consider the function:
f(x)=3sin(2(x-30))-1
determine the first three x-intercepts to the right of the origin, rounded to the nearest tenth of a degree.
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.
To determine the x-intercepts (or zeros) of a function, you need to find the values of x where the function equals zero. In this case, we need to find the first three x-intercepts to the right of the origin for the function f(x) = 3sin(2(x-30))-1.
To find the first x-intercept, we can start by setting the function equal to zero:
0 = 3sin(2(x-30))-1
To solve this equation, begin by isolating the sin term:
3sin(2(x-30)) = 1
Next, divide both sides by 3:
sin(2(x-30)) = 1/3
Now, to find the inverse sine of both sides, you should use an inverse trigonometric function such as arcsin (sin^(-1)):
2(x-30) = sin^(-1)(1/3)
To solve for x, divide both sides by 2 and add 30:
x - 30 = (1/2) * sin^(-1)(1/3) + 30
x = (1/2) * sin^(-1)(1/3) + 30
This gives you the x-coordinate of the first x-intercept to the right of the origin. To find the second and third x-intercepts, you can repeat the same process but use a different number of radians for the inverse sine term, as you are looking for the next two x-intercepts. Round the x-values to the nearest tenth of a degree to get your final answer.
That's better. For f(x) = 0, you need
sin(2x-60) = 1/3
Let sinθ = 1/3, so θ = 19.4°
2x - 60 = 19.4°
2x = 60 + 19.4°
x = 30 + 9.7°
Figuring that sin is positive in 1st and 2nd quadrants,
x = 180n + 120 - 9.7 = 180n + 110.3
for any integer n