Solve each equation for exact solutions over the interval [0, 2π).
(cot x―1) (√3 cot x + 1) = 0

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Recognize that the equation is factored as \((\cot x - 1)(\sqrt{3} \cot x + 1) = 0\). To find the solutions, set each factor equal to zero separately: \(\cot x - 1 = 0\) and \(\sqrt{3} \cot x + 1 = 0\).

Solve the first equation \(\cot x - 1 = 0\) for \(\cot x\): add 1 to both sides to get \(\cot x = 1\).

Solve the second equation \(\sqrt{3} \cot x + 1 = 0\) for \(\cot x\): subtract 1 and then divide by \(\sqrt{3}\) to get \(\cot x = -\frac{1}{\sqrt{3}}\).

Recall that \(\cot x = \frac{\cos x}{\sin x}\) and use the unit circle or known special angles to find all \(x\) in \([0, 2\pi)\) where \(\cot x = 1\) and \(\cot x = -\frac{1}{\sqrt{3}}\). Remember that \(\cot x\) is positive in the first and third quadrants and negative in the second and fourth quadrants.

List all exact solutions for \(x\) from both equations within the interval \([0, 2\pi)\), using the reference angles corresponding to the values of \(\cot x\) found.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cotangent Function and Its Properties

The cotangent function, cot(x), is the reciprocal of the tangent function, defined as cot(x) = cos(x)/sin(x). It is periodic with period π and undefined where sin(x) = 0. Understanding cotangent's behavior and values is essential for solving equations involving cot(x).

Solving Trigonometric Equations by Factoring

When a trigonometric equation is factored into a product of expressions equal to zero, each factor can be set to zero separately. This approach breaks the problem into simpler equations, allowing for the determination of all possible solutions within the given interval.

Finding Exact Solutions on the Interval [0, 2π)

Exact solutions refer to precise values of x that satisfy the equation, often expressed in terms of π. Restricting solutions to [0, 2π) means finding all angles within one full rotation of the unit circle, considering the periodicity and domain restrictions of the trigonometric functions involved.

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