Ohio Assessments for Educators (OAE) Mathematics Practice Exam

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What is the result of sinαcosβ using the sum-to-product identities?

1/2(sin(α+β)-sin(α-β))
1/2(sin(α+β)+sin(α-β))
sin(α)cos(β)
sin(α+β)

The correct answer is: 1/2(sin(α+β)+sin(α-β))

Using the sum-to-product identities, the expression sin(α)cos(β) can be transformed. The appropriate identity states that the product of sine and cosine can be expressed as: sin(α)cos(β) = 1/2(sin(α + β) + sin(α - β)). This identity effectively combines the sine and cosine functions into a sum of sine functions, which helps in simplifying various problems in trigonometry. The "1/2" factor is crucial, as it ensures that the equation remains balanced and represents the correct values of the sine functions added. Therefore, the result of sin(α)cos(β) using the sum-to-product identities is indeed represented by the transformation into this specific form with the proper sine expressions added. This shows how versatile trigonometric identities can be in expressing relationships between angles in a different way, which is particularly useful in higher-level mathematics and applications.