Question

99 ! тригонометрия! 10 sin^2(x/3)-12 sin(x/3)cos(/3)-11 cos^2(x/3)=1 cos(3π/2+x)-5cosx=0 √3ctg(π/3-4x)=3

Answer 1

$1)\; \; 10sin^2 \frac{x}{3} -12sin \frac{x}{3} \cdot cos\frac{x}{3} -11cos^2\frac{x}{3} =1\\\\ 10sin^2\frac{x}{3}-12sin \frac{x}{3} \cdot cos\frac{x}{3} -11cos^2\frac{x}{3}=sin^2\frac{x}{3} +cos^2\frac{x}{3} \\\\9sin^2 \frac{x}{3} -12sin \frac{x}{3}\cdot cos \frac{x}{3} -12cos^2 \frac{x}{3} =0\; |:3cos^2 \frac{x}{3} \ne 0\\\\3tg^2 \frac{x}{3} -4tg\frac{x}{3} -4=0\\\\D/4=4+12=16$

$a)\; \; tg\frac{x}{3} = \frac{2-4}{3} =- \frac{2}{3} \\\\ \frac{x}{3}=-arctg\frac{2}{3}+\pi n\; ,\; n\in Z\\\\\underline {x=-3arctg\frac{2}{3}+3\pi n,\; n\in Z}\\\\b)\; \; tg \frac{x}{3}= \frac{2+4}{3}=2$

$\frac{x}{3}=arctg2+\pi k,\; k\in Z\\\\\underline {x=3arctg2+3\pi k,\; k\in Z}$

$2)\; \; cos( \frac{3\pi }{2} +x)-5cosx=0\\\\sinx-5cosx=0\; |:cosx\ne 0\\\\tgx-5=0\\\\tgx=5\\\\\underline {x=arctg5+\pi n,\; n\in Z}\\\\3)\; \; \sqrt3\cdot ctg(\frac{\pi}{3}-4x)=3\\\\ctg( \frac{\pi }{3} -4x)=\sqrt3\\\\ \frac{\pi }{3} -4x=arcctg\sqrt3+\pi n\; ,\; n\in Z\\\\ \frac{\pi }{3} -4x= \frac{\pi }{6}+\pi n,\; n\in Z\\\\4x= \frac{\pi }{3}- \frac{\pi }{6} -\pi n=\frac{\pi}{6}-\pi n,\; n\in Z\\\\\underline {x=\frac{\pi}{24}-\frac{\pi n}{4}\; ,\; n\in Z}$

$(Mozno\; pisat\; tak:\; \; x=\frac{\pi}{24}+\frac{\pi n}{4}\; ,\; n\in Z)$