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

Verified step by step guidance

1. 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.
2. Recall that \( \cos \theta = -1 \) occurs at specific angles. Specifically, \( \cos \theta = -1 \) when \( \theta = \pi + 2k\pi \), where \( k \) is any integer.
3. Set \( 2x = \pi + 2k\pi \) to match the form where cosine equals \( -1 \). This gives the equation \( 2x = \pi + 2k\pi \).
4. Solve for \( x \) by dividing both sides by 2: \( x = \frac{\pi}{2} + k\pi \).
5. 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.

Key Concepts

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.

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.

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.