Question

Математика ЕГЭ задание 13

Answer 1

log₂(2sin²x-15cos(3π/2-x))=3
2sin²x+15sinx=2³
2sin²x+15sinx-8=0
sinx=m, -1≤m≤1
2m²+15m-8=0
D=15²+4*2*8=225+64=289=17²
m=(-15+17)/2*2=2/4=1/2
m=(-15-17)/2*2=-32/4=-8 - посторонний корень
sinx=1/2
x=π/6+2πn, n∈Z
x=5π/6+2πn, n∈Z

-5π/2≤π/6+2πn≤-π
-15π/6≤π/6+2πn≤-6π/6
-16π/6≤2πn≤-7π/6
-8/6≤n≤-7/12
n=-1 ⇒ x=π/6-2π=-11π/6

-5π/2≤5π/6+2πn≤-π
-15π/6≤5π/6+2πn≤-6π/6
-20π/6≤2πn≤-11π/6
-10/6≤n≤-11/12
n=-1 ⇒ x=5π/6-2π=-7π/6

Ответ: а) x=π/6+2πn, n∈Z; x=5π/6+2πn, n∈Z
б) x=-11π/6; x=7π/6