Trigonify

Topics · 4 identities

Sum to Product identities

Turning sums and differences of sines and cosines into products.

Sum to product formula: sin A + sin B

sin⁡A+sin⁡B=2sin⁡A+B2cos⁡A−B2\sin A + \sin B = 2 \sin\tfrac{A+B}{2} \cos\tfrac{A-B}{2}

sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2). Going from a sum to a product is what lets you solve equations by factoring: a product is zero when either factor is, so the whole solution set falls out at once.

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Sum to product formula: sin A − sin B

sin⁡A−sin⁡B=2cos⁡A+B2sin⁡A−B2\sin A - \sin B = 2 \cos\tfrac{A+B}{2} \sin\tfrac{A-B}{2}

sin A − sin B = 2 cos((A + B)/2) sin((A − B)/2). The sine and cosine swap places compared with the sum version - the half-sum takes the cosine and the half-difference takes the sine.

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Sum to product formula: cos A + cos B

cos⁡A+cos⁡B=2cos⁡A+B2cos⁡A−B2\cos A + \cos B = 2 \cos\tfrac{A+B}{2} \cos\tfrac{A-B}{2}

cos A + cos B = 2 cos((A + B)/2) cos((A − B)/2). Two cosines added give a product of two cosines, with no leading minus - which is what distinguishes it from the difference version.

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Sum to product formula: cos A − cos B

cos⁡A−cos⁡B=−2sin⁡A+B2sin⁡A−B2\cos A - \cos B = -2 \sin\tfrac{A+B}{2} \sin\tfrac{A-B}{2}

cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2). The leading minus sign is the one people drop: a difference of cosines becomes a negative product of sines, not a positive one.

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