a) Xét \(\Delta ABC:\)
\(\cos A=\dfrac{b^2+c^2-a^2}{2bc}=\dfrac{7^2+5^2-\left(4\sqrt{2}\right)^2}{2.7.5}=\dfrac{3}{5}.\\ \cos B=\dfrac{a^2+c^2-b^2}{2ac}=\dfrac{\left(4\sqrt{2}\right)^2+5^2-7^2}{2.4\sqrt{2}.5}=\dfrac{\sqrt{2}}{20}.\\ \cos C=\dfrac{a^2+b^2-c^2}{2ab}=\dfrac{\left(4\sqrt{2}\right)^2+7^2-5^2}{2.4\sqrt{2}.7}=\dfrac{\sqrt{2}}{2}.\)
b) Xét \(\Delta MBN:\)
\(MN^2=MB^2+BN^2-2.MB.BN.\cos B.\\ \Rightarrow MN^2=\left(\dfrac{1}{3}AB\right)^2+\left(\dfrac{1}{2}BC\right)^2-2.\dfrac{1}{3}AB.\dfrac{1}{2}BC.\dfrac{\sqrt{2}}{20}.\\ =\dfrac{1}{9}.5^2+\dfrac{1}{4}.\left(4\sqrt{2}\right)^2-\dfrac{2}{3}.5.\dfrac{1}{2}.4\sqrt{2}.\dfrac{\sqrt{2}}{20}.\\ =\dfrac{91}{9}.\)
c) Ta có:
\(\cos A=\dfrac{3}{5}.\\ \Rightarrow\widehat{A}\approx53,13^o.\\ \Rightarrow\sin A=\dfrac{4}{5}.\)
\(S_{\Delta BMN}=\dfrac{1}{2}bc\sin A.\\ =\dfrac{1}{2}.7.5.\dfrac{4}{5}=14\text{(đvdt).}\)
