Be sure that you've familiarized yourself with the second set of formulas presented in this section by working C5–C8 in the Concept and Vocabulary Check. In Exercises 9–22, express each sum or difference as a product. If possible, find this product's exact value. sin 7x ﹣ sin 3x

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Recall the sine difference identity for expressing the difference of sines as a product: $\sin A - \sin B = 2 \cos \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right)$.

Identify $A$ and $B$ in the given expression: here, $A = 7x$ and $B = 3x$.

Apply the formula by substituting $A$ and $B$: $\sin 7x - \sin 3x = 2 \cos \left( \frac{7x + 3x}{2} \right) \sin \left( \frac{7x - 3x}{2} \right)$.

Simplify the arguments inside the cosine and sine functions: $\cos \left( \frac{10x}{2} \right) = \cos 5x$ and $\sin \left( \frac{4x}{2} \right) = \sin 2x$.

Write the final product form: $\sin 7x - \sin 3x = 2 \cos 5x \sin 2x$. If needed, evaluate the exact value by substituting a specific value for $x$.

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Sum-to-Product Formulas

Sum-to-product formulas transform sums or differences of sine or cosine functions into products of trigonometric functions. For example, the difference of sines can be expressed as 2 cos((A+B)/2) sin((A−B)/2). This simplifies complex expressions and is essential for rewriting sin 7x − sin 3x as a product.

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Verifying Identities with Sum and Difference Formulas

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They allow the manipulation and simplification of expressions, such as converting sums or differences into products, which is crucial for solving problems like sin 7x − sin 3x.

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Fundamental Trigonometric Identities

Exact Values of Trigonometric Functions

Exact values refer to the precise values of trigonometric functions at specific angles, often expressed in terms of square roots and fractions. After expressing sin 7x − sin 3x as a product, finding the exact value involves evaluating these functions at given angles, which is important for completing the problem.

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