Answer:
x ∈ {0.063170, 0.722228, 1.633966, 2.293024}
Step-by-step explanation:
It is probably convenient to replace csc(4x) with its equivalent, 1/sin(4x). Then add 4 and multiply by sin(4x)/4.
1/sin(4x) -4 = 0
1/sin(4x) = 4
1/4 = sin(4x)
Using the inverse sine function, we find ...
4x = arcsin(1/4) +2kπ . . . . in radians
and the supplementary solution, ...
4x = π -arcsin(1/4) +2kπ
Dividing these by 4 gives the possible values of x.
x = arcsin(1/4)/4 +kπ/2
x = π/4 -arcsin(1/4) +kπ/2
The four smallest positive solutions will be found where k = 0 or 1.
x ∈ {0.063170, 0.722228, 1.633966, 2.293024}
_____
I like a graphing calculator for solving these quickly and easily.
