Sum-to-Product Identities

Section 7.5 Sum-to-Product Identities

In this section, we will be made aware of the identities to express products of trigonometric functions as sums and sums of trigonometric functions as products. This is only intended to be a reference section since PCC doesn’t formally cover these topics in the course. You should know that the sum-to-product and product-to-sum identities exist when you might need to use them.

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Subsection Textbook Reference

This relates to content in §9.4 of Algebra and Trigonometry 2e.

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Subsection Definitions

Definition 7.5.1. Product-to-Sum Identities.

\(\displaystyle \cos(\alpha)\cos(\beta)=\frac{1}{2}\left[\cos(\alpha-\beta)+\cos(\alpha + \beta)\right]\)

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\(\displaystyle \sin(\alpha)\cos(\beta)=\frac{1}{2}\left[\sin(\alpha+\beta)+\sin(\alpha - \beta)\right]\)

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\(\displaystyle \sin(\alpha)\sin(\beta)=\frac{1}{2}\left[\cos(\alpha-\beta)-\cos(\alpha + \beta)\right]\)

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\(\displaystyle \cos(\alpha)\sin(\beta)=\frac{1}{2}\left[\sin(\alpha+\beta)-\sin(\alpha - \beta)\right]\)

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Definition 7.5.2. Sum-to-Product Identities.

\(\displaystyle \sin(\alpha)+\sin(\beta)=2\sin\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right)\)

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\(\displaystyle \sin(\alpha)-\sin(\beta)=2\sin\left(\frac{\alpha - \beta}{2}\right) \cos\left(\frac{\alpha + \beta}{2}\right)\)

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\(\displaystyle \sin(\alpha)+\sin(\beta)=-2\sin\left(\frac{\alpha + \beta}{2}\right) \sin\left(\frac{\alpha - \beta}{2}\right)\)

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\(\displaystyle \sin(\alpha)+\sin(\beta)=2\cos\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right)\)

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An example of when one of these identities might be useful is if you were trying to solve the equation \(\sin{4\theta} - \sin(2\theta)\) by converting it into the form \(\sin(\theta)\cos(3\theta) = 0\text{.}\) Try to finish solving this equation in Exercises .

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