In Exercises 11–24, find all solutions of each equation.tan x = 1

Verified step by step guidance

1. Recognize that \( \tan x = 1 \) implies that the angle \( x \) is where the tangent function equals 1.
2. Recall that the tangent function, \( \tan x \), is equal to 1 at specific angles in the unit circle. These angles are \( \frac{\pi}{4} \) and \( \frac{5\pi}{4} \) within the interval \([0, 2\pi)\).
3. Understand that the tangent function has a period of \( \pi \), meaning it repeats every \( \pi \) radians.
4. To find all solutions, express \( x \) in terms of the general solution: \( x = \frac{\pi}{4} + n\pi \) and \( x = \frac{5\pi}{4} + n\pi \), where \( n \) is any integer.
5. Conclude that the solutions are all angles of the form \( x = \frac{\pi}{4} + n\pi \) and \( x = \frac{5\pi}{4} + n\pi \) for any integer \( n \).

Key Concepts

Tangent Function

The tangent function, denoted as tan(x), is a fundamental trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It can also be expressed as tan(x) = sin(x)/cos(x). The function is periodic with a period of π, meaning it repeats its values every π radians.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan or tan^(-1), are used to find angles when the value of a trigonometric function is known. For example, if tan(x) = 1, then x can be found using x = arctan(1). The principal value of arctan(1) is π/4, but due to the periodic nature of the tangent function, there are infinitely many solutions.

General Solution of Trigonometric Equations

The general solution of a trigonometric equation provides all possible angles that satisfy the equation. For tan(x) = 1, the solutions can be expressed as x = π/4 + nπ, where n is any integer. This accounts for the periodicity of the tangent function, allowing us to find all angles that yield the same tangent value.