# for the four smallest positive solutions?

The trigonometric equation is csc(5x) - 3 = 0.

csc(5x) = 3

Using reciprocal identity : csc x = 1/sin x.

1/sin(5x) = 3

sin(5x) = 1/3

sin(sin-1(x)) = x   for every x in the interval [-1, 1].

sin-1(sin(x)) = x   for every x in the interval [ - π/2, π/2].

sin(5x) = sin(sin-1(1/3))

sin(5x) = sin(19.470)

The genaral solution of sin(θ) = sin(α) is θ = nπ + (- 1)nα, where n is an integer.

5x = nπ + (- 1)n*19.470

⇒ x = [ nπ + (- 1)n*19.470 ]/5.

If n = 0, then x = [ 0*π + (- 1)0*19.47 ]/5 = 19.47/5 = 3.8940,

If n = 1, then x = [ 1*π + (- 1)1*19.47 ]/5 = (3.14 - 19.47)/5 = - 16.33/5 = - 3.2660,

If n = 2, then x = [ 2*π + (- 1)2*19.47 ]/5 = (6.28 + 19.47)/5 = 25.75/5 = 5.150,

If n = 3, then x = [ 3*π + (- 1)3*19.47 ]/5 = (9.42 - 19.47)/5 = - 10.05/5 = - 2.010,

If n = 4, then x = [ 4*π + (- 1)4*19.47 ]/5 = (12.56 + 19.47)/5 = 32.03/5 = 6.4060,

If n = 5, then x = [ 5*π + (- 1)5*19.47 ]/5 = (15.7 - 19.47)/5 = - 3.77/5 = - 0.7540, and

If n = 6, then x = [ 6*π + (- 1)6*19.47 ]/5 = (18.84 +19.47)/5 = 38.31/5 = 7.6620.

Therefore, the four smallest positive solutions are x = 3.8940, x = 5.150, x = 6.40060, and x = 7.6620.