To determine the x-intercepts of the function f(x) = 3sin(2(x-30))-1, we need to find the values of x for which f(x) is equal to zero.
The x-intercept occurs when the function crosses the x-axis, which means f(x) = 0. So, we need to solve the equation:
0 = 3sin(2(x-30))-1
Let's solve this equation step-by-step:
Step 1: Add 1 to both sides of the equation:
1 = 3sin(2(x-30))
Step 2: Divide both sides by 3:
1/3 = sin(2(x-30))
Step 3: Take the inverse sine on both sides to isolate (2(x-30)):
sin^(-1)(1/3) = 2(x-30)
Step 4: Solve for x:
x - 30 = sin^(-1)(1/3)/2
Step 5: Add 30 to both sides of the equation:
x = sin^(-1)(1/3)/2 + 30
To find the first three x-intercepts to the right of the origin, we can substitute the values of n = 1, 2, and 3 into the equation to find x, where n represents the first, second, and third x-intercepts after the origin.
Let's calculate the x-intercepts:
For n = 1:
x = sin^(-1)(1/3)/2 + 30
x ≈ 9.3 degrees
For n = 2:
x = sin^(-1)(1/3)/2 + 30
x ≈ 63.8 degrees
For n = 3:
x = sin^(-1)(1/3)/2 + 30
x ≈ 118.3 degrees
Therefore, the first three x-intercepts to the right of the origin, rounded to the nearest tenth of a degree, are approximately 9.3 degrees, 63.8 degrees, and 118.3 degrees.