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11TH•Mathematics•Trigonometric Identities•10.7 Product–Sum Formulas

10.7 Product to Sum (and Sum to Product)

Convert products of sines and cosines to sums, and sums to products.

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10.7 Product to Sum (and Sum to Product)

From the addition formulas:

2sin⁡αcos⁡β=sin⁡(α+β)+sin⁡(α−β)2cos⁡αsin⁡β=sin⁡(α+β)−sin⁡(α−β)2cos⁡αcos⁡β=cos⁡(α+β)+cos⁡(α−β)−2sin⁡αsin⁡β=cos⁡(α+β)−cos⁡(α−β)\begin{align*} 2\sin\alpha\cos\beta &= \sin(\alpha+\beta) + \sin(\alpha-\beta) \\ 2\cos\alpha\sin\beta &= \sin(\alpha+\beta) - \sin(\alpha-\beta) \\ 2\cos\alpha\cos\beta &= \cos(\alpha+\beta) + \cos(\alpha-\beta) \\ -2\sin\alpha\sin\beta &= \cos(\alpha+\beta) - \cos(\alpha-\beta) \end{align*}

With P=α+βP = \alpha+\beta, Q=α−βQ = \alpha-\beta:

sin⁡P+sin⁡Q=2sin⁡P+Q2cos⁡P−Q2sin⁡P−sin⁡Q=2cos⁡P+Q2sin⁡P−Q2cos⁡P+cos⁡Q=2cos⁡P+Q2cos⁡P−Q2cos⁡P−cos⁡Q=−2sin⁡P+Q2sin⁡P−Q2\begin{align*} \sin P + \sin Q &= 2\sin\dfrac{P+Q}{2}\cos\dfrac{P-Q}{2} \\ \sin P - \sin Q &= 2\cos\dfrac{P+Q}{2}\sin\dfrac{P-Q}{2} \\ \cos P + \cos Q &= 2\cos\dfrac{P+Q}{2}\cos\dfrac{P-Q}{2} \\ \cos P - \cos Q &= -2\sin\dfrac{P+Q}{2}\sin\dfrac{P-Q}{2} \end{align*}

Example 14

2sin⁡7θcos⁡3θ=sin⁡10θ+sin⁡4θ2\sin 7\theta\cos 3\theta = \sin 10\theta + \sin 4\theta.

Example 15

sin⁡19∘cos⁡11∘+sin⁡71∘sin⁡11∘=12\sin 19^\circ\cos 11^\circ + \sin 71^\circ\sin 11^\circ = \dfrac{1}{2}

(Rewrite as half the sum of product-to-sum forms; simplify using cos⁡82∘=sin⁡8∘\cos 82^\circ = \sin 8^\circ.)

Example 16

sin⁡5x+sin⁡7x=2sin⁡6xcos⁡x\sin 5x + \sin 7x = 2\sin 6x\cos x.

Example 17

cos⁡θ+cos⁡3θ+cos⁡5θ+cos⁡7θ=4cos⁡θcos⁡2θcos⁡4θ\cos\theta + \cos 3\theta + \cos 5\theta + \cos 7\theta = 4\cos\theta\cos 2\theta\cos 4\theta.

Example 18

cos⁡20∘cos⁡40∘cos⁡80∘=18\cos 20^\circ\cos 40^\circ\cos 80^\circ = \dfrac{1}{8}.

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10.4 Double, Half and Triple Angles