Solve each equation for exact solutions over the interval [0, 2π).
tan² x + 3 = 0

Verified step by step guidance

1. Start with the given equation: \(\tan^{2} x + 3 = 0\).
2. Isolate the \(\tan^{2} x\) term by subtracting 3 from both sides: \(\tan^{2} x = -3\).
3. Recall that \(\tan^{2} x\) represents the square of the tangent function, which is always greater than or equal to zero for all real \(x\) because squaring any real number cannot produce a negative result.
4. Since \(\tan^{2} x = -3\) implies a negative value, recognize that there are no real values of \(x\) for which this equation holds true within the interval \([0, 2\pi)\).
5. Conclude that the equation has no exact solutions in the interval \([0, 2\pi)\) because the square of tangent cannot be negative.

Key Concepts

Properties of the Tangent Function

The tangent function, tan(x), is defined as sin(x)/cos(x) and has a period of π. It is undefined where cos(x) = 0, and its values can be positive or negative. Understanding its behavior and domain is essential when solving equations involving tan²(x).

Solving Quadratic Equations in Trigonometric Functions

Equations like tan²(x) + 3 = 0 can be treated as quadratic in terms of tan(x). Solving involves isolating tan²(x) and checking for real solutions, considering that squares of real numbers are non-negative, which affects the existence of solutions.

Interval and Exact Solutions in Trigonometry

Finding exact solutions over [0, 2π) requires identifying all angles within one full rotation that satisfy the equation. Solutions must be expressed in exact form (like π/4), and understanding the periodicity and symmetry of trig functions helps determine all valid solutions.