Question

Решите уравнения:
1) tg(x-3pi/4)+1=0
2) ctg(3,5pi+x)=0
3) 3sinx-8=sqrt2

Answer 1

$1) tg bigg(x - dfrac{3pi}{4} bigg) + 1 = 0$

$tg bigg(x - dfrac{3pi}{4} bigg) = -1$

$x - dfrac{3pi}{4} = arctg(-1)$

$x - dfrac{3pi}{4} = -dfrac{pi}{4}$

$x - dfrac{3pi}{4} = -dfrac{pi}{4} + pi n, n in Z$

$x = -dfrac{pi}{4} + dfrac{3pi}{4} + pi n, n in Z$

$x = dfrac{pi}{2} + pi n, n in Z$

$2) ctg (3,5 pi + x) = 0$

$-tg x = 0$

$tg x = 0$

$x = pi n, n in Z$

$3) 3sin x - 8 = sqrt{2}$

ОДЗ: $x in [-1; 1]$

$3sin x = sqrt{2} + 8$

$sin x approx dfrac{1,41 + 8}{3} = dfrac{9,41}{3} approx 3,14$

$oslash$ (Нет решений).

Answer 2

1)tg(x-3π/4)+1=0
tg(x-3π/4)=-1
x-3π/4=-π/4+πk
x=3π/4-π/4+πk
x=2π/4+πk
x=π/2+πk
2)ctg(3,5π+x)=0
3,5π+x=π/2+πk
x=-3,5π+0,5π+πk
x=-3π+πk

3)3sinx-8=√2
sinx€[-1;1]
3sinx=8+√2
sinx=((8+√2)/3)>1
x€∅