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

Thank you!

  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}

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

d) 48

step-by-step explanation:

16,32,48,64,80,96,112,128,144,160,176,192,208,224,240,256,272,288,304,320,336

b

step-by-step explanation: