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. cos 75° ﹣ cos 15°

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Recall the cosine difference-to-product identity: \(\cos A - \cos B = -2 \sin \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right)\).

Identify the angles in the problem: \(A = 75^\circ\) and \(B = 15^\circ\).

Calculate the average of the angles: \(\frac{A + B}{2} = \frac{75^\circ + 15^\circ}{2} = 45^\circ\).

Calculate half the difference of the angles: \(\frac{A - B}{2} = \frac{75^\circ - 15^\circ}{2} = 30^\circ\).

Substitute these values into the identity to express \(\cos 75^\circ - \cos 15^\circ\) as a product: \(-2 \sin 45^\circ \sin 30^\circ\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Sum-to-Product Formulas

Sum-to-product formulas convert sums or differences of trigonometric functions into products, simplifying expressions and calculations. For cosine, the difference formula is cos A - cos B = -2 sin((A+B)/2) sin((A-B)/2). This transformation is essential for rewriting cos 75° - cos 15° as a product.

Exact Values of Special Angles

Certain angles like 15°, 30°, 45°, 60°, and 75° have known exact trigonometric values involving square roots. Knowing these values allows precise evaluation of expressions without approximations, which is crucial when finding the exact value of the product form of cos 75° - cos 15°.

Angle Addition and Subtraction

Understanding how to add and subtract angles is fundamental in applying sum-to-product formulas. Calculating (A+B)/2 and (A-B)/2 correctly ensures accurate transformation of the original expression. This skill helps in breaking down complex trigonometric expressions into simpler components.