Final answer:
To solve the trigonometric equation, invert the sine term to find the angles and account for the periodicity of the sine function to find the smallest positive solutions.
Step-by-step explanation:
To solve the equation 8sin(π/3x)=5, we start by isolating the sine term:
sin(π/3x) = 5/8
Next, we find the inverse sine (also known as arcsin) to find the initial angles that satisfy this equation. Because the sine function is periodic, we will have multiple solutions.
Let θ be the angle such that sin(θ) = 5/8. Then:
θ = arcsin(5/8)
We know that sine is positive in the first and second quadrants, so we will have one angle in each quadrant:
- θ1 = arcsin(5/8)
- θ2 = π - arcsin(5/8)
Furthermore, since the argument of the sine function includes a multiple of x (π/3x), we need to account for the periodicity of sine, which is 2π. So, our general solutions for x will be:
- x1 = 3/π (θ1 + 2nπ), where n is an integer
- x2 = 3/π (θ2 + 2nπ), where n is an integer
By considering the four smallest positive solutions for x, we will use n = 0, 1 for both θ1 and θ2, calculating the respective values of x1 and x2 for these n's. This will give us the distinct smallest positive solutions for x.
It is crucial to check if these solutions make sense within the context of the question and to see if they fulfill the original equation's requirements.