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Law of sines

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 sines (der Sinussatz). It holds in every triangle, also without a right angle.
📄 Open worksheet (PDF, German)

The video

In the video the law of sines comes from the height of the triangle. Then you calculate a side and an angle. You also see why there are sometimes two solutions.

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

Contents of the video

  1. 0:00 Introduction
  2. 0:25 The height
  3. 0:53 The law of sines
  4. 1:17 Calculating a side
  5. 1:44 Calculating an angle
  6. 2:14 Careful, mistake!
  7. 2:45 Try it yourself

The most important points

Law of sines

In every triangle: \(\dfrac{a}{\sin\alpha} = \dfrac{b}{\sin\beta} = \dfrac{c}{\sin\gamma}\).

The height h shows: h = b · \sin\alpha = a · \sin\beta.
The height \(h\) shows: \(h = b \cdot \sin\alpha = a \cdot \sin\beta\).

Application

A side: \(a = 5\), \(\alpha = 40^\circ\), \(\beta = 60^\circ\) gives \(b = 5 \cdot \sin(60^\circ) : \sin(40^\circ) \approx 6.74\). An angle: \(\sin\beta = b \cdot \sin\alpha : a\).