How do you write cosine with the exponential function?

From Euler’s formula e^{iφ} = cos φ + i·sin φ it follows that cos φ = ½(e^{iφ} + e^{−iφ}). Geometrically this is the midpoint between e^{iφ} and its mirror image e^{−iφ} on the unit circle.

Why is there a 1/(2i) in the sine formula?

The midpoint of e^{iφ} and −e^{−iφ} lies on the imaginary axis at i·sin φ. Dividing by i turns it into the real number sin φ, hence sin φ = (e^{iφ} − e^{−iφ})/(2i).

Is the result a complex number then?

No. The intermediate steps are complex, but the mean of the two twins is real for every angle. The imaginary parts cancel.

What does the helix view show?

With time as a third axis, e^{iωt} becomes a helix and e^{−iωt} a helix turning the other way. Their mean is a flat cosine wave in the real plane.

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Euler-Zwillinge: Kosinus und Sinus aus e^{iφ}

About this app

e^{iφ} und e^{−iφ} laufen gegenläufig über den Einheitskreis; der Mittelpunkt ihrer Verbindung liegt immer auf der reellen Achse bei cos φ, mit −e^{−iφ} auf der imaginären Achse bei i·sin φ. In 3D werden die Zwillinge zu gegenläufigen Schrauben, ihr Mittel zur flachen Kosinuswelle.

With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.

The idea

Euler’s formula lets you write cosine and sine with the complex exponential: cos φ = ½(e^{iφ} + e^{−iφ}) and sin φ = (e^{iφ} − e^{−iφ})/(2i). It looks like a complicated way to say something simple. Geometrically it is an average: e^{iφ} lies on the unit circle, and its complex conjugate twin e^{−iφ} is its mirror image in the real axis. The midpoint of the segment between them always lies on the real axis, at exactly cos φ. These two formulas are the key to the complex Fourier series.

What the app shows

On the unit circle, e^{iφ} and e^{−iφ} turn in opposite directions. Their connecting segment and its midpoint are highlighted, and the curve over φ is drawn alongside: in cosine mode the midpoint on the real axis, in sine mode the midpoint of e^{iφ} and −e^{−iφ} on the imaginary axis. The helix view adds a time axis: the two pointers become helices turning in opposite directions, and their mean is a flat real cosine wave. The camera moves from the front view, where you see the circle, to the side view with the wave.

Try it yourself

Drag the angle φ from 0 to 2π and watch the midpoint swing back and forth on the real axis, although both pointers circle in the complex plane. Switch to sine and check why you have to divide by 2i: the midpoint then lies on the imaginary axis. Open the helix view and turn the camera until the two helices become a flat wave.

Frequently asked questions

How do you write cosine with the exponential function?: From Euler’s formula e^{iφ} = cos φ + i·sin φ it follows that cos φ = ½(e^{iφ} + e^{−iφ}). Geometrically this is the midpoint between e^{iφ} and its mirror image e^{−iφ} on the unit circle.
Why is there a 1/(2i) in the sine formula?: The midpoint of e^{iφ} and −e^{−iφ} lies on the imaginary axis at i·sin φ. Dividing by i turns it into the real number sin φ, hence sin φ = (e^{iφ} − e^{−iφ})/(2i).
Is the result a complex number then?: No. The intermediate steps are complex, but the mean of the two twins is real for every angle. The imaginary parts cancel.
What does the helix view show?: With time as a third axis, e^{iωt} becomes a helix and e^{−iωt} a helix turning the other way. Their mean is a flat cosine wave in the real plane.

Subject: Fourier analysis

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