Solve the following equation on the interval [0, 2\pi):

\[
\text{cot}(3x) = \sqrt{3}
\]

The equation Cot(3x)=√3 has multiple solutions on the interval [0,2π) obtained by solving for the angles at which the tangent function equals 1/√3 and then dividing by 3 to account for the '3x'. ;

Answered by adipratapsingh12 | 2024-06-19

To solve the equation cot(3x) = √3 on the interval [0, 2π), we need to find the values of x that satisfy the equation. Cotangent is the reciprocal of tangent, so we can rewrite the equation as tan(3x) = 1/√3. The tangent function has a period of π, so we can find the solutions within the given interval by finding the values of x for which tan(3x) is equal to 1/√3.
We can start by finding the reference angle for 1/√3. Since the tangent function is positive in the first and third quadrants, we need to find the reference angle in the first quadrant. To do this, we can use the inverse tangent function, atan, to find the reference angle:
Reference angle = atan(1/√3) = π/6
Next, we can use the periodicity of the tangent function to find the values of x that satisfy the equation:
3x = π/6 + nπ, where n is an integer
x = (π/6 + nπ)/3, where n is an integer
Since the interval is specified as [0, 2π), we need to find the values of x that fall within this interval:
x = (π/6)/3, −(π/6)/3, (π/6 + π)/3, −(π/6 + π)/3
x = π/18, −π/18, 7π/18, −7π/18
Therefore, the solutions to the equation cot(3x) = √3 on the interval [0, 2π) are x = π/18, −π/18, 7π/18, and −7π/18.

Answered by IanMckellen | 2024-06-24

The solutions to the equation cot ( 3 x ) = 3 ​ on the interval [ 0 , 2 π ) are x = 18 π ​ , x = 18 7 π ​ , and x = 18 13 π ​ .
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Answered by IanMckellen | 2024-10-10