Question

Тригонометрия. один пример. , подробное решение! 1) 8 sin²x + sinx cosx + cos²x - 4 = 0

Answer 1

$8\sin^2x+\cos x\sin x+\cos^2x-4=0;\\ a) \cos x=0=\sin x=\pm1=8+0+0-4 \neq 0;\\ b)\cos x \neq 0\\ \frac{8\sin^2x+\cos x\sin x+\cos^2x-4}{\cos^2x}=0;\\ 8tg^2x+tgx+1- \frac{4}{\cos^2x}=0;\\ | \frac{1}{\cos^2x}=\frac{\sin^2x+\cos^x}{\cos^2x}=1+tg^2x|\\ 8tg^2x+tgx+1- 4-4tg^2x=0\\ 4tg^2x+tgx-3=0; \ \ t=tgx;\\ 4t^2+t-3=0;\\ D=b^2-4ac=1+4\cdot3\cdot4=1+48=49=7^2;\\$
$t_{1}= \frac{-b-\sqrt{D}}{2\cdota}= \frac{-1-7}{8}=-1;=tgx_{1}=-1; x_{1}= \frac{3\pi}{4}+\pi n, n\in Z\\ t_{1}= \frac{-b+\sqrt{D}}{2\cdota}= \frac{-1+7}{8}= \frac{3}{4} ;=tgx_{2}=-1; x_{2}= arctg \frac{3}{4} +\pi n, n\in Z\\$