0\; ,\; \; b=\frac{\sqrt3}{2}>0\; \; \to \; \; \varphi \in (0,\frac{\pi }{2})\\\\tg\varphi =\frac{b}{a}=\frac{\sqrt3/2}{1/2}=\sqrt3\; \; \to \; \; \varphi =arctg\sqrt3=\frac{\pi }{3}\in (0,\frac{\pi }{2})\\\\|z|=r=\sqrt{a^2+b^2}=\sqrt{\frac{1}{4}+\frac{3}{4}}=\sqrt{\frac{4}{4}}=1\\\\z=1\cdot (cos\frac{\pi }{3}+i\, sin\frac{\pi }{3})=cos\frac{\pi }{3}+i\, sin\frac{\pi }{3}" alt="2)\; \; z=\frac{1}{2}+\frac{\sqrt3\, i}{2}\\\\a=\frac{1}{2}>0\; ,\; \; b=\frac{\sqrt3}{2}>0\; \; \to \; \; \varphi \in (0,\frac{\pi }{2})\\\\tg\varphi =\frac{b}{a}=\frac{\sqrt3/2}{1/2}=\sqrt3\; \; \to \; \; \varphi =arctg\sqrt3=\frac{\pi }{3}\in (0,\frac{\pi }{2})\\\\|z|=r=\sqrt{a^2+b^2}=\sqrt{\frac{1}{4}+\frac{3}{4}}=\sqrt{\frac{4}{4}}=1\\\\z=1\cdot (cos\frac{\pi }{3}+i\, sin\frac{\pi }{3})=cos\frac{\pi }{3}+i\, sin\frac{\pi }{3}" align="absmiddle" class="latex-formula">
0\; ,\; \; b=1>0\; \; \to \; \; \varphi \in (0;\frac{\pi }{2})\\\\tg\varphi =\frac{b}{a}=\frac{1}{\sqrt3}\; \; \to \; \; \varphi =arctg\frac{1}{\sqrt3}=\frac{\pi }{6}\in (0;\frac{\pi}{2})\\\\|z|=r=\sqrt{a^2+b^2}=\sqrt{3+1}=2\\\\z=2\cdot (cos\frac{\pi}{6}+i\, sin\frac{\pi}{6})" alt="1)\; \; z=\sqrt3+i\\\\a=\sqrt3>0\; ,\; \; b=1>0\; \; \to \; \; \varphi \in (0;\frac{\pi }{2})\\\\tg\varphi =\frac{b}{a}=\frac{1}{\sqrt3}\; \; \to \; \; \varphi =arctg\frac{1}{\sqrt3}=\frac{\pi }{6}\in (0;\frac{\pi}{2})\\\\|z|=r=\sqrt{a^2+b^2}=\sqrt{3+1}=2\\\\z=2\cdot (cos\frac{\pi}{6}+i\, sin\frac{\pi}{6})" align="absmiddle" class="latex-formula">