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a.
\(P=cos120^0+cos120^0+cos120^0=-\dfrac{3}{2}\)
b.
\(A=\dfrac{\dfrac{sinx}{cosx}-\dfrac{cosx}{cosx}}{\dfrac{sinx}{cosx}+\dfrac{cosx}{cosx}}=\dfrac{tanx-1}{tanx+1}=\dfrac{2-1}{2+1}=\dfrac{1}{3}\)
c.
\(A=\dfrac{cos\left(720+30\right)+sin\left(360+60\right)}{sin\left(-360+30\right)-cos\left(-360-30\right)}=\dfrac{cos30+sin60}{sin30-cos30}=-3-\sqrt{3}\)
\(cos\left(\overrightarrow{AC};\overrightarrow{BA}\right)=cos\left(\overrightarrow{AC};\overrightarrow{AB'}\right)=cos\widehat{CAB'}=cos135^o\)\(=\dfrac{\sqrt{2}}{2}\).
\(sin\left(\overrightarrow{AC};\overrightarrow{BD}\right)=sin90^o=1\) do \(AC\perp BD\).
\(cos\left(\overrightarrow{AB};\overrightarrow{CD}\right)=cos180^o=-1\) do hai véc tơ \(\overrightarrow{AB};\overrightarrow{CD}\) ngược hướng.
a, \(AC=\dfrac{AB}{sin45^o}=\dfrac{a}{\dfrac{\sqrt{2}}{2}}=a\sqrt{2}\)
\(\overrightarrow{AB}.\overrightarrow{AC}=AB.AC.cos\widehat{BAC}=a.a\sqrt{2}.cos45^o=a^2\)
b, \(\left(\overrightarrow{AB}+\overrightarrow{AD}\right)\left(\overrightarrow{BD}+\overrightarrow{BC}\right)=\overrightarrow{AC}\left(\overrightarrow{BD}+\overrightarrow{BC}\right)\)
\(=\overrightarrow{AC}.\overrightarrow{BD}+\overrightarrow{AC}.\overrightarrow{BC}\)
\(=AC.BD.cos90^o+AC.AD.cos45^o\)
\(=a\sqrt{2}.a\sqrt{2}.0+a\sqrt{2}.a.\dfrac{\sqrt{2}}{2}=a^2\)
c, \(\overrightarrow{AB}.\overrightarrow{BD}=AB.BD.cos135^o=-a.a\sqrt{2}.\dfrac{\sqrt{2}}{2}=-a^2\)
d, \(\left(\overrightarrow{AC}-\overrightarrow{AB}\right)\left(2\overrightarrow{AD}-\overrightarrow{AB}\right)=\overrightarrow{BC}.\left(\overrightarrow{AD}+\overrightarrow{BD}\right)\)
\(=\overrightarrow{BC}.\overrightarrow{AD}+\overrightarrow{BC}.\overrightarrow{BD}\)
\(=AD^2+BC.BD.cos45^o\)
\(=a^2+a.a\sqrt{2}.\dfrac{\sqrt{2}}{2}=2a^2\)
e, \(\left(\overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{AD}\right)\left(\overrightarrow{DA}+\overrightarrow{DB}+\overrightarrow{DC}\right)\)
\(=\left(\overrightarrow{AC}+\overrightarrow{AC}\right)\left(\overrightarrow{DB}+\overrightarrow{DB}\right)\)
\(=4.\overrightarrow{AC}.\overrightarrow{DB}=4.AC.DB.cos90^o=0\)
+) \(\left( {\overrightarrow {AB} ,\overrightarrow {AC} } \right) = \widehat {ABC} = 60^\circ \)
+) Dựng hình bình hành ABCD, ta có: \(\overrightarrow {AD} = \overrightarrow {BC} \)
\( \Rightarrow \left( {\overrightarrow {AB} ,\overrightarrow {BC} } \right) = \left( {\overrightarrow {AB} ,\overrightarrow {AD} } \right) = \widehat {BAD} = 120^\circ \)
+), Ta có: ABC là tam giác đều, H là trung điểm BC nên \(AH \bot BC\)
\(\left( {\overrightarrow {AH} ,\overrightarrow {BC} } \right) = \left( {\overrightarrow {AH} ,\overrightarrow {AD} } \right) = \widehat {HAD} = 90^\circ \)
+) Hai vectơ \(\overrightarrow {BH} \) và \(\overrightarrow {BC} \)cùng hướng nên \(\left( {\overrightarrow {BH} ,\overrightarrow {BC} } \right) = 0^\circ \)
+) Hai vectơ \(\overrightarrow {HB} \) và \(\overrightarrow {BC} \)ngược hướng nên \(\left( {\overrightarrow {HB} ,\overrightarrow {BC} } \right) = 180^\circ \)
\(\left|\overrightarrow{MA}+\overrightarrow{BC}+\overrightarrow{AB}\right|=\left|\overrightarrow{AC}+\overrightarrow{BA}\right|\)
\(\Leftrightarrow\left|\overrightarrow{MA}+\overrightarrow{AC}\right|=\left|\overrightarrow{BC}\right|\)
\(\Leftrightarrow\left|\overrightarrow{MC}\right|=\left|\overrightarrow{BC}\right|\)
\(\Leftrightarrow MC=BC\)
\(\Rightarrow\) Tập hợp M là đường tròn tâm C bán kính BC
Do tam giác ABC vuông tại A và \(\widehat{B}=30^o\) \(\Rightarrow C=60^o\)
\(\Rightarrow\left(\overrightarrow{AB},\overrightarrow{BC}\right)=150^o;\)\(\left(\overrightarrow{BA},\overrightarrow{BC}\right)=30^o;\left(\overrightarrow{AC},\overrightarrow{CB}\right)=120^o\)
\(\left(\overrightarrow{AB},\overrightarrow{AC}\right)=90^o;\left(\overrightarrow{BC},\overrightarrow{BA}\right)=30^o\).Do vậy:
a) \(\cos\left(\overrightarrow{AB},\overrightarrow{BC}\right)+\sin\left(\overrightarrow{BA},\overrightarrow{BC}\right)+\tan\frac{\left(\overrightarrow{AC},\overrightarrow{CB}\right)}{2}\)
\(=\cos150^o+\sin30^o+\tan60^o\)
\(=-\frac{\sqrt{3}}{2}+\frac{1}{2}+\sqrt{3}\)
\(=\frac{\sqrt{3}+1}{2}\)
b) \(\sin\left(\overrightarrow{AB},\overrightarrow{AC}\right)+\cos\left(\overrightarrow{BC},\overrightarrow{AB}\right)+\cos\left(\overrightarrow{CA},\overrightarrow{BA}\right)\)
\(=\sin90^o+\cos30^o+\cos0^o\)
\(=1+\frac{\sqrt{3}}{2}\)
\(=\frac{2+\sqrt{3}}{2}\)