Find all solutions of each equation. tan x = 1

검증된 단계별 안내

Recall that the equation \( \tan x = 1 \) means we are looking for all angles \( x \) where the tangent function equals 1.

Identify the principal solution by remembering that \( \tan x = 1 \) at \( x = \frac{\pi}{4} \) (or 45 degrees) in the first quadrant.

Since the tangent function has a period of \( \pi \), all solutions can be expressed as \( x = \frac{\pi}{4} + k\pi \), where \( k \) is any integer.

Write the general solution explicitly: \[ x = \frac{\pi}{4} + k\pi, \quad k \in \mathbb{Z} \]

This formula gives all angles \( x \) for which \( \tan x = 1 \), covering all possible solutions.

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주요 개념

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Definition and Properties of the Tangent Function

The tangent function, tan(x), is defined as the ratio of sine to cosine: tan(x) = sin(x)/cos(x). It is periodic with period π, meaning tan(x + π) = tan(x). Understanding its behavior and domain restrictions is essential for solving equations involving tan(x).

Solving Basic Trigonometric Equations

To solve equations like tan(x) = 1, identify the principal angle where the equation holds true, then use the function's periodicity to find all solutions. For tangent, solutions repeat every π radians, so general solutions are expressed as x = θ + nπ, where n is any integer.

Reference Angles and Quadrant Analysis

Reference angles help determine the exact solutions of trigonometric equations by relating angles to their acute counterparts. Since tan(x) = 1 at 45° (π/4 radians) and in the third quadrant where tangent is positive, recognizing these quadrants aids in finding all valid solutions.