Ch. 3 - Trigonometric Identities and Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problem 13

Chapter 3, Problem 13

Find all solutions of each equation. tan x = 1

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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