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a: \(6\sqrt{3}=\sqrt{108}>\sqrt{54}=3\sqrt{6}\)
\(\Rightarrow5^{6\sqrt{3}}>5^{3\sqrt{6}}\)
b: \(\sqrt{2}\cdot2^{\dfrac{2}{3}}=2^{\dfrac{1}{2}}\cdot2^{\dfrac{2}{3}}=2^{\dfrac{1}{2}+\dfrac{2}{3}}=2^{\dfrac{7}{6}}\)
\(\left(\dfrac{1}{2}\right)^{-\dfrac{4}{3}}=2^{\left(-1\right)\cdot\left(-\dfrac{4}{3}\right)}=2^{\dfrac{4}{3}}\)
mà \(\dfrac{7}{6}< \dfrac{8}{6}=\dfrac{4}{3}\).
nên \(\sqrt{2}\cdot2^{\dfrac{2}{3}}< \left(\dfrac{1}{2}\right)^{-\dfrac{4}{3}}\).
\(a,cos\left(\dfrac{5\pi}{12}\right)=cos\left(\dfrac{\pi}{4}+\dfrac{\pi}{6}\right)=cos\left(\dfrac{\pi}{4}\right)cos\left(\dfrac{\pi}{6}\right)-sin\left(\dfrac{\pi}{4}\right)sin\left(\dfrac{\pi}{6}\right)=\dfrac{\sqrt{2}}{2}\cdot\dfrac{\sqrt{3}}{2}-\dfrac{\sqrt{2}}{2}\cdot\dfrac{1}{2}=\dfrac{\sqrt{6}-\sqrt{2}}{4}\\ sin\left(\dfrac{5\pi}{12}\right)=sin\left(\dfrac{\pi}{4}+\dfrac{\pi}{6}\right)=sin\left(\dfrac{\pi}{4}\right)cos\left(\dfrac{\pi}{6}\right)+cos\left(\dfrac{\pi}{4}\right)sin\left(\dfrac{\pi}{6}\right)=\dfrac{\sqrt{2}}{2}\cdot\dfrac{\sqrt{3}}{2}+\dfrac{\sqrt{2}}{2}\cdot\dfrac{1}{2}=\dfrac{\sqrt{6}+\sqrt{2}}{4}\\ tan\left(\dfrac{5\pi}{12}\right)=\dfrac{sin\left(\dfrac{5\pi}{12}\right)}{cos\left(\dfrac{5\pi}{12}\right)} =2-\sqrt{3}\\ cot\left(\dfrac{5\pi}{12}\right)=\dfrac{1}{tan\left(\dfrac{5\pi}{12}\right)}=\dfrac{1}{2-\sqrt{3}}\)
\(b,cos\left(-555^o\right)=cos\left(3\pi+\dfrac{\pi}{12}\right)=-cos\left(\dfrac{\pi}{12}\right)=-cos\left(\dfrac{\pi}{3}-\dfrac{\pi}{4}\right)=-\left[cos\left(\dfrac{\pi}{3}\right)cos\left(\dfrac{\pi}{4}\right)+sin\left(\dfrac{\pi}{3}\right)sin\left(\dfrac{\pi}{4}\right)\right]=-\dfrac{\sqrt{6}+\sqrt{2}}{4}\\ sin\left(-555^o\right)=sin\left(3\pi+\dfrac{\pi}{12}\right)=sin\left(\dfrac{\pi}{12}\right)=sin\left(\dfrac{\pi}{3}-\dfrac{\pi}{4}\right)=sin\left(\dfrac{\pi}{3}\right)cos\left(\dfrac{\pi}{4}\right)-cos\left(\dfrac{\pi}{3}\right)sin\left(\dfrac{\pi}{4}\right)=\dfrac{\sqrt{3}}{2}\cdot\dfrac{\sqrt{2}}{2}-\dfrac{1}{2}\cdot\dfrac{\sqrt{2}}{2}=\dfrac{\sqrt{6}-\sqrt{2}}{4}\\ tan\left(-555^o\right)=\dfrac{sin\left(-555^o\right)}{cos\left(-555^o\right)}=-2+\sqrt{3}\\ cot\left(-555^o\right)=\dfrac{1}{tan\left(-555^o\right)}=\dfrac{1}{-2+\sqrt{3}}=-2-\sqrt{3}\)
a) \(log_50,5=-0,439677\)
c) \(In\left(\dfrac{3}{2}\right)=0,405465\)
\(a,\sqrt{42}=\sqrt{3\cdot14}>\sqrt{3\cdot12}=6\\ \sqrt[3]{51}=\sqrt[3]{17}< \sqrt[3]{3\cdot72}=6\\ \Rightarrow\sqrt{42}>\sqrt[3]{51}\\ b,16^{\sqrt{3}}=4^{2\sqrt{3}}\\ 18>12\Rightarrow3\sqrt{2}>2\sqrt{3}\Rightarrow4^{3\sqrt{2}}>4^{2\sqrt{3}}\\ \Rightarrow4^{3\sqrt{2}}>16^{\sqrt{3}}\)
\(c,\left(\sqrt{16}\right)^6=16^3=4^6=4^2\cdot4^4=4^2\cdot16^2\\ \left(\sqrt[3]{60}\right)^6=60^2=4^2\cdot15^2\\ 4^2\cdot16^2>4^2\cdot15^2\Rightarrow\sqrt{16}>\sqrt[3]{60}\Rightarrow0,2^{\sqrt{16}}< 0,2^{\sqrt[3]{60}}\)
\(2\sqrt{3}=\sqrt{12}< \sqrt{18}=3\sqrt{2}\)
=>\(2^{2\sqrt{3}}< 2^{3\sqrt{2}}\)
\(\begin{array}{l}\cos 75^\circ = \frac{{\sqrt 6 - \sqrt 2 }}{4}\\\tan \left( { - \frac{{19\pi }}{6}} \right) = - \frac{{\sqrt 3 }}{3}\end{array}\)
a) \(\cos \alpha = - 0,75\)
\( \Leftrightarrow \alpha ={138^ \circ }35'36''\) hay \(\alpha =2,4188584\) rad
b) \(\tan \alpha = 2,46\)
\( \Leftrightarrow \alpha ={67^ \circ }52'01''\) hay \(\alpha =1,1846956\) rad
c) \(\cot \alpha = -6,18\)
\( \Leftrightarrow \alpha ={ -9^ \circ }11'30''\) hay \(\alpha = -0,1604\) rad
\(5^x=3\Leftrightarrow x=log_53\\ 3^y=5\Leftrightarrow y=log_35\\ \Rightarrow xy=log_53\cdot log_35=1\)