In elementary algebra, transforming a polynomial into a product (factoring) is a resource of enormous practical importance:
$x^2 - 2x = x(x - 2)$ highlighting
$x^2 - 4 = (x + 2)(x - 2)$ difference of squares
$x^2 \pm 4x + 4 = (x \pm 2)^2$ perfect square trinomials
$x^3 + 8 = (x + 2)(x^2 - 2x + 4)$ sum of cubes
$x^3 - 8 = (x - 2)(x^2 + 2x + 4)$ difference of cubes
$x^3 \pm 3x^2 + 3x \pm 1 = (x \pm 1)^3$ perfect cubes

We will often apply these features to trigonometry, with transformations like $\operatorname{sen}^2 x - 2\operatorname{sen} x = \operatorname{sen} x(\operatorname{sen} x - 2)$ and $\operatorname{sen}^2 x - \cos^2 x = (\operatorname{sen} x + \cos x)(\operatorname{sen} x - \cos x)$. In addition to algebraic resources, trigonometry has formulas that allow you to complete a factorization — in the example above, also factor $\operatorname{sen} x + \cos x$ and $\operatorname{sen} x - \cos x$.

The story

Formulas that transform products into sums are called Werner formulas, named after the German priest and mathematician Johannes Werner (1468–1522), who used them in astronomy problems. At the end of the 16th century, astronomers such as Tycho Brahe and his assistant Paul Wittich realized that they allowed the exchange of multiplications of large numbers, such as the sines of tablets, for additions and subtractions, which were much faster: the method became known as prostapheresis (from the Greek “addition and subtraction”). Shortly afterwards, Napier’s logarithms would do the same job in an even simpler way.

Tycho Brahe’s prostapheresis: to multiply two numbers, look for them on a cosine table, add and subtract the arcs and take the average of two cosines.
  1. 1580
    Paul Wittich takes prostapheresis to the Tycho observatory in Hven.
  2. 1614
    Napier publishes the logarithms: multiplying becomes adding.
  3. c. 1622
    Oughtred invents the slide rule.
  4. 1972
    Pocket scientific calculators arrive.
  1. Nasce no castelo da família, na Escânia, então parte da Dinamarca.
  2. Estuda na universidade. Um eclipse do Sol, em 1560, o faz decidir-se pela astronomia.
  3. Perde parte do nariz num duelo de espadas com um primo e passa a usar uma prótese de metal.
  4. Constrói Uraniborg e depois Stjerneborg, observatórios com instrumentos gigantes. Em 1580, Paul Wittich lhe ensina a prostaférese.
  5. Matemático imperial de Rodolfo II. Contrata Johannes Kepler, que herdará as suas observações.
Os lugares de Tycho Brahe, do castelo da família à corte de Praga. Clique numa etapa ou use Percorrer a viagem. Map: © OpenStreetMap contributors

Sources: Tycho Brahe (MacTutor) · Werner (MacTutor) · Wittich (MacTutor).

In everyday life

  • Tune the guitar. Two strings almost in tune, at 440 Hz and 442 Hz, added together sound like $2\operatorname{sen}(2\pi \cdot 441\,t)\cos(2\pi \cdot 1\,t)$: a note of 441 Hz whose volume pulses 2 times per second. When the pulsation disappears, the strings are in tune.
  • Telephone keypad. Each key adds two tones: the 1 key, for example, adds 697 Hz and 1209 Hz. The control panel identifies the key separating the sum.
  • AM Radio. Multiplying the sound by the station’s waveform generates, using the product-in-sum formulas, “sum” and “difference” frequencies — the side bands that the radio tunes into.
Two strings almost in tune: the sum of the waves becomes a product, and the volume pulsates with the difference in frequencies.

Theory

Werner’s formulas

We know that

$$\cos(a + b) = \cos a\cos b - \operatorname{sen} a\operatorname{sen} b \;(1) \qquad \cos(a - b) = \cos a\cos b + \operatorname{sen} a\operatorname{sen} b \;(2)$$$$\operatorname{sen}(a + b) = \operatorname{sen} a\cos b + \operatorname{sen} b\cos a \;(3) \qquad \operatorname{sen}(a - b) = \operatorname{sen} a\cos b - \operatorname{sen} b\cos a \;(4)$$

Adding and subtracting these equalities two by two:

PropriedadeWerner's formulas

$$(1) + (2):\quad \cos(a + b) + \cos(a - b) = 2\cos a\cos b$$$$(1) - (2):\quad \cos(a + b) - \cos(a - b) = -2\operatorname{sen} a\operatorname{sen} b$$$$(3) + (4):\quad \operatorname{sen}(a + b) + \operatorname{sen}(a - b) = 2\operatorname{sen} a\cos b$$$$(3) - (4):\quad \operatorname{sen}(a + b) - \operatorname{sen}(a - b) = 2\operatorname{sen} b\cos a$$

Product transformation formulas

Using Werner’s formulas $a + b = p$ and $a - b = q$, therefore $a = \dfrac{p + q}{2}$ and $b = \dfrac{p - q}{2}$, we obtain:

TeoremaTransformation into product

$$\cos p + \cos q = 2\cos\frac{p + q}{2}\cos\frac{p - q}{2}$$$$\cos p - \cos q = -2\operatorname{sen}\frac{p + q}{2}\operatorname{sen}\frac{p - q}{2}$$$$\operatorname{sen} p + \operatorname{sen} q = 2\operatorname{sen}\frac{p + q}{2}\cos\frac{p - q}{2}$$$$\operatorname{sen} p - \operatorname{sen} q = 2\operatorname{sen}\frac{p - q}{2}\cos\frac{p + q}{2}$$

PropriedadeSum and difference of tangents

$$\operatorname{tg} p + \operatorname{tg} q = \frac{\operatorname{sen} p}{\cos p} + \frac{\operatorname{sen} q}{\cos q} = \frac{\operatorname{sen} p\cos q + \operatorname{sen} q\cos p}{\cos p\cos q} = \frac{\operatorname{sen}(p + q)}{\cos p\cos q}$$$$\operatorname{tg} p - \operatorname{tg} q = \frac{\operatorname{sen} p\cos q - \operatorname{sen} q\cos p}{\cos p\cos q} = \frac{\operatorname{sen}(p - q)}{\cos p\cos q}$$

ExemploFactoring sin x + cos x

As $\cos x = \operatorname{sen}\left(\frac{\pi}{2} - x\right)$:

$$\operatorname{sen} x + \cos x = \operatorname{sen} x + \operatorname{sen}\left(\frac{\pi}{2} - x\right) = 2\operatorname{sen}\frac{\pi}{4}\cos\left(x - \frac{\pi}{4}\right) = \sqrt{2}\cos\left(x - \frac{\pi}{4}\right).$$

This explains the graph of exercise 61: the sum of sine and cosine is a cosine of amplitude $\sqrt{2}$, offset from $\frac{\pi}{4}$.

Ative o JavaScript para ver esta figura interativa.

The exercises in this section are numbers 171 to 192 from chapter list.