Trigonify

Topics · 3 identities

Product to Sum

Turning products of sines and cosines into sums.

Product to sum formula: sin α cos β

sin⁡αcos⁡β=12[sin⁡(α+β)+sin⁡(α−β)]\sin\alpha \cos\beta = \tfrac{1}{2}[\sin(\alpha+\beta) + \sin(\alpha-\beta)]

sin α cos β = ½[sin(α + β) + sin(α − β)]. Products of trig functions are awkward to integrate and to graph; sums are not. This is also the identity behind acoustic beats, where two close frequencies multiply into a slow envelope.

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Product to sum formula: cos α cos β

cos⁡αcos⁡β=12[cos⁡(α−β)+cos⁡(α+β)]\cos\alpha \cos\beta = \tfrac{1}{2}[\cos(\alpha-\beta) + \cos(\alpha+\beta)]

cos α cos β = ½[cos(α − β) + cos(α + β)]. Both terms carry a plus sign here, unlike the sine-times-sine version. It comes straight from adding the cosine sum and difference formulas.

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Product to sum formula: sin α sin β

sin⁡αsin⁡β=12[cos⁡(α−β)−cos⁡(α+β)]\sin\alpha \sin\beta = \tfrac{1}{2}[\cos(\alpha-\beta) - \cos(\alpha+\beta)]

sin α sin β = ½[cos(α − β) − cos(α + β)]. Note that a product of two sines turns into cosines, and that the second term is subtracted - the one sign in the product-to-sum family worth double-checking.

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