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Topos Uranos
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Sum-to-product formulas and the harmonic form

Summary

This lesson derives from the formulas for the sine and the cosine of a sum and of a difference those that convert products of sines and cosines into sums and, conversely, sums into products, and it interprets the latter on the unit circle by means of the midpoint of a chord. It proves that every expression asin⁡x+bcos⁡xa \sin x + b \cos x can be written in the harmonic form Rsin⁡(x+φ)R \sin(x + \varphi), with R=a2+b2R = \sqrt{a^2 + b^2} and a unique angle φ\varphi in [0,2π)\ico{0}{2\pi}, derives from this its range in two ways, one of which is the Cauchy-Schwarz inequality, and concludes that the superposition of two oscillations of the same frequency is another oscillation. It explains beats as the product of a slow oscillation by a fast one and, by multiplying by a factor that makes the sum telescoping, computes in closed form the sums of sines and of cosines of angles in arithmetic progression.

Learning objectives

  1. Derive the product-to-sum and sum-to-product formulas from the sum and difference formulas, and interpret the latter on the unit circle.
  2. Write every expression asin⁡x+bcos⁡xa \sin x + b \cos x in harmonic form, prove the existence and the uniqueness of its amplitude and of its angle, and use it to find ranges, maxima and minima.
  3. Recognize the superposition of two oscillations of the same frequency as an oscillation, and describe beats by means of their envelope.
  4. Compute sums of sines and of cosines in arithmetic progression by means of telescoping sums, and apply these transformations to simplifying expressions and to proving identities.

From products to sums

In the lesson on the sum and difference formulas the following four equalities were proved, for all real numbers α\alpha and β\beta:

sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β,sin⁡(α−β)=sin⁡αcos⁡β−cos⁡αsin⁡β\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta, \qquad \sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta
cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β,cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta, \qquad \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta

The two formulas of each pair contain the same two products and differ only in the sign of the second one. If this is so, when they are added or subtracted one of the products cancels and the other is doubled: each product of a sine or a cosine by another is expressed as a half-sum or a half-difference of sines or of cosines. These formulas, called prosthaphaeresis formulas (from the Greek, “addition and subtraction”), served the astronomers of the late sixteenth century, Tycho Brahe among them, to reduce the multiplication of numbers with many digits to additions and subtractions, by consulting tables of sines, before John Napier published his tables of logarithms in 1614.

TheoremProduct to sum

For all real numbers α\alpha and β\beta, the product sin⁡αcos⁡β\sin \alpha \cos \beta is the half-sum of sin⁡(α+β)\sin(\alpha + \beta) and sin⁡(α−β)\sin(\alpha - \beta); the product cos⁡αsin⁡β\cos \alpha \sin \beta is the half-difference of those same sines; the product cos⁡αcos⁡β\cos \alpha \cos \beta is the half-sum of cos⁡(α−β)\cos(\alpha - \beta) and cos⁡(α+β)\cos(\alpha + \beta), and the product sin⁡αsin⁡β\sin \alpha \sin \beta is their half-difference.

sin⁡αcos⁡β=12(sin⁡(α+β)+sin⁡(α−β)),cos⁡αsin⁡β=12(sin⁡(α+β)−sin⁡(α−β))\sin \alpha \cos \beta = \frac{1}{2} \left( \sin(\alpha + \beta) + \sin(\alpha - \beta) \right), \qquad \cos \alpha \sin \beta = \frac{1}{2} \left( \sin(\alpha + \beta) - \sin(\alpha - \beta) \right)
cos⁡αcos⁡β=12(cos⁡(α−β)+cos⁡(α+β)),sin⁡αsin⁡β=12(cos⁡(α−β)−cos⁡(α+β))\cos \alpha \cos \beta = \frac{1}{2} \left( \cos(\alpha - \beta) + \cos(\alpha + \beta) \right), \qquad \sin \alpha \sin \beta = \frac{1}{2} \left( \cos(\alpha - \beta) - \cos(\alpha + \beta) \right)

Each formula results from adding or subtracting the two formulas of a pair. The figure illustrates them with α=2x\alpha = 2x and β=x\beta = x: the product, drawn thick, coincides with the half-sum or the half-difference of the two oscillations of frequencies 33 and 11, drawn thin.

