Question

Find the principal solution of the following equation:

`sqrt(3)` cosecθ + 2 = 0 

Answer 1

`sqrt(3)` cosec θ + 2 = 0 

`sqrt(3)` cosec θ = - 2 

cosec θ = `(-2)/sqrt(3)`

sin θ =  `-sqrt(3)/(2)`

sin θ = `-sin(pi/3)`

sin θ = `sin(pi + pi/3)` and sin θ = `sin (2pi - pi/3)`

sin θ = `sin ((4pi)/3)` and sin θ  = `sin ((5pi)/3)`

= `[therefore 0≤ (4pi)/3 < 2pi and 0≤ (5pi)/3< 2pi]`

θ = `(4pi)/(3)` and θ = `(5pi)/(3)`

Principal solutions one  `(4pi)/(3)` and  `(5pi)/(3)`.