Trigonometrie plana si sferica

Trigonometrie plana si sferica

Ecuatii si inecuatii trigonometrice

  1. ecuatia $$\sin x=a$$
    • daca $$|a|\le1\Rightarrow x=k\pi+(-1)^k\arcsin a,\ k\in\mathbb{Z}$$
    • daca $$|a|>1\Rightarrow$$ nu exista solutii
  2. inecuatia $$\sin x>a$$
    • daca $$a\ge1\Rightarrow$$ nu exista solutii
    • daca $$a<-1\Rightarrow$$ multimea solutiilor este $$\mathbb{R}$$
    • daca $$-1\le a<1\Rightarrow$$ multimea solutiilor este
      @d\bigcup_{k\in\mathbb{Z}}\left(2k\pi+\arcsin a,(2k+1)\pi-\arcsin a\right)@d
  3. inecuatia $$\sin x<a$$
    • daca $$a\le-1\Rightarrow$$ nu exista solutii
    • daca $$a>1\Rightarrow$$ multimea solutiilor este $$\mathbb{R}$$
    • daca $$-1<a\le1\Rightarrow$$ multimea solutiilor este
      @d\bigcup_{k\in\mathbb{Z}}\left((2k-1)\pi-\arcsin a,2k\pi+\arcsin a\right)@d
  4. ecuatia $$\cos x=a$$
    • daca $$|a|\le1\Rightarrow x=2k\pi\pm\arccos a,\ k\in\mathbb{Z}$$
    • daca $$|a|>1\Rightarrow$$ nu exista solutii
  5. inecuatia $$\cos x>a$$
    • daca $$a\ge1\Rightarrow$$ nu exista solutii
    • daca $$a<-1\Rightarrow$$ multimea solutiilor este $$\mathbb{R}$$
    • daca $$-1\le a<1\Rightarrow$$ multimea solutiilor este
      @d\bigcup_{k\in\mathbb{Z}}\left(2k\pi-\arccos a,2k\pi+\arccos a\right)@d
  6. inecuatia $$\cos x<a$$
    • daca $$a\le-1\Rightarrow$$ nu exista solutii
    • daca $$a>1\Rightarrow$$ multimea solutiilor este $$\mathbb{R}$$
    • daca $$-1<a\le1\Rightarrow$$ multimea solutiilor este
      @d\bigcup_{k\in\mathbb{Z}}\left(2k\pi+\arccos a,2(k+1)\pi-\arccos a\right)@d
  7. ecuatia $$\operatorname{tg}x=a\Rightarrow x=k\pi+\operatorname{arctg} a,\ k\in\mathbb{Z}$$
  8. inecuatia $$\operatorname{tg}x>a\Rightarrow$$ multimea solutiilor este @d\bigcup_{k\in\mathbb{Z}}\left(k\pi+\operatorname{arctg} a,k\pi+\frac{\pi}{2}\right)@d
  9. inecuatia $$\operatorname{tg}x<a\Rightarrow$$ multimea solutiilor este @d\bigcup_{k\in\mathbb{Z}}\left(k\pi-\frac{\pi}{2},k\pi+\operatorname{arctg} a\right)@d
  10. ecuatia $$\operatorname{ctg}x=a\Rightarrow x=k\pi+\operatorname{arcctg} a,\ k\in\mathbb{Z}$$
  11. inecuatia $$\operatorname{ctg}x>a\Rightarrow$$ multimea solutiilor este @d\bigcup_{k\in\mathbb{Z}}\left(k\pi,k\pi+\operatorname{arcctg} a\right)@d
  12. inecuatia $$\operatorname{ctg}x<a\Rightarrow$$ multimea solutiilor este @d\bigcup_{k\in\mathbb{Z}}\left(k\pi+\operatorname{arcctg} a,k\pi+\pi\right)@d

   

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