An equation is given. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.)
Find the solutions in the interval:
√3tan(3θ)-1 Interval [0,2pi)
To find the solutions in the interval [0, 2π) for the equation √3tan(3θ) - 1 = 0, we need to solve for θ.
Step 1: Start with the given equation:
√3tan(3θ) - 1 = 0
Step 2: Add 1 to both sides of the equation:
√3tan(3θ) = 1
Step 3: Take the square root of both sides of the equation:
tan(3θ) = 1/√3
Step 4: Find the reference angle:
The reference angle is the angle whose tangent value is 1/√3. The tangent function is positive in both the first and third quadrants.
Therefore, the reference angle is π/6 (30 degrees).
Step 5: Set up the equations for the solutions:
To find the solutions in the interval [0, 2π), we need to consider both the first and third quadrants.
The general solutions for tan(3θ) = 1/√3 are:
3θ = π/6 + kπ (First quadrant)
3θ = π + π/6 + kπ (Third quadrant)
Step 6: Solve for θ:
For the first quadrant:
3θ = π/6 + kπ
Divide both sides by 3:
θ = (π/6 + kπ)/3
For the third quadrant:
3θ = π + π/6 + kπ
Divide both sides by 3:
θ = (π + π/6 + kπ)/3
Step 7: Calculate the solutions in the interval [0, 2π):
To find the solutions in the given interval, substitute values for k that yield angles within the interval [0, 2π).
For the first equation, when k = 0:
θ = (π/6 + 0π)/3 = π/18
For the second equation, when k = 0:
θ = (π + π/6 + 0π)/3 = 7π/18
Since these solutions are both within the interval [0, 2π), we have found the solutions:
θ = π/18, 7π/18
Therefore, the solutions in the interval [0, 2π) for the equation √3tan(3θ) - 1 = 0 are π/18 and 7π/18.
To find the solutions to the equation √3tan(3θ) - 1 in the given interval [0, 2π), you can follow these steps:
Step 1: Solve the equation √3tan(3θ) - 1 = 0.
Start by adding 1 to both sides of the equation: √3tan(3θ) = 1.
Step 2: Isolate the tangent function.
Divide both sides by √3: tan(3θ) = 1/√3.
Step 3: Find the reference angle.
The reference angle is the angle formed between the x-axis and the line representing the tangent function. In this case, the reference angle is π/6.
Step 4: Determine the possible values for 3θ.
Since the tangent function is positive in the first and third quadrants, the equation tan(3θ) = 1/√3 has two sets of solutions. We'll use the reference angle to find the values for 3θ in each set.
In the first quadrant (0 ≤ 3θ < 2π):
3θ = π/6 + 2πk, where k is any integer from 0 to 2π divided by π/6.
Simplify the expression for k: k = 0, 1, 2, 3, 4, 5, 6.
In the third quadrant (π ≤ 3θ < 2π):
3θ = π + π/6 + 2πk, where k is any integer from 0 to 2π divided by π/6.
Simplify the expression for k: k = 5, 6, 7, 8, 9, 10, 11.
Step 5: Solve for θ.
Divide each value of 3θ by 3 to find the corresponding values of θ.
For the first quadrant solutions:
θ = (π/6)/3 + (2π/3)k, where k is 0, 1, 2, 3, 4, 5, 6.
For the third quadrant solutions:
θ = (π + π/6)/3 + (2π/3)k, where k is 5, 6, 7, 8, 9, 10 11.
Step 6: Check the interval.
Make sure that all the solutions for θ fall within the given interval [0, 2π). If any solutions are outside this interval, they should be excluded.
So, the solutions for θ in the interval [0, 2π) for the equation √3tan(3θ) - 1 = 0 are:
θ = (π/6)/3 + (2π/3)k, where k = 0, 1, 2, 3, 4, 5.
θ = (π + π/6)/3 + (2π/3)k, where k = 5, 6, 7, 8, 9, 10.
Note: Make sure to round the terms to three decimal places where appropriate.
Dhejen
I'm assuming you meant
√3tan(3θ)-1 = 0
√3tan(3θ) =1
tan(3θ) = 1/√3
3θ = π/6 + kπ
θ = π/18 + kπ/3
I'll let you list all 6 solutions