Question

find all solutions of the equation in the interval [0,2π) csc(theta)+√2=0

Answer 1

This is just like an algebra problem such as (x-3)(x+2) = 0
and you solve each parenthetical for 0, since 0 multiplied by anything is 0 and thus the equation would be correct
so x-3 = 0
x = 3
x+2 = 0
x= -2
those are the two solutions to that algebra example

But since trig functions repeat, they give you an interval so you don't find solutions that are the same angle but a different number of revolutions, such as 60 and 420 (π/3 and 7π/3)

0≤ θ <2π is the same as 0-359 degrees, not 360 since 0 and 360 are the same here

now,
cscθ - 2 = 0
cscθ = 2

csc is 1/sin
so for csc to be 2, sin must be .5
from your unit circle (which I assume you know because you have this problem) you'll see that sin is .5 at θ = 30 and 150 degrees, or π/6 and 5π/6

now
cotθ + 1 = 0
cotθ = -1

tan is opposite over adjacent, so cotangent is adjacent over opposite. that means the two legs are equal, and that only happens with right triangles with a 45 degree angle. But it also has to be negative. The four 45 degree positions are 45, 135, 225, and 315. cotangent or tangent are negative at 135 and 315, so those are the solutions

so in the end, 30, 150, 135, and 315 are the solutions
or
π/6, 5π/6, 3π/4, and 7π/4

Answer 2

welcome 2 os!

Answer 3

and this is for csc(theta) = -√2 not just csc(theta) = 2 ? because I found a chart that shows csc(theta) = -√2 is 5π/4 and 7π/4 only