3tan(3x) = sqrt(3)
Solve the equation for exact solutions of the interval [0,2pi).
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To solve the given equation 3tan(3x) = sqrt(3) for the interval [0, 2pi), follow these steps:
Step 1: Divide both sides of the equation by 3 to isolate the tangent function:
tan(3x) = sqrt(3)/3
Step 2: Take the inverse tangent (arctan) of both sides to eliminate the tangent function:
arctan(tan(3x)) = arctan(sqrt(3)/3)
Step 3: Simplify the left side by using the property: arctan(tan(y)) = y, where -pi/2 < y < pi/2:
3x = arctan(sqrt(3)/3)
Step 4: Find the general solution for 3x by considering the periodicity of the tangent function:
3x = arctan(sqrt(3)/3) + nπ, where n is an integer
Step 5: Solve for x by dividing both sides of the equation by 3:
x = (arctan(sqrt(3)/3) + nπ)/3
Step 6: Determine the exact solutions within the given interval [0, 2pi):
Since n can be any integer, we need to find all possible values of x from n = 0 to n = 6 (inclusive) and substitute them into the equation to obtain the solutions within the interval [0, 2pi).
For n = 0:
x = (arctan(sqrt(3)/3))/3
For n = 1:
x = (arctan(sqrt(3)/3) + π)/3
For n = 2:
x = (arctan(sqrt(3)/3) + 2π)/3
For n = 3:
x = (arctan(sqrt(3)/3) + 3π)/3
For n = 4:
x = (arctan(sqrt(3)/3) + 4π)/3
For n = 5:
x = (arctan(sqrt(3)/3) + 5π)/3
For n = 6:
x = (arctan(sqrt(3)/3) + 6π)/3
These are all the exact solutions of the equation for the interval [0, 2pi).
To solve the equation 3tan(3x) = √3, we need to isolate the variable x. Let's break it down step by step:
Step 1: Rewrite the equation
Since tan(3x) is the only trigonometric function, we want to isolate it on one side of the equation. Divide both sides of the equation by 3: tan(3x) = √3/3.
Step 2: Take the inverse tangent
To eliminate the tangent function, take the inverse tangent (or arctan) of both sides of the equation: arctan(tan(3x)) = arctan(√3/3).
Step 3: Simplify the left side
Note that arctan(tan(θ)) = θ for θ ∈ [0, π), but when θ is outside of this interval, we need to adjust it. Since the given interval is [0, 2π), we need to adjust the solutions for θ ∈ (π, 2π). Thus, we can rewrite the equation as 3x = arctan(√3/3) + π.
Step 4: Solve for x
Now, we can solve for x by dividing both sides of the equation by 3: x = (arctan(√3/3) + π)/3.
Therefore, the exact solutions of the given equation in the interval [0, 2π) are x = (arctan(√3/3) + π)/3 and x = (arctan(√3/3) + π + 2π)/3.