![image](https://tex.z-dn.net/?f=cos150%5E%7B%5Ccirc+%7D%3D%5Cfrac%7B%5Coverline+%7Ba%7D%5Ccdot+%5Coverline+%7Bb%7D%7D%7B%7C%5Coverline+%7Ba%7D%7C%5Ccdot+%7C%5Coverline+%7Bb%7D%7C%7D%5C%5C%5C%5C-%5Cfrac%7B%5Csqrt3%7D%7B2%7D%3D%5Cfrac%7B-2%2B2%2B4z%7D%7B%5Csqrt%7B1%2B4%2B16%7D%5Ccdot+%5Csqrt%7B4%2B1%2Bz%5E2%7D%7D%5C%5C%5C%5C%5Cfrac%7B4z%7D%7B%5Csqrt%7B21%7D%5Ccdot+%5Csqrt%7B5%2Bz%5E2%7D%7D%3D-%5Cfrac%7B%5Csqrt3%7D%7B2%7D%5C%5C%5C%5C-8z%3D3%5Csqrt7%5Ccdot+%5Csqrt%7B5%2Bz%5E2%7D%5C%5C%5C%5C-8z%3D3%5Csqrt%7B35%2B7z%5E2%7D%5C%3B+%5C%3B+%5Cto+%5C%3B+%5C%3B+-8z%3E0%2C%5C%3B+%5Cto+%5C%3B+z%3C0%5C%5C%5C%5C64z%5E2%3D9%2835%2B7z%5E2%29%5C%5C%5C%5Cz%5E2%3D315%5C%5C%5C%5Cz%3D%5Cpm+%5Csqrt%7B315%7D%3D%5Cpm+3%5Csqrt%7B35%7D%5C%5C%5C%5COnvet%3A%5C%3B+z%3D-3%5Csqrt%7B35%7D)
0,\; \to \; z<0\\\\64z^2=9(35+7z^2)\\\\z^2=315\\\\z=\pm \sqrt{315}=\pm 3\sqrt{35}\\\\Onvet:\; z=-3\sqrt{35}" alt="cos150^{\circ }=\frac{\overline {a}\cdot \overline {b}}{|\overline {a}|\cdot |\overline {b}|}\\\\-\frac{\sqrt3}{2}=\frac{-2+2+4z}{\sqrt{1+4+16}\cdot \sqrt{4+1+z^2}}\\\\\frac{4z}{\sqrt{21}\cdot \sqrt{5+z^2}}=-\frac{\sqrt3}{2}\\\\-8z=3\sqrt7\cdot \sqrt{5+z^2}\\\\-8z=3\sqrt{35+7z^2}\; \; \to \; \; -8z>0,\; \to \; z<0\\\\64z^2=9(35+7z^2)\\\\z^2=315\\\\z=\pm \sqrt{315}=\pm 3\sqrt{35}\\\\Onvet:\; z=-3\sqrt{35}" align="absmiddle" class="latex-formula">