Question

Solve csc(4x) − 4 = 0 for the four smallest positive solutions

Thank you!

Answer 1

x ∈ {0.063170, 0.722228, 1.633966, 2.293024}

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}

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I like a graphing calculator for solving these quickly and easily.