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

Verified step by step guidance

1. Recognize that the left side of the equation, \( \sin 2x \cos x + \cos 2x \sin x \), is in the form of the sum-to-product identity: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).
2. Rewrite the left side using the identity: \( \sin(2x + x) = \sin 3x \).
3. The equation now becomes \( \sin 3x = \frac{\sqrt{2}}{2} \).
4. Determine the angles where \( \sin \theta = \frac{\sqrt{2}}{2} \) within the interval \([0, 2\pi)\). These angles are \( \frac{\pi}{4} \) and \( \frac{3\pi}{4} \).
5. Solve for \( x \) by setting \( 3x = \frac{\pi}{4} + 2k\pi \) and \( 3x = \frac{3\pi}{4} + 2k\pi \), where \( k \) is an integer, and then divide by 3 to find \( x \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identities, angle sum and difference identities, and double angle identities. Understanding these identities is crucial for simplifying trigonometric expressions and solving equations, as they allow for the transformation of one form of a trigonometric function into another.

Double Angle Formulas

Double angle formulas are specific trigonometric identities that express trigonometric functions of double angles (e.g., sin(2x), cos(2x)) in terms of single angles. For instance, sin(2x) = 2sin(x)cos(x) and cos(2x) = cos²(x) - sin²(x). These formulas are essential for solving equations involving angles that are multiples of a given angle, as they help to rewrite the equation in a more manageable form.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. In this context, the interval [0, 2π) indicates that the solutions to the equation should be found within the range starting from 0 up to, but not including, 2π. Understanding interval notation is important for determining the valid solutions of trigonometric equations, as trigonometric functions are periodic and can have multiple solutions within a given interval.