \frac{\pi }{3}} \right)\]
\[\frac{{\sqrt 3 }}{6} - \frac{{\sqrt 2 }}{2}\]
\[\frac{{\sqrt 3 }}{3} + \frac{1}{2}\]
\[\frac{{\sqrt 3 }}{3} - \frac{1}{2}\]
\[\frac{{\sqrt 3 }}{6} + \frac{{\sqrt 2 }}{2}\]
Thu gọn biểu thức\[{\rm{P = si}}{{\rm{n}}^{\rm{6}}}\left( {\rm{x}} \right){\rm{ + co}}{{\rm{s}}^{\rm{6}}}\left( {\rm{x}} \right)\]
\[{\rm{P = 1 + 3co}}{{\rm{s}}^{\rm{2}}}\left( {{\rm{2x}}} \right)\]
\[{\rm{P = 1 + }}\frac{{\rm{3}}}{{\rm{4}}}{\rm{si}}{{\rm{n}}^{\rm{2}}}\left( {{\rm{2x}}} \right)\]
\[{\rm{P = 1}} - \frac{{\rm{3}}}{{\rm{4}}}{\rm{si}}{{\rm{n}}^{\rm{2}}}\left( {{\rm{2x}}} \right)\]
\[{\rm{P = 1}} - {\rm{3co}}{{\rm{s}}^{\rm{2}}}\left( {{\rm{2x}}} \right)\]
Biểu thức\[{\rm{Q = }}\frac{{{\rm{1 + sin}}\left( {{\rm{4a}}} \right) - {\rm{cos}}\left( {{\rm{4a}}} \right)}}{{{\rm{1 + sin}}\left( {{\rm{4a}}} \right){\rm{ + cos}}\left( {{\rm{4a}}} \right)}}\]bằng biểu thức nào sau đây:
\[{\rm{A = sin}}\left( {{\rm{2a}}} \right)\]
\[{\rm{B = cos}}\left( {{\rm{2a}}} \right)\]
\[{\rm{C = tan}}\left( {{\rm{2a}}} \right)\]
\[{\rm{D = cot}}\left( {{\rm{2a}}} \right)\]
Cho góc nhọn a, b thỏa mãn\[{\rm{tan}}\left( {\rm{a}} \right){\rm{ = }}\frac{{\rm{1}}}{{\rm{7}}}{\rm{, tan}}\left( {\rm{b}} \right){\rm{ = }}\frac{{\rm{3}}}{{\rm{4}}}\]. Tính a + b
\[\frac{{\rm{\pi }}}{{\rm{3}}}\]
\[ - \frac{{\rm{\pi }}}{{\rm{3}}}\]
\[\frac{{\rm{\pi }}}{{\rm{4}}}\]
\( - \frac{{\rm{\pi }}}{{\rm{4}}}\)
Cho \[{\rm{cot}}\left( {\rm{\alpha }} \right){\rm{ = }}\frac{{\rm{2}}}{{\rm{3}}}\]. Tính\[{\rm{sin}}\left( {{\rm{2\alpha + }}\frac{{{\rm{7\pi }}}}{{\rm{4}}}} \right)\]
\[\frac{{17\sqrt 2 }}{{26}}\]
\[ - \frac{{17\sqrt 2 }}{{26}}\]
\[\frac{{\sqrt 2 }}{{26}}\]
\[ - \frac{{\sqrt 2 }}{{26}}\]
Rút gọn biểu thức\[{\rm{A = co}}{{\rm{s}}^{\rm{2}}}\left( {\rm{\alpha }} \right){\rm{ + co}}{{\rm{s}}^{\rm{2}}}\left( {{\rm{\alpha + \beta }}} \right) - {\rm{2cos}}\left( {\rm{\alpha }} \right){\rm{cos}}\left( {\rm{\beta }} \right){\rm{cos}}\left( {{\rm{\alpha + \beta }}} \right)\]ta được kết quả
\[{\rm{co}}{{\rm{s}}^{\rm{2}}}\left( {\rm{\alpha }} \right)\]
\[{\rm{co}}{{\rm{s}}^{\rm{2}}}\left( {\rm{\beta }} \right)\]
\[{\rm{si}}{{\rm{n}}^{\rm{2}}}\left( {\rm{\alpha }} \right)\]
\[{\rm{si}}{{\rm{n}}^{\rm{2}}}\left( {\rm{\beta }} \right)\]
Cho góc lượng giác \(\alpha \)thỏa mãn \[\frac{{{\rm{sin}}\left( {{\rm{2\alpha }}} \right){\rm{ + sin}}\left( {{\rm{5\alpha }}} \right) - {\rm{sin}}\left( {{\rm{3\alpha }}} \right)}}{{{\rm{2co}}{{\rm{s}}^{\rm{2}}}\left( {{\rm{2\alpha }}} \right){\rm{ + cos}}\left( {\rm{\alpha }} \right) - {\rm{1}}}}{\rm{ = }} - {\rm{2}}\]. Tính \(\sin \left( \alpha \right)\).
– 1
0
1
\(\frac{{ - 1}}{2}\)
Tính tổng \[{\rm{S = si}}{{\rm{n}}^{\rm{2}}}{{\rm{5}}^{\rm{0}}}{\rm{ + si}}{{\rm{n}}^{\rm{2}}}{\rm{1}}{{\rm{0}}^{\rm{0}}}{\rm{ + si}}{{\rm{n}}^{\rm{2}}}{\rm{1}}{{\rm{5}}^{\rm{0}}}{\rm{ + }}...{\rm{ + si}}{{\rm{n}}^{\rm{2}}}{\rm{8}}{{\rm{5}}^{\rm{0}}}\]
S = 17
\[{\rm{S = }}\frac{{{\rm{17}}}}{{\rm{2}}}\]
S = 1
S = 0
Tính các góc của tam giác ABC biết\[\left( {{\rm{1 + }}\frac{{\rm{1}}}{{{\rm{sinA}}}}} \right)\left( {{\rm{1 + }}\frac{{\rm{1}}}{{{\rm{sinB}}}}} \right)\left( {{\rm{1 + }}\frac{{\rm{1}}}{{{\rm{sinC}}}}} \right){\rm{ = }}{\left( {{\rm{1 + }}\frac{{\rm{1}}}{{\sqrt[{\rm{3}}]{{{\rm{sinA}}{\rm{.sinB}}{\rm{.sinC}}}}}}} \right)^{\rm{3}}}\]
\[\widehat A{\rm{ = }}\widehat B{\rm{ = }}\widehat C{\rm{ = 6}}{{\rm{0}}^{\rm{0}}}\]
\[\widehat A{\rm{ = 9}}{{\rm{0}}^0}{\rm{; }}\widehat B{\rm{ = 6}}{{\rm{0}}^{\rm{0}}};\,\,\widehat C{\rm{ = 3}}{{\rm{0}}^{\rm{0}}}\]
\[\widehat A{\rm{ = 9}}{{\rm{0}}^0}{\rm{; }}\widehat B{\rm{ = 3}}{{\rm{0}}^{\rm{0}}};\,\,\widehat C{\rm{ = 6}}{{\rm{0}}^{\rm{0}}}\]
\[\widehat A{\rm{ = 9}}{{\rm{0}}^0}{\rm{; }}\widehat B{\rm{ = 4}}{{\rm{5}}^{\rm{0}}};\,\,\widehat C{\rm{ = 45}}{{\rm{0}}^{\rm{0}}}\]
Nếu \[{\rm{tan}}\left( {\rm{\alpha }} \right)\] và \[{\rm{tan}}\left( {\rm{\beta }} \right)\] là nghiệm của phương trình \[{{\rm{x}}^{\rm{2}}} - {\rm{px + q = 0, (q}} \ne 1)\] thì giá trị của biểu thức \[{\rm{Q = co}}{{\rm{s}}^{\rm{2}}}\left( {{\rm{\alpha + \beta }}} \right){\rm{ + psin}}\left( {{\rm{\alpha + \beta }}} \right){\rm{cos}}\left( {{\rm{\alpha + \beta }}} \right){\rm{ + qsi}}{{\rm{n}}^{\rm{2}}}\left( {{\rm{\alpha + \beta }}} \right)\] bằng
q
p
0
1
Cho tam giác ABC có các góc thỏa mãn sin(A) + sin(B) = cos(A) + cos(B) . Tính số đo góc C của tam giác ABC
300
900
600
400
