Bài 59: Chứng minh rằng với mọi α,β,γ ta có:
\(cos(α + β).sin(α – β) + cos(β + γ).sin(β – γ) \)
\(+ cos(γ + α).sin(γ – α) = 0\)
Đáp án
Ta có:
\(\eqalign{
& cos\left( {\alpha + \beta } \right).sin\left( {\alpha – \beta } \right){\rm{ }}cos\left( {\beta + \gamma } \right).sin\left( {\beta – \gamma } \right) \cr&+ cos\left( {\gamma + \alpha } \right).sin\left( {\gamma – \alpha } \right) \cr
& = {1 \over 2}(\sin 2\alpha – \sin 2\beta ) + {1 \over 2}(\sin 2\beta – \sin 2\gamma )\cr& + {1 \over 2}(\sin 2\gamma – \sin 2\alpha ) = 0 \cr} \)
Bài 60: Nếu \(\sin \alpha + \cos \alpha = {1 \over 2}\) thì sin2α bằng:
\(\eqalign{
& (A)\,{3 \over 8}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(B)\, – {3 \over 4} \cr
& (C)\,{1 \over {\sqrt 2 }}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(D)\,{3 \over 4} \cr} \)
Đáp án
Ta có:
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\(\eqalign{
& \sin \alpha + \cos \alpha = {1 \over 2} \Rightarrow 1 + 2\sin \alpha \cos \alpha = {1 \over 4} \cr
& \Rightarrow \sin 2\alpha = – {3 \over 4} \cr} \)
Chọn (B)
Bài 61: Với mọi \(α\), \(\sin ({{3\pi } \over 2} + \alpha )\) bằng:
(A) sinα
(B) –sinα
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(C) –cos α
(D) cosα
Đáp án
Ta có:
\(\sin ({{3\pi } \over 2} + \alpha ) = \sin (\pi + {\pi \over 2} + \alpha ) \)
\(= – \sin ({\pi \over 2}\, + \alpha ) = – \cos \alpha \)
Chọn (C)
Bài 62: \({{\sin {\pi \over {15}}\cos {\pi \over 10} + \sin {\pi \over {10}}\cos {\pi \over 15}} \over {\cos {{2\pi } \over {15}}\cos {\pi \over {15}} – \sin {{2\pi } \over {15}}\sin {\pi \over {15}}}}\) bằng:
\((A)\,\sqrt 3 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\;(B)\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\)
\((C)\, – 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(D)\, {1 \over 2}\)
Đáp án
Ta có:
\(\eqalign{
& {{\sin {\pi \over {15}}\cos {\pi \over 10} + \sin {\pi \over {10}}\cos {\pi \over 15}} \over {\cos {{2\pi } \over {15}}\cos {\pi \over {15}} – \sin {{2\pi } \over {15}}\sin {\pi \over {15}}}} = {{\sin ({\pi \over {15}} + {\pi \over {10}})} \over {\cos ({{2\pi } \over {15}} + {\pi \over 5})}} \cr
& = {{\sin {\pi \over 6}} \over {\cos {\pi \over 3}}} = 1 \cr} \)
Chọn (B)