Proof

  1. sin⁡(α+β)+sin⁡(α−β)=2sin⁡αcos⁡β→sin⁡αcos⁡β=12(sin⁡(α+β)+sin⁡(α−β))\sin(\alpha + \beta) + \sin(\alpha - \beta) = \resaltar{2 \sin \alpha \cos \beta} \rightarrow \dato{sc}{\sin \alpha \cos \beta = \frac{1}{2} \left( \sin(\alpha + \beta) + \sin(\alpha - \beta) \right)}

    We add the formulas for the sine of the sum and the sine of the difference: the terms cos⁡αsin⁡β\cos \alpha \sin \beta cancel. Dividing by 22 gives the first formula.

  2. sin⁡(α+β)−sin⁡(α−β)=2cos⁡αsin⁡β→cos⁡αsin⁡β=12(sin⁡(α+β)−sin⁡(α−β))\sin(\alpha + \beta) - \sin(\alpha - \beta) = \resaltar{2 \cos \alpha \sin \beta} \rightarrow \cos \alpha \sin \beta = \frac{1}{2} \left( \sin(\alpha + \beta) - \sin(\alpha - \beta) \right)

    If instead of adding them we subtract them, the terms sin⁡αcos⁡β\sin \alpha \cos \beta cancel.

  3. cos⁡(α−β)+cos⁡(α+β)=2cos⁡αcos⁡β→cos⁡αcos⁡β=12(cos⁡(α−β)+cos⁡(α+β))\cos(\alpha - \beta) + \cos(\alpha + \beta) = \resaltar{2 \cos \alpha \cos \beta} \rightarrow \dato{cc}{\cos \alpha \cos \beta = \frac{1}{2} \left( \cos(\alpha - \beta) + \cos(\alpha + \beta) \right)}

    With the cosine formulas we proceed in the same way. When they are added, the products of sines cancel.

  4. cos⁡(α−β)−cos⁡(α+β)=2sin⁡αsin⁡β→sin⁡αsin⁡β=12(cos⁡(α−β)−cos⁡(α+β))\cos(\alpha - \beta) - \cos(\alpha + \beta) = \resaltar{2 \sin \alpha \sin \beta} \rightarrow \dato{ss}{\sin \alpha \sin \beta = \frac{1}{2} \left( \cos(\alpha - \beta) - \cos(\alpha + \beta) \right)}

    When the formula for the cosine of the sum is subtracted from the formula for the cosine of the difference, the products of cosines cancel.

  5. sin⁡αcos⁡α=12sin⁡(2α),cos⁡2α=12(1+cos⁡(2α)),sin⁡2α=12(1−cos⁡(2α))\sin \alpha \cos \alpha = \frac{1}{2} \sin(2\alpha), \quad \cos^2 \alpha = \frac{1}{2} \left( 1 + \cos(2\alpha) \right), \quad \sin^2 \alpha = \frac{1}{2} \left( 1 - \cos(2\alpha) \right)

    With β=α\beta = \alpha, and since sin⁡0=0\sin 0 = 0 and cos⁡0=1\cos 0 = 1, we recover the double-angle formula for the sine and the power-reducing formulas of the lesson on the sum and difference formulas. In the figure, sin⁡2x\sin^2 x oscillates with period π\pi around 12\frac{1}{2}.

The formulas make it possible to compute products that do not reduce to special values factor by factor. For example, with α=5π12\alpha = \frac{5\pi}{12} and β=π12\beta = \frac{\pi}{12}, the sum of the angles is π2\frac{\pi}{2} and their difference is π3\frac{\pi}{3}:

cos⁡5π12cos⁡π12=12(cos⁡π3+cos⁡π2)=12(12+0)=14\cos \frac{5\pi}{12} \cos \frac{\pi}{12} = \frac{1}{2} \left( \cos \frac{\pi}{3} + \cos \frac{\pi}{2} \right) = \frac{1}{2} \left( \frac{1}{2} + 0 \right) = \frac{1}{4}

However, the main interest of these formulas lies elsewhere: a product of two oscillations, which is hard to picture, is the superposition of two oscillations whose frequencies are the sum and the difference of the frequencies of the factors. Thus, sin⁡(3x)cos⁡x=12(sin⁡(4x)+sin⁡(2x))\sin(3x) \cos x = \frac{1}{2} \left( \sin(4x) + \sin(2x) \right).