Решите Sinx+cosx/cosx+cos3x=1

10. Найти сумму корней на промежутке √3-5tgx=0, на [0˚; 72˚]

11. (tgx+1) (sinx-1) = 0, на [-50°; 350°] найти сумму корней на промежутке

9

(sinx+cosx-cosx-cos3x) / (cosx+cos3x) = 0

sinx-cos3x=0

sinx-sin (π/2-3x) = 0

-2sin (π/4+x) cos (π/4+2x) = 0

sin (π/4+x) = 0⇒π/4+x=πn⇒x=-π/4+πn, n∈z

cos (π/4+2x) = 0⇒π/4+2x=πk⇒2x=-π/4+2πk⇒x=-π/8+πk, k∈z

cosx+cos3x≠0

2cos2ccosx≠0

cos2x≠0⇒2x≠π/2+πm⇒x≠π/4+πm/2; m∈z (общее для cos2x и cosx)

10

tgx=√3/5

x=arctg√3/5+πn, n∈z

x=arctg√3/5∈[0; 72]

11

tgx=-1⇒x=-π/4+πn, n∈z

sinx=1⇒x=π/2+2πk, k∈z

x={-π/4; 3π/4; 7π/4; 0; π/2}

-π/4+3π/4+7π/4+0+π/2=11π/4