Основано на упр. 1 для самостоят.работы, стр. 6 Найди область определения функции: y=\sqrt{1-\tg^{2}x}; 2\pi n\lt x\lt \dfrac{\pi}{2} + 2\pi n, n\in \Z. -\cfrac{\pi}{4}+\pi n\le x\le \cfrac{\pi}{4} + \pi n, n\in \Z. -\cfrac{\pi}{4}+\pi n\ge x\ge \cfrac{\pi}{4} + \pi n, n\in \Z. y=\sqrt{\ctg^{2}x-3}; \pi n\lt x\le \cfrac{\pi}{6} + \pi n, \space \cfrac{5\pi}{6}+\pi n\le x\lt \pi(1+n), n\in \Z. -\cfrac{\pi}{6}+\pi n\ge x\ge \cfrac{\pi}{6} + \pi n, n\in \Z. 2\pi n\lt x\lt \cfrac{\pi}{4} + 4\pi n, n\in \Z. y=\sqrt{3\sin^{2}x-\cos^{2}x}; -\cfrac{\pi}{6}+\pi n\ge x\ge \cfrac{\pi}{6} + \pi n, n\in \Z. 2\pi n\lt x\lt \cfrac{\pi}{6} + 5\pi n, n\in \Z. \cfrac{\pi}{6} + \pi n\le x\le \cfrac{5\pi}{6} + \pi n, n\in \Z. y=\sqrt{\sin^{2}x-3\cos^{2}x}; \cfrac{\pi}{3}+\pi n\ge x\ge \cfrac{2\pi}{3} + \pi n, n\in \Z. \cfrac{\pi}{3} + \pi n\le x\le \cfrac{2\pi}{3} + \pi n, n\in \Z. 2\pi n\lt x\lt \cfrac{\pi}{3} + 2\pi n, n\in \Z. y=\sqrt{\cos2x-\sin x}; \cfrac{5\pi}{6} + 2\pi n\le x\le \cfrac{13\pi}{6} + 2\pi n, n\in \Z. \cfrac{5\pi}{6}+\pi n\ge x\ge \cfrac{13\pi}{6} + \pi n, n\in \Z. 5\pi n\lt x\lt \cfrac{13\pi}{6} + 2\pi n, n\in \Z. y=\sqrt{3\cos2x+7\cos x}; -\arccos\cfrac{1}{3} + 2\pi n\le x\le \arccos\cfrac{1}{3} + 2\pi n, n\in \Z. \cfrac{1}{3}+2\pi n\ge x\ge \cfrac{13\pi}{6} + \pi n, n\in \Z. \arccos\cfrac{1}{3} +\pi n\lt x\lt \arccos\cfrac{1}{6} + 2\pi n, n\in \Z. y=\sqrt{1-4\sin x-4\cos^{2}x}; -\cfrac{5}{6} + 2\pi n\le x\le -\cfrac{\pi}{6} + 2\pi n, n\in \Z. \cfrac{1}{6}+2\pi n\ge x\ge \cfrac{5\pi}{6} + \pi n, n\in \Z. -\cfrac{5\pi}{6} +2\pi n\le x\le -\cfrac{\pi}{6} + 2\pi n, n\in \Z. y=\sqrt{2\sin 2x-2\sin^{2}x-1}; -\arctg\cfrac{1}{3} + 2\pi n\le x\le -\cfrac{\pi}{4} + 2\pi n, n\in \Z. \arctg\,\cfrac{1}{3}+\pi n\le x\le \cfrac{\pi}{4} + \pi n, n\in \Z. \arccos\cfrac{\pi}{4} +2\pi n\le x\le -\cfrac{\pi}{6} + 2\pi n, n\in \Z.
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Задание

Основано на упр. 1 для самостоят.работы, стр. 6

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Найди область определения функции:

\(y=\sqrt{1-\tg^{2}x}\) ;

  • \(2\pi n\lt x\lt \dfrac{\pi}{2} + 2\pi n\) , \(n\in \Z\) .
  • \(-\cfrac{\pi}{4}+\pi n\le x\le \cfrac{\pi}{4} + \pi n\) , \(n\in \Z\) .
  • \(-\cfrac{\pi}{4}+\pi n\ge x\ge \cfrac{\pi}{4} + \pi n\) , \(n\in \Z\) .

\(y=\sqrt{\ctg^{2}x-3}\) ;

  • \(\pi n\lt x\le \cfrac{\pi}{6} + \pi n\) , \(\space \cfrac{5\pi}{6}+\pi n\le x\lt \pi(1+n)\) , \(n\in \Z\) .
  • \(-\cfrac{\pi}{6}+\pi n\ge x\ge \cfrac{\pi}{6} + \pi n\) , \(n\in \Z\) .
  • \(2\pi n\lt x\lt \cfrac{\pi}{4} + 4\pi n\) , \(n\in \Z\) .

\(y=\sqrt{3\sin^{2}x-\cos^{2}x}\) ;

  • \(-\cfrac{\pi}{6}+\pi n\ge x\ge \cfrac{\pi}{6} + \pi n\) , \(n\in \Z\) .
  • \(2\pi n\lt x\lt \cfrac{\pi}{6} + 5\pi n\) , \(n\in \Z\) .
  • \(\cfrac{\pi}{6} + \pi n\le x\le \cfrac{5\pi}{6} + \pi n\) , \(n\in \Z\) .

\(y=\sqrt{\sin^{2}x-3\cos^{2}x}\) ;

  • \(\cfrac{\pi}{3}+\pi n\ge x\ge \cfrac{2\pi}{3} + \pi n\) , \(n\in \Z\) .
  • \(\cfrac{\pi}{3} + \pi n\le x\le \cfrac{2\pi}{3} + \pi n\) , \(n\in \Z\) .
  • \(2\pi n\lt x\lt \cfrac{\pi}{3} + 2\pi n\) , \(n\in \Z\) .

\(y=\sqrt{\cos2x-\sin x}\) ;

  • \(\cfrac{5\pi}{6} + 2\pi n\le x\le \cfrac{13\pi}{6} + 2\pi n\) , \(n\in \Z\) .
  • \(\cfrac{5\pi}{6}+\pi n\ge x\ge \cfrac{13\pi}{6} + \pi n\) , \(n\in \Z\) .
  • \(5\pi n\lt x\lt \cfrac{13\pi}{6} + 2\pi n\) , \(n\in \Z\) .

\(y=\sqrt{3\cos2x+7\cos x}\) ;

  • \(-\arccos\cfrac{1}{3} + 2\pi n\le x\le \arccos\cfrac{1}{3} + 2\pi n\) , \(n\in \Z\) .
  • \(\cfrac{1}{3}+2\pi n\ge x\ge \cfrac{13\pi}{6} + \pi n\) , \(n\in \Z\) .
  • \(\arccos\cfrac{1}{3} +\pi n\lt x\lt \arccos\cfrac{1}{6} + 2\pi n\) , \(n\in \Z\) .

\(y=\sqrt{1-4\sin x-4\cos^{2}x}\) ;

  • \(-\cfrac{5}{6} + 2\pi n\le x\le -\cfrac{\pi}{6} + 2\pi n\) , \(n\in \Z\) .
  • \(\cfrac{1}{6}+2\pi n\ge x\ge \cfrac{5\pi}{6} + \pi n\) , \(n\in \Z\) .
  • \(-\cfrac{5\pi}{6} +2\pi n\le x\le -\cfrac{\pi}{6} + 2\pi n\) , \(n\in \Z\) .

\(y=\sqrt{2\sin 2x-2\sin^{2}x-1}\) ;

  • \(-\arctg\cfrac{1}{3} + 2\pi n\le x\le -\cfrac{\pi}{4} + 2\pi n\) , \(n\in \Z\) .
  • \(\arctg\,\cfrac{1}{3}+\pi n\le x\le \cfrac{\pi}{4} + \pi n\) , \(n\in \Z\) .
  • \(\arccos\cfrac{\pi}{4} +2\pi n\le x\le -\cfrac{\pi}{6} + 2\pi n\) , \(n\in \Z\) .

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