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 by using the Pythagorean identity: \( \sin^2 x + \cos^2 x = 1 \).
2. Square both sides of the given equation: \( (\sin x + \cos x)^2 = 1^2 \).
3. Expand the left side: \( \sin^2 x + 2\sin x \cos x + \cos^2 x = 1 \).
4. Substitute the Pythagorean identity into the equation: \( 1 + 2\sin x \cos x = 1 \).
5. Simplify the equation to find \( \sin x \cos x = 0 \), and solve for \( x \) in the interval \([0, 2\pi)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Common identities include the Pythagorean identity, reciprocal identities, and co-function identities. These identities are essential for simplifying trigonometric expressions and solving equations, as they allow us to rewrite functions in different forms.

Sine and Cosine Functions

The sine and cosine functions are fundamental trigonometric functions that relate the angles of a right triangle to the ratios of its sides. The sine function represents the ratio of the opposite side to the hypotenuse, while the cosine function represents the ratio of the adjacent side to the hypotenuse. Understanding their properties, such as their ranges and periodicity, is crucial for solving trigonometric equations.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the angles that satisfy a given trigonometric equation. This often requires using identities to manipulate the equation into a more solvable form. Solutions are typically found within a specified interval, such as [0, 2π), and may involve multiple solutions due to the periodic nature of trigonometric functions.