targas-edu.de
DE ☾ A+

The cosine function

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

On the unit circle the point \(P\) has a height and a width. The height is the sine, the width is the cosine (der Kosinus). You learn the graph and how it is related to the sine.
📄 Open worksheet (PDF, German)

The video

In the video you see the cosine on the circle and as a curve. Then you compare sine and cosine and see the Pythagorean theorem in the circle.

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

Contents of the video

  1. 0:00 Introduction: x-coordinate
  2. 0:22 The graph
  3. 0:40 Sine and cosine
  4. 1:01 The Pythagorean theorem
  5. 1:26 Careful, mistake!
  6. 1:42 Try it yourself

The most important points

Cosine on the unit circle

\(\cos\alpha\) is the \(x\)-coordinate of \(P\), the distance from the \(y\)-axis. \(\cos 0^\circ = 1\), \(\cos 90^\circ = 0\), \(\cos 180^\circ = -1\).

The cosine is the width of the point P.
The cosine is the width of the point \(P\).

The graph

The cosine curve starts at \(1\). Period \(2\pi\), values between \(-1\) and \(1\).

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

Sine and cosine

\(\cos x = \sin(x + \frac{\pi}{2})\): the cosine curve is the sine curve shifted to the left by \(\frac{\pi}{2}\). The graph is symmetric about the \(y\)-axis.

Sine and cosine compared.
Sine and cosine compared.

The Pythagorean theorem

On the unit circle \(\cos\alpha\) and \(\sin\alpha\) are the legs, the radius \(1\) is the hypotenuse. So \(\sin^2\alpha + \cos^2\alpha = 1\). Example: \(0{,}6^2 + 0{,}8^2 = 0{,}36 + 0{,}64 = 1\).

Careful: Do not mix them up: the sine is the height, the cosine is the width. At \(0^\circ\) we have \(\sin 0^\circ = 0\), but \(\cos 0^\circ = 1\).

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
Page 1/4
The worksheet – Page 2
Page 2/4
The worksheet – Page 3
Page 3/4
Show the worksheet solutions
The worksheet – Page 4
Page 4/4

Try it yourself

Task

Calculate \(\cos\pi\) and \(\cos\frac{\pi}{3}\). It holds that \(\sin x = 0{,}6\) in the first quarter. How big is \(\cos x\)?

Show solution

\(\cos\pi = -1\). \(\cos\frac{\pi}{3} = \frac{1}{2}\). \(\cos^2 x = 1 - 0{,}6^2 = 0{,}64\), so \(\cos x = 0{,}8\).

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: die Kosinusfunktion (Einheitskreis, Graph, Zusammenhang mit dem Sinus). 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.