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The sine function

This page was translated from German with AI. It has not been checked by a native speaker. Original page in German

A point moves around the unit circle. Its height is called the sine (der Sinus). If you plot the height against the angle, a wave appears: the graph of the sine function.
📄 Open worksheet (PDF, German)

The video

In the video the height of the point is carried over to the coordinate system step by step. Then you see how \(a\) and \(b\) change the wave.

Subtitles are available in ten languages. Choose them at the bottom right of the video.

Contents of the video

  1. 0:00 Introduction: unit circle
  2. 0:27 The graph
  3. 0:48 Properties
  4. 1:11 Amplitude and period
  5. 1:35 Careful, mistake!
  6. 1:57 Try it yourself

The most important points

Sine on the unit circle

\(\sin\alpha\) is the height (the \(y\)-coordinate) of the point \(P\). \(\sin 90^\circ = 1\), \(\sin 180^\circ = 0\), \(\sin 270^\circ = -1\).

The sine is the height of the point P.
The sine is the height of the point \(P\).

The graph

On the \(x\)-axis is the angle in radians, on the \(y\)-axis the sine. The points form the sine curve.

From the circle to the graph.
From the circle to the graph.

Properties

Period \(2\pi\) (die Periode). The values lie between \(-1\) and \(1\). Zeros: \(0\), \(\pi\), \(2\pi\), … Maximum \((\frac{\pi}{2}|1)\), minimum \((\frac{3\pi}{2}|-1)\).

Amplitude and period

\(y = a \cdot \sin(b \cdot x)\): \(a\) stretches in the \(y\)-direction (amplitude, die Amplitude), \(b\) changes the period to \(\frac{2\pi}{b}\). Example \(y = 3\sin(2x)\): amplitude \(3\), period \(\pi\).

\sin x, 2\sin x and \sin 2x compared.
\(\sin x\), \(2\sin x\) and \(\sin 2x\) compared.

Careful: In GeoGebra sin(90°) is \(1\). sin(90) without the degree sign calculates in radians and gives about \(0{,}89\). With \(\pi\) you leave out the degree sign, with degrees you write it.

The worksheet

Here you can see the worksheet for the video (in German). First it explains everything again, then there are practice tasks at three levels. You can also open it as a PDF and print it.

📄 Open worksheet (PDF, German)

The worksheet – Page 1
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The worksheet – Page 2
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The worksheet – Page 3
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Show the worksheet solutions
The worksheet – Page 4
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Try it yourself

Task

Calculate \(\sin\frac{\pi}{6}\) and \(\sin\pi\). What is the period of \(y = \sin(3x)\)?

Show solution

\(\sin\frac{\pi}{6} = \frac{1}{2}\). \(\sin\pi = 0\). The period is \(\frac{2\pi}{3}\).

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: die Sinusfunktion (Einheitskreis, Graph, Amplitude und Periode). Stelle mir nacheinander fünf kurze Aufgaben. Warte immer auf meine Antwort. Sag mir dann, ob sie richtig ist, und erkläre kurz, warum. Benutze einfache deutsche Sätze.

How this was made: The idea, structure and examples of the series are mine. I made them with the help of AI: a language model (LLM) built the slides of the video following my instructions. The voice was generated with Qwen3-TTS, modelled on my own voice.

The subtitles, the translations, the worksheet and this page were also made with the help of AI. The translations have not been checked by native speakers.

The videos, worksheets, texts and prompts in this series are licensed under CC BY 4.0: you may use, change and share them if you name the source.