Solve csc(5 x) - 3 = 0

Question

for the four smallest positive solutions?

Answer 1

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

Add 3 to each side.

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.