In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). sin x + cos x = 1

Verified step by step guidance

1. Start with the given equation: \(\sin x + \cos x = 1\).
2. Square both sides of the equation to use the Pythagorean identity: \((\sin x + \cos x)^2 = 1^2\).
3. Expand the left side using the formula \((a + b)^2 = a^2 + 2ab + b^2\): \(\sin^2 x + 2 \sin x \cos x + \cos^2 x = 1\).
4. Use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) to simplify the equation: \(1 + 2 \sin x \cos x = 1\).
5. Subtract 1 from both sides to isolate the product term: \(2 \sin x \cos x = 0\), then solve for \(x\) by setting \(\sin x \cos x = 0\) and finding all solutions in \([0, 2\pi)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. In this problem, identities like the Pythagorean identity or the sum-to-product formulas help transform and simplify the equation sin x + cos x = 1 to a more solvable form.

Solving Trigonometric Equations

Solving trigonometric equations involves manipulating the equation using identities and algebraic techniques to isolate the variable. The goal is to find all angle values x within the given interval [0, 2π) that satisfy the equation, considering the periodic nature of sine and cosine functions.

Interval and Periodicity of Trigonometric Functions

The interval [0, 2π) represents one full cycle of sine and cosine functions. Understanding the periodicity ensures that all solutions within this range are found, and no extraneous solutions outside the interval are considered. This is crucial for correctly interpreting the solution set.