Solve each equation for x, where x is restricted to the given interval.
y = 1/2 cot 3 x , for x in [0, π/3]

Verified step by step guidance

1. Start by rewriting the equation: \( y = \frac{1}{2} \cot(3x) \).
2. Multiply both sides by 2 to isolate the cotangent function: \( 2y = \cot(3x) \).
3. Take the arccotangent (inverse cotangent) of both sides to solve for \( 3x \): \( 3x = \cot^{-1}(2y) \).
4. Divide both sides by 3 to solve for \( x \): \( x = \frac{1}{3} \cot^{-1}(2y) \).
5. Ensure that the solution for \( x \) falls within the interval \([0, \frac{\pi}{3}]\).

Key Concepts

Cotangent Function

The cotangent function, denoted as cot(x), is the reciprocal of the tangent function. It is defined as cot(x) = cos(x)/sin(x). Understanding cotangent is essential for solving equations involving this function, as it relates to the angles and sides of a right triangle, and its periodic nature affects the solutions within specified intervals.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the values of the variable that satisfy the equation. This often requires using identities, inverse functions, and understanding the periodicity of trigonometric functions. In this case, we need to isolate x and consider the specific interval [0, π/3] to find valid solutions.

Interval Restrictions

Interval restrictions define the range of values for which a solution is valid. In this problem, x is restricted to the interval [0, π/3], meaning we only consider solutions that fall within this range. This is crucial for determining the appropriate angles that satisfy the equation while adhering to the specified limits.