find the solutions of the equation that are in the interval [0,2pi). csc^2 θ = 4csc θ

Answer 1

Let x = csc(theta)

then

x^2 -4x = 0

(x)(x-4) = 0

x = 0, 4 = cosecant of theta

cosecant is never zero, so x=4 is the only solution

cscx can never be between -1 and 1

the angle whose cosecant = 4 is the same angle whose sine = 1/4

cscx and sinx are inverses of each other, cscx = 1/sinx

use the inverse sine function on a calculator

sin^-1(4) = 14.48 degrees = 0.25 radians

= 0.08pi radians

Answer 2

First, move everything to the left side then factor:

csc² θ - 4csc θ = 0

csc θ (csc θ - 4) = 0

Therefore, you have:

csc θ = 0, csc θ = 4

The only solution is:

θ = sin^-1(1/4) = 0.2527