In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅).𝝅 𝝅sin ( x + ------ ) + sin ( x - ------ ) = 14 4

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1. Recognize that the equation involves the sum of two sine functions with phase shifts. Use the sum-to-product identities to simplify the expression: \( \sin(A) + \sin(B) = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \).
2. Identify \( A = x + \frac{\pi}{4} \) and \( B = x - \frac{\pi}{4} \). Calculate \( A + B = 2x \) and \( A - B = \frac{\pi}{2} \).
3. Substitute these into the sum-to-product identity: \( \sin(x + \frac{\pi}{4}) + \sin(x - \frac{\pi}{4}) = 2 \sin(x) \cos(\frac{\pi}{4}) \).
4. Since \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \), the equation becomes \( \sqrt{2} \sin(x) = 1 \).
5. Solve for \( \sin(x) \) by dividing both sides by \( \sqrt{2} \), and find the values of \( x \) in the interval \([0, 2\pi)\) that satisfy the equation.

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 identity, angle sum and difference identities, and double angle identities. These identities are essential for simplifying trigonometric expressions and solving equations, as they allow for the transformation of complex expressions into more manageable forms.

Sine Function Properties

The sine function is a periodic function that oscillates between -1 and 1, with a period of 2π. It is defined for all real numbers and can be represented on the unit circle, where the sine of an angle corresponds to the y-coordinate of the point on the circle. Understanding the properties of the sine function, including its behavior and symmetry, is crucial for solving equations involving sine.

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. Recognizing the significance of interval notation is important for determining valid solutions and ensuring they fall within the specified range.