De Moivre, Euler and trigonometry

Once the complex exponential is in hand, trigonometry stops being a list of formulas to memorise: it can be recomputed in three lines.

De Moivre's formula

Writing (e^(iθ))^n = e^(inθ) in trigonometric form:

(cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)

This is de Moivre's formula. It expresses cos(nθ) and sin(nθ) in terms of cos θ and sin θ. For n = 2:

(cos θ + i sin θ)² = cos²θ - sin²θ + 2i sin θ cos θ
                   = cos 2θ + i sin 2θ

Identifying real and imaginary parts — one complex equality is worth two real ones — recovers at a stroke:

cos 2θ = cos²θ - sin²θ           and           sin 2θ = 2 sin θ cos θ

The double-angle formulas no longer need memorising: they come back in one line. The same computation with n = 3 gives cos 3θ = 4cos³θ - 3cos θ.

Euler's formulas

In the other direction, adding and subtracting e^(iθ) and e^(-iθ):

e^(iθ)  = cos θ + i sin θ
e^(-iθ) = cos θ - i sin θ

             e^(iθ) + e^(-iθ)                    e^(iθ) - e^(-iθ)
   cos θ = --------------------        sin θ = --------------------
                    2                                 2i

These two formulas enable linearisation: turning a product or a power of trigonometric functions into a sum, which makes an integral computable.

              e^(iθ) + e^(-iθ)  ²      e^(2iθ) + 2 + e^(-2iθ)      1 + cos 2θ
cos²θ =     ( ------------------ )  =  ------------------------  = ------------
                      2                          4                      2

Integrating cos²θ was impossible as it stood; in the form (1 + cos 2θ)/2 it is immediate. This is the everyday use of these formulas in analysis and physics.

The dictionary

What you want to do                      The tool
---------------------------------------  ------------------------
expand cos(nθ) in powers of cos          de Moivre
linearise cos^n θ into a sum of cos(kθ)  Euler
compute a trigonometric integral         Euler, then integrate
sum cosines                              switch to exponentials

An example of a sum

How do you compute cos 0 + cos θ + cos 2θ + … + cos nθ? Switching to exponentials, the sum becomes the real part of a geometric series with ratio e^(iθ), whose sum is known. What looked like a pile of cosines reduces to a closed formula.

This is exactly the mechanism exploited by Fourier analysis and signal processing: decompose a phenomenon into a sum of complex exponentials, compute on the exponentials, then come back to the real world.

Summary

• De Moivre: (cos θ + i sin θ)^n = cos nθ + i sin nθ — for expanding.
• One complex equality gives two real ones (real and imaginary parts).
• Euler: cos θ = (e^(iθ) + e^(-iθ))/2 and sin θ = (e^(iθ) - e^(-iθ))/2i — for linearising.
• Linearisation makes powers of cosine and sine integrable.
• Sums of cosines become complex geometric series.