Final answer:
To find the four smallest positive solutions of the equation csc(4x)-2 = 0, we solve for 4x using the values of π/6 and 5π/6 where sine equals 1/2 and use periodicity to obtain solutions. Dividing these by 4 gives x values of π/24, 5π/24, 13π/24, and 17π/24.
Step-by-step explanation:
To solve the equation csc(4x)-2 = 0 for the four smallest positive solutions, we must first isolate the cosecant function:
- csc(4x) = 2
- sin(4x) = 1/csc(4x) = 1/2
Now, we look for angles where the sine function has a value of 1/2. These occur at 4x = π/6 or 4x = 5π/6 (in the first and second quadrants, where sine is positive).
The sine function has a period of 2π, which means the function repeats every 2π radians. To find all solutions, we add whole multiples of 2π to the initial angles:
- 4x = π/6 + n∙(2π), n ∈ ℕ
- 4x = 5π/6 + n∙(2π), n ∈ ℕ
When we divide by 4 to solve for x, we get:
- x = π/24 + n∙(π/2), n ∈ ℕ
- x = 5π/24 + n∙(π/2), n ∈ ℕ
The four smallest positive solutions will be for n = 0 and n = 1 for each equation, resulting in:
- x = π/24
- x = 5π/24
- x = π/24 + π/2 = 13π/24
- x = 5π/24 + π/2 = 17π/24