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 6x + sin 2x

검증된 단계별 안내

Recognize that the expression is a sum of two sine functions: \(\sin 6x + \sin 2x\).

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

Identify \(A = 6x\) and \(B = 2x\), then substitute these into the formula to rewrite the sum as a product.

Calculate the average and half-difference of the angles: \(\frac{6x + 2x}{2} = 4x\) and \(\frac{6x - 2x}{2} = 2x\).

Express the original sum as \(2 \sin 4x \cos 2x\), which is the product form of the given sum.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Sum-to-Product Formulas

Sum-to-product formulas transform sums or differences of sine or cosine functions into products. For sine, the formula sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2) is used to rewrite the expression as a product, simplifying calculations and enabling exact value determination.

Angle Manipulation and Substitution

Understanding how to manipulate angles by addition, subtraction, and division is essential. In this problem, recognizing that 6x and 2x can be combined as (6x + 2x)/2 and (6x − 2x)/2 allows the use of sum-to-product formulas effectively.

Exact Values of Trigonometric Functions

After expressing the sum as a product, finding the exact value requires knowledge of special angles and their sine and cosine values. Familiarity with exact values for angles like 30°, 45°, 60°, or their radian equivalents helps in evaluating the product precisely.