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

Verified step by step guidance

1. Start by isolating the cotangent term in the equation: 2 \(\cot\) x + 1 = -1. Subtract 1 from both sides to get 2 \(\cot\) x = -2.
2. Divide both sides of the equation by 2 to solve for \(\cot\) x: \(\cot\) x = \(\frac{-2}{2}\) = -1.
3. Recall that \(\cot\) x = \(\frac{\cos x}{\sin x}\). The equation \(\cot\) x = -1 means \(\frac{\cos x}{\sin x}\) = -1, or equivalently, \(\cos\) x = -\(\sin\) x.
4. Rewrite the equation as \(\cos\) x + \(\sin\) x = 0. To find exact solutions, consider the unit circle or use the identity \(\tan\) x = \(\frac{\sin x}{\cos x}\). Since \(\cot\) x = -1, then \(\tan\) x = -1.
5. Find all values of x in the interval [0, 2\(\pi\)) where \(\tan\) x = -1. These correspond to angles where the tangent function equals -1, typically in the second and fourth quadrants.

Key Concepts

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 domain restrictions is essential for solving equations involving cot x.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angles within the given interval that satisfy the equation. This often requires algebraic manipulation and using inverse trigonometric functions, while considering the function's periodicity to find all solutions.

Interval Notation and Solution Sets

When solving trigonometric equations, solutions are often restricted to a specific interval, such as [0, 2π). Understanding how to express solutions within this interval and how to handle periodicity ensures that all valid solutions are identified and correctly reported.