Trigonometrie plana si sferica

Trigonometrie plana si sferica

Functiile trigonometrice in complex

Functii elementare reale pot fi prelungite in complex: functia polinomiala, functia rationala, functia radical, functia exponentiala, functia logaritmica, functia putere complexa.

Functiile trigonometrice si hiperbolice in complex se definesc cu ajutorul functiei exponentiale si prelungesc in complex functiile corespunzatoare reale:
    \begin{eqnarray}
    \cos z=\frac{1}{2}\left(e^{iz}+e^{-iz}\right)&\sin z=\frac{1}{2i}\left(e^{iz}-e^{-iz}\right)\\
        \operatorname{ch} z=\frac{1}{2}\left(e^{z}+e^{-z}\right)&\operatorname{sh} z=\frac{1}{2}\left(e^{z}-e^{-z}\right)
    \end{eqnarray}Proprietati:

  • $$\cos(iz)=\operatorname{ch} z$$ si $$\sin(iz)=i\operatorname{sh} z$$
  • $$\cos^2z+\sin^2z=1$$ si $$\operatorname{ch}^2z-\operatorname{sh}^2z=1$$
  • $$\sin(z_1\pm z_2)=\sin z_1\cos z_2\pm\cos z_1\sin z_2$$
  • $$\cos(z_1\pm z_2)=\cos z_1\cos z_2\mp\sin z_1\sin z_2$$
  • $$\operatorname{sh}(z_1\pm z_2)=\operatorname{sh} z_1\operatorname{ch} z_2\pm\operatorname{ch} z_1\operatorname{sh} z_2$$
  • $$\operatorname{ch}(z_1\pm z_2)=\operatorname{ch} z_1\operatorname{ch} z_2\pm\operatorname{sh} z_1\operatorname{sh} z_2$$
  • Functiile trigonometrice $$\sin$$ si $$\cos$$ sunt periodice de perioada $$2\pi$$, iar functiile hiperbolice $$\operatorname{sh}$$ si $$\operatorname{ch}$$ sunt periodice de perioada $$2\pi i$$
  • Functiile $$\cos$$ si $$\operatorname{ch}$$ sunt pare, iar functiile $$\sin$$ si $$\operatorname{sh}$$ sunt impare
  • $$\sin\left(\frac{\pi}{2}-z\right)=\cos z$$ si $$\cos\left(\frac{\pi}{2}-z\right)=\sin z$$

Se pot defini si functiile $$\operatorname{tg} z=\frac{\sin z}{\cos z}$$, $$\operatorname{ctg} z=\frac{\cos z}{\sin z}$$, $$\operatorname{th} z=\frac{\operatorname{sh} z}{\operatorname{ch} z}$$, $$\operatorname{cth} z=\frac{\operatorname{ch} z}{\operatorname{sh} z}$$.

Functiile inverse trigonometrice si inverse hiperbolice in complex se definesc cu ajutorul functiei logaritmice in complex:

  • $$\displaystyle\arcsin z=\frac{1}{i}\operatorname{Ln}\left(iz+\sqrt{1-z^2}\right)$$
  • $$\displaystyle\arccos z=\frac{1}{i}\operatorname{Ln}\left(z+\sqrt{z^2-1}\right)$$
  • $$\displaystyle\operatorname{arctg} z=\frac{1}{2i}\operatorname{Ln}\frac{i-z}{i+z}$$
  • $$\displaystyle\operatorname{arcctg} z=\frac{1}{2i}\operatorname{Ln}\frac{z+i}{z-i}$$
  • $$\displaystyle\operatorname{argsh} z=\operatorname{Ln}\left(z+\sqrt{z^2+1}\right)$$
  • $$\displaystyle\operatorname{argch} z=\operatorname{Ln}\left(z+\sqrt{z^2-1}\right)$$
  • $$\displaystyle\operatorname{argth} z=\frac{1}{2}\operatorname{Ln}\frac{1+z}{1-z}$$
  • $$\displaystyle\operatorname{argcth} z=\frac{1}{2}\operatorname{Ln}\frac{z+1}{z-1}$$
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