from {0,2pie} what are the solutions of tan x/4= square root of 3/3
To find the solutions of the equation tan(x/4) = √3/3 in the interval [0, 2π], we need to follow these steps:
1. Start by isolating x/4. Multiply both sides of the equation by 4 to get rid of the fraction:
tan(x/4) * 4 = (√3/3) * 4
tan(x) = 4√3/3
2. Next, take the inverse tangent (or arctan) of both sides of the equation to cancel out the tangent function:
arctan(tan(x)) = arctan(4√3/3)
x = arctan(4√3/3)
3. Now we have the value of x in radians. To find the solutions within the given interval [0, 2π], we need to consider the periodic nature of the tangent function.
4. The tangent function has a period of π, which means the values repeat every π radians. Therefore, we can find the general solutions by adding or subtracting π to the value of x we obtained.
x = arctan(4√3/3) + kπ, where k is an integer
5. Finally, we need to check which solutions of x fall within the interval [0, 2π]. Substitute each value of k and see if the resulting angles fall within that range. Round the angles to the desired precision.
And that's how you can find the solutions of the equation tan(x/4) = √3/3 in the interval [0, 2π].