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Law of cosines

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

You learn the law of cosines (der Kosinussatz). It holds in every triangle and generalises the theorem of Pythagoras.
📄 Open worksheet (PDF, German)

The video

In the video you see when you need the law of cosines: with two sides and the angle between them, or with three sides. Then you calculate a side and an angle.

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

Contents of the video

  1. 0:00 Introduction
  2. 0:21 The law of cosines
  3. 0:49 Pythagoras as a special case
  4. 1:09 Calculating a side
  5. 1:43 Calculating an angle
  6. 2:22 Careful, mistake!
  7. 2:48 Try it yourself

The most important points

Law of cosines

\(c^2 = a^2 + b^2 - 2ab \cdot \cos\gamma\). The side \(c\) lies opposite \(\gamma\). The same for \(a\) and \(b\).

The law of cosines in \triangle ABC.
The law of cosines in \(\triangle ABC\).

Special case and application

For \(\gamma = 90^\circ\) we have \(\cos 90^\circ = 0\): the theorem of Pythagoras remains. Side: \(a = 5\), \(b = 7\), \(\gamma = 60^\circ\) gives \(c = \sqrt{39} \approx 6.24\). Angle: \(\cos\gamma = (a^2 + b^2 - c^2) : 2ab\).

Calculating a side: c^2 = 39, c \approx 6.24.
Calculating a side: \(c^2 = 39\), \(c \approx 6.24\).
Mistake: forgetting the root (c = 39). Correct is c = \sqrt{39}.
Mistake: forgetting the root (\(c = 39\)). Correct is \(c = \sqrt{39}\).

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

Try it yourself

Task

\(a = 4\), \(b = 5\) and \(\gamma = 120^\circ\). How long is \(c\)?

Show solution

\(c^2 = 16 + 25 - 40 \cdot (-0.5) = 61\), so \(c \approx 7.81\).

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: Kosinussatz (zwei Seiten und Winkel dazwischen, drei Seiten, Pythagoras als Sonderfall). Give me five short tasks, one after the other. Always wait for my answer. Then tell me if it is right and explain briefly why. Use simple German sentences.

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.