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5.4. Product-to-Sum and Sum-to-Product Formulas

Another class of useful trigonometric formulas are the **product-to-sum****and*** sum-to-product formulas*. Product-to-sum formulas are formulas for products of sine and cosine with different arguments, namely sin θ sin φ, sin θ cos φ, and cos θ cos φ. It is easy to derive these formulas from our sum and difference formulas for sine and cosine, namely Equations (5.2.1), (5.2.2), (5.2.6) and (5.2.7). Adding Equations (5.2.1) and (5.2.6), for instance, we obtain sin(θ + φ) + sin(θ - φ) = 2 sin θ cos φ, whence we have

**(5.4.1)**sin θ cos φ = [sin(θ + φ) + sin(θ - φ)] / 2

Similarly, adding Equations (5.2.2) and (5.2.7) and dividing the result by 2, we obtain

**(5.4.2)**cos θ cos φ = [cos(θ + φ) + cos(θ - φ)] / 2

Finally, subtracting Equation (5.2.2) from (5.2.7) and dividing the result by 2 yields

**(5.4.3)**sin θ sin φ = [cos(θ - φ) - cos(θ + φ)] / 2

As in the case of the double-angle formulas, we obtain useful formulas from these by letting φ = θ. Equation (5.4.2) yields

**(5.4.4)**cos^{2}θ = (1 + cos 2θ) / 2

and (5.4.3) yields

**(5.4.5)**sin^{2}θ = (1 - cos 2θ) / 2

Note that (5.4.1) yields sin θ cos θ = (sin 2θ) / 2, which we already derived in another way in Section 5.3 as Equation (5.3.1), apart from a factor of 2.

Sum-to-product formulas express the sums sin α + sin β, cos α + cos β, and cos β - cos α as products of sines and/or cosines. Once again, these formulas are easy to derive. First we let α = θ + φ and β = θ - φ, which together imply θ = (α + β) / 2 and φ = (α - β) / 2. Upon making this change of variables on both sides of Equation (5.4.1), multiplying both sides by 2 and switching sides, we obtain the following sum-to-product formula for the sum of two sines:

**(5.4.6)**sin α + sin β = 2 sin((α + β) / 2) cos((α - β) / 2)

By similar manipulations of Equations (5.4.2) and (5.4.3) we obtain the following additiona sum-to-product formulas:

**(5.4.7)**cos α + cos β = 2 cos((α + β) / 2) cos((α - β) / 2)

**(5.4.8)**cos β - cos α = 2 sin((α + β) / 2) sin((α - β) / 2)

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