Question

Find two solutions of the following equations, giving your answers in radians
sin ϴ = 1/2
cosec ϴ = 2sqrt(3)/3
cot ϴ = -1

Answer 1

In your trig repertoire , you should know the 3 basic trig values of angles 0°, 30°, 45°, 60°, 90°
by knowing the ratio of sides of the 30-60-90 and the 45-45-90 degree triangles.
This is very very important in any further study of trig.

I will do the only "hard" one:

csc Ø = 2√3/3
sinØ = 3/(2√3) , since cscØ = 1/sinØ
3/(2√3)
= 3/(2√3) * √3/√3
= 3√3/6
= √3/2 <---- recognize that sin 60° = √3/2
so sinØ = √3/2 ----> Ø = 60°

But the sine is positive in quads I and II
so Ø is also equal to 180°-60° = 120°
of course in radians
Ø = π/3 or Ø = 2π/3

Answer 2

In your trig repertoire , you should know the 3 basic trig values of angles 0°, 30°, 45°, 60°, 90°
by knowing the ratio of sides of the 30-60-90 and the 45-45-90 degree triangles. ..Make some flash cards, and memorize these NOW. Or, you can be a dead duck at the end of the course.