Folosim urmatoarele valori ale functiilor trigonometrice:
[tex]\displaystyle\bf\\~~~~~~~~~30^o~~~45^o~~~60^0\\\\sin~~~~\frac{1}{2}~~~~\frac{\sqrt{2}}{2}~~~\frac{\sqrt{3}}{2}\\\\cos~~~\frac{\sqrt{3}}{2}~~~\frac{\sqrt{2}}{2}~~~~\frac{1}{2}\\\\tg~~~~\frac{\sqrt{3}}{3}~~~~~1~~~~\sqrt{3}\\\\ctg~~~\sqrt{3}~~~~~1~~~~\frac{\sqrt{3}}{3}\\\\\text{\bf Rezolvare:}\\\\a)\\\sin60^o\cos30^0-\sin30^o\cos60^o=\frac{\sqrt{3}}{2}\times\frac{\sqrt{3}}{2}-\frac{1}{2}\times\frac{1}{2}=\frac{3}{4}-\frac{1}{4}=\frac{2}{4}=\boxed{\bf\frac{1}{2}}[/tex]
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[tex]\displaystyle\bf\\b)\\tg~45^o~tg~30^o+\sin45^o~tg~60^o-2\sin60^o\cos45^o=\\\\=1\times\frac{\sqrt{3}}{3}+\frac{\sqrt{2}}{2}\times\sqrt{3}-2\times\frac{\sqrt{3}}{2}\times\frac{\sqrt{2}}{2}=\\\\=\frac{\sqrt{3}}{3}+\frac{\sqrt{2}\times\sqrt{3}}{2}-\frac{2\times\sqrt{3}\times\sqrt{2}}{2\times2}=\\\\=\frac{\sqrt{3}}{3}+\frac{\sqrt{2}\times\sqrt{3}}{2}-\frac{\sqrt{3}\times\sqrt{2}}{2}=\boxed{\bf\frac{\sqrt{3}}{3}}[/tex]
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[tex]\displaystyle\bf\\c)\\(1+tg^260^o)\times\sin45^o-\cos45^o:\sin^260^o=\\\\=\left(1+\Big(\sqrt{3}\Big)^2 \right)\times\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\times\left(\frac{\sqrt{3}}{2}\right)^2=\\\\=\left(1+3 \right)\times\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\times\frac{3}{4}=\\\\=4\times\frac{\sqrt{2}}{2}-\frac{3\sqrt{2}}{8}=2\sqrt{2}-\frac{3\sqrt{2}}{8}=\\\\=\frac{16\sqrt{2}-3\sqrt{2}}{8}=\boxed{\bf\frac{13\sqrt{2}}{8}}[/tex]
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[tex]\displaystyle\bf\\d)\\3\sin30^o-4\cos60^o+2tg~45^o=\\\\3\times\frac{1}{2}-4\times\frac{1}{2}+2\times1=\frac{3}{2}-2+2=\boxed{\bf\frac{3}{2}}\\\\e)\\4ctg~30^o-2tg~45^o:\sin60^o+ctg~45^o:\sin30^o=\\\\=4\times\sqrt{3}-2\times1:\frac{\sqrt{3}}{2}+1:\frac{1}{2}=\\\\=4\sqrt{3}-2\times\frac{2}{\sqrt{3}}+1\times\frac{2}{1}=\\\\=4\sqrt{3}-\frac{4\sqrt{3}}{3}+2=\frac{12\sqrt{3}-4\sqrt{3}}{3}+2=\boxed{\bf\frac{8\sqrt{3}}{3}+2}[/tex]