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 75° + sin 15°

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Recognize that the expression involves the sum of two sine functions: \(\sin 75^\circ + \sin 15^\circ\).

Recall the sum-to-product identity for sine functions: \(\sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right)\).

Apply the identity by setting \(A = 75^\circ\) and \(B = 15^\circ\), then calculate the averages: \(\frac{A+B}{2} = \frac{75^\circ + 15^\circ}{2}\) and \(\frac{A-B}{2} = \frac{75^\circ - 15^\circ}{2}\).

Rewrite the original sum as a product using the identity: \(2 \sin \left( \frac{75^\circ + 15^\circ}{2} \right) \cos \left( \frac{75^\circ - 15^\circ}{2} \right)\).

If required, evaluate the sine and cosine values at these angles to find the exact value of the product.

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주요 개념

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Sum-to-Product Formulas

Sum-to-product formulas convert sums or differences of sine or cosine functions into products, simplifying expressions. For sine, the formula sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2) is used to rewrite sums as products, aiding in evaluation and further manipulation.

Exact Values of Special Angles

Certain angles like 15°, 30°, 45°, 60°, and 75° have known exact sine and cosine values derived from geometric constructions or half-angle formulas. Knowing these values allows precise calculation of trigonometric expressions without approximations.

Angle Addition and Subtraction

Understanding how to add and subtract angles is essential when applying sum-to-product formulas, as these require computing (A+B)/2 and (A−B)/2. This skill ensures correct substitution and simplification of trigonometric expressions.