TABELA NAJCZĘŚCIEJ OBLICZANYCH ARGUMENTÓW LICZB ZESPOLONYCH
| \(\displaystyle\cos \varphi\) | \(\displaystyle\sin \varphi\) | \(\displaystyle\varphi\) |
|---|---|---|
| \(1\) | \(0\) | \(0\) |
| \(0\) | \(1\) | \(\displaystyle\frac{\pi}2=90^o\) |
| \(-1\) | \(0\) | \(\displaystyle\pi=180^o\) |
| \(0\) | \(-1\) | \(\displaystyle\frac{3\pi}2=270^o\) |
| \(\displaystyle\cos \varphi\) | \(\displaystyle\sin \varphi\) | \(\displaystyle\varphi\) |
|---|---|---|
| \(\displaystyle\frac{\sqrt3}{2}\) | \(\displaystyle\frac12\) | \(\displaystyle\frac{\pi}6=30^o\) |
| \(\displaystyle-\frac{\sqrt3}{2}\) | \(\displaystyle\frac12\) | \(\displaystyle\frac{5\pi}6=150^o\) |
| \(\displaystyle-\frac{\sqrt3}{2}\) | \(\displaystyle-\frac{1}2\) | \(\displaystyle\frac{7\pi}6=210^o\) |
| \(\displaystyle\frac{\sqrt3}{2}\) | \(\displaystyle-\frac{1}2\) | \(\displaystyle\frac{11\pi}6=330^o\) |
| \(\displaystyle\cos \varphi\) | \(\displaystyle\sin \varphi\) | \(\displaystyle\varphi\) |
|---|---|---|
| \(\displaystyle\frac{\sqrt2}{2}\) | \(\displaystyle\frac{\sqrt2}{2}\) | \(\displaystyle\frac{\pi}4=45^o\) |
| \(\displaystyle-\frac{\sqrt2}{2}\) | \(\displaystyle\frac{\sqrt2}{2}\) | \(\displaystyle\frac{3\pi}4=135^o\) |
| \(\displaystyle-\frac{\sqrt2}{2}\) | \(\displaystyle-\frac{\sqrt2}{2}\) | \(\displaystyle\frac{5\pi}4=225^o\) |
| \(\displaystyle\frac{\sqrt2}{2}\) | \(\displaystyle-\frac{\sqrt2}{2}\) | \(\displaystyle\frac{7\pi}4=315^o\) |
| \(\displaystyle\cos \varphi\) | \(\displaystyle\sin \varphi\) | \(\displaystyle\varphi\) |
|---|---|---|
| \(\displaystyle\frac{1}{2}\) | \(\displaystyle\frac{\sqrt3}{2}\) | \(\displaystyle\frac{\pi}3=60^o\) |
| \(\displaystyle-\frac{1}{2}\) | \(\displaystyle\frac{\sqrt3}{2}\) | \(\displaystyle\frac{2\pi}3=120^o\) |
| \(\displaystyle-\frac{1}{2}\) | \(\displaystyle-\frac{\sqrt3}{2}\) | \(\displaystyle\frac{4\pi}3=240^o\) |
| \(\displaystyle\frac{1}{2}\) | \(\displaystyle-\frac{\sqrt3}{2}\) | \(\displaystyle\frac{5\pi}3=300^o\) |
Przykład
Znajdź argument liczby zespolonej : \(z=1-i\sqrt3\)
Najpierw obliczamy moduł liczby zespolonej \(\displaystyle \left | z \right |=\sqrt{a^{2}+b^{2}}\) gdzie : \(a=1 \, ,\,\, b=-\sqrt3\)
\(\displaystyle \left | z \right |=\sqrt{1^{2}+(-\sqrt3)^{2}}=\sqrt{1+3}=2\)
Obliczamy wartości funkcji sinus i cosinus \(cos\varphi=\frac{a}{|z|} \,\,\ sin\varphi=\frac{b}{|z|} \)
\(\displaystyle cos\varphi=\frac{1}{2} \,\,\ sin\varphi=-\frac{\sqrt3}{2} \)
Na podstawie otrzymanych wartości odczytujemy z tabeli, że argument liczby zespolonej wynosi :
\(\displaystyle \varphi=300^o=\frac{5\pi}3\)