In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). sin ( x + 𝝅/4) + sin ( x - 𝝅/4 ) = 1

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1. Recognize that the equation involves the sum of two sine functions with arguments that differ by a constant. Recall the sine sum-to-product identity: \(\sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right)\).
2. Identify \(A = x + \frac{\pi}{4}\) and \(B = x - \frac{\pi}{4}\). Substitute these into the identity to rewrite the left side of the equation as \(2 \sin \left( \frac{(x + \frac{\pi}{4}) + (x - \frac{\pi}{4})}{2} \right) \cos \left( \frac{(x + \frac{\pi}{4}) - (x - \frac{\pi}{4})}{2} \right)\).
3. Simplify the arguments inside the sine and cosine functions: the sine argument becomes \(\frac{2x}{2} = x\), and the cosine argument becomes \(\frac{\frac{\pi}{4} + \frac{\pi}{4}}{2} = \frac{\pi}{4}\).
4. Rewrite the equation as \(2 \sin x \cos \frac{\pi}{4} = 1\). Since \(\cos \frac{\pi}{4}\) is a known value, express it explicitly to simplify the equation further.
5. Solve for \(\sin x\) by dividing both sides by \(2 \cos \frac{\pi}{4}\), then find all solutions for \(x\) in the interval \([0, 2\pi)\) where \(\sin x\) equals that value.

Key Concepts

Sum-to-Product Identities

Sum-to-product identities transform sums of sine or cosine functions into products, simplifying the solving process. For example, sin(A) + sin(B) = 2 sin((A+B)/2) cos((A−B)/2). Applying this helps rewrite the given equation into a more manageable form.

Solving Trigonometric Equations on a Given Interval

Solving trigonometric equations involves finding all angle solutions within a specified interval, here [0, 2π). This requires considering the periodicity of sine and cosine functions and checking all possible solutions that satisfy the equation within the interval.

Basic Properties of the Sine Function

Understanding the sine function’s range (−1 to 1), periodicity (2π), and symmetry is essential. These properties help determine valid solutions and simplify expressions, especially when combined with angle addition or subtraction inside the sine function.