Question

Решить уравнение cos x - корень из 3sin x=2cos 7x,записать число корней на промежутке [0 градусов и 180]

Answer 1

2(1/2cosx-√3/2sinx)=2cos7x
2cos(x+π/3)=2cos7x
cos7x-cos(x+π/3)=0
-2sin(4x+π/6)sin(3x-π/6)=0
sin(4x+π/6)=0⇒4x+π/6=πn⇒4x=-π/6+πn⇒x=-π/24+πn/4
0≤-π/24+πn/4≤π
0≤-1+6n≤24
1≤6n≤25
1/6≤n≤25/9
n=1⇒x=-π/24+π/4=5π/24
n=2⇒x=-π/24+π/2=11π/24
n=3⇒x=-π/24+3π/4=17π/24
n=4⇒x=-π/24+π=23/π/24
sin(3x-π/6)=0⇒3x-π/6=πn⇒3x=π/6+πn⇒x=π/18+πn/3
0≤π/18+πn/3≤π
0≤1+18n≤18
-1≤18n≤17
-1/18≤n≤17/18
n=0⇒x=π/18