Ch. 3 - Trigonometric Identities and Equations

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Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problema 3.RE.54

Capitolo 3, Problema 3.RE.54

In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. cos 2x = -1

Guida verificata passo dopo passo

Recognize that the equation is \( \cos 2x = -1 \). Our goal is to find all values of \( x \) in the interval \( [0, 2\pi) \) that satisfy this equation.

Recall that \( \cos \theta = -1 \) occurs at specific angles. Specifically, \( \cos \theta = -1 \) when \( \theta = \pi + 2k\pi \), where \( k \) is any integer.

Set \( 2x = \pi + 2k\pi \) to match the form where cosine equals \( -1 \). This gives the equation \( 2x = \pi + 2k\pi \).

Solve for \( x \) by dividing both sides by 2: \( x = \frac{\pi}{2} + k\pi \).

Find all values of \( x \) within the interval \( [0, 2\pi) \) by substituting integer values of \( k \) and checking which \( x \) values lie in the interval.

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Double-Angle Identity for Cosine

The double-angle identity expresses cos(2x) in terms of x, commonly as cos(2x) = 2cos²(x) - 1 or cos(2x) = 1 - 2sin²(x). This identity allows rewriting or solving equations involving cos(2x) by relating it to single-angle trigonometric functions.

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Double Angle Identities

Solving Trigonometric Equations on a Given Interval

Solving trig equations on [0, 2π) means finding all angle solutions within one full rotation. It requires considering the periodicity of trig functions and identifying all angles that satisfy the equation within the specified domain.

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How to Solve Linear Trigonometric Equations

Exact and Approximate Values of Trigonometric Functions

Exact values refer to well-known angles where trig functions have simple radical or fractional values (e.g., cos(π) = -1). Approximate values are numerical estimates rounded to a specified decimal place, used when exact values are not easily expressible.

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Introduction to Trigonometric Functions

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