In Exercises 55–58, use a graph to solve each equation for -2π ≤ x ≤ 2π. tan x = -1

Verified step by step guidance

1. Recall that the equation to solve is \(\tan x = -1\) over the interval \(-2\pi \leq x \leq 2\pi\).
2. Understand that the tangent function has a period of \(\pi\), meaning its values repeat every \(\pi\) units along the x-axis.
3. Identify the reference angle where \(\tan x = 1\), which is \(\frac{\pi}{4}\), since \(\tan \frac{\pi}{4} = 1\).
4. Since we want \(\tan x = -1\), the solutions will be angles where the tangent is negative. Tangent is negative in the second and fourth quadrants, so the solutions correspond to angles \(x = -\frac{3\pi}{4} + k\pi\) and \(x = -\frac{7\pi}{4} + k\pi\) for integers \(k\).
5. Use the graph of \(y = \tan x\) to visually identify all points where the curve crosses the line \(y = -1\) within the interval \(-2\pi\) to \(2\pi\). These x-values are the solutions to the equation.

Key Concepts

Graph of the Tangent Function

The tangent function, tan(x), is periodic with period π and has vertical asymptotes where cos(x) = 0. Its graph repeats every π units and crosses zero at multiples of π. Understanding its shape and behavior helps identify solutions to equations like tan(x) = -1 within a given interval.

Solving Trigonometric Equations Using Graphs

Solving equations graphically involves plotting the function and identifying x-values where the function equals a given value. For tan(x) = -1, you find points on the tangent curve that intersect the horizontal line y = -1 within the specified domain, providing approximate or exact solutions.

Domain and Periodicity in Trigonometric Solutions

The domain restriction (-2π ≤ x ≤ 2π) limits the solutions to a specific interval. Since tan(x) has period π, solutions repeat every π units. Recognizing this periodicity allows you to find all solutions within the interval by adding integer multiples of π to a principal solution.