Question

ЕГЭ по профильной математике 13 задание
4cos^2x+2(√2-1)sin(π/2-x)-√2=0
помогите решить

Answer 1

$4 cos^{2} (x) + 2( sqrt{2} - 1) sin( frac{pi}{2} - x) - sqrt{2} = 0 \ 4 cos^{2} (x) + 2( sqrt{2} - 1) cos(x) - sqrt{2} = 0 \ cos(x) = t \ 4 {t}^{2} + 2( sqrt{2} - 1)t - sqrt{2} = 0 \ D = (2 (sqrt{2} - 1)) ^{2} + 4 times 4 sqrt{2} = 4(2 - 2 sqrt{2} + 1) + \ + 16 sqrt{2} = 8 - 8 sqrt{2} + 4 + 16 sqrt{2} = 8 + 8 sqrt{2} + 4 = \ = (2 + 2 sqrt{2} )^{2} \ t_{1} = frac{ - 2sqrt{2} + 2 - 2 - 2 sqrt{2} }{8} = - frac{ sqrt{2} }{2} \ t_{2} = frac{ - 2 sqrt{2} + 2 + 2 + 2 sqrt{2} }{8} = frac{1}{2}$

$1) cos(x) = - frac{ sqrt{2} }{2} \ x = frac{3pi}{4} + 2pi n \ x = frac{5pi}{4} + pi n, : n in mathbb Z$

$2) cos(x) = frac{1}{2} \ x = ± frac{pi}{3} + 2pi m, : m in mathbb Z$