Ohio Assessments for Educators (OAE) Mathematics Practice Exam

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What is the expanded form of sinαsinβ?

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

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

The expanded form of sinαsinβ can be understood through the application of trigonometric identities, specifically the product-to-sum identities. The identity for the product of sine functions states that: sinαsinβ = 1/2(cos(α - β) - cos(α + β)). This identity allows you to express the product of two sine functions as a combination of cosine functions. The reasoning behind using cosine stems from the relationship between these trigonometric functions and their representation on the unit circle, where the angles can be added or subtracted together to represent different interactions. In the context of the provided choices, this identity clearly aligns with the first choice, as it directly reflects the correct formulation of the sine product. Understanding this identity is essential for solving problems involving transformations of trigonometric functions, particularly when simplifying expressions in calculus or advanced algebra topics where these functions frequently arise. The other choices represent different trigonometric identities or combinations that do not apply to the product of sinα and sinβ in the context of this question.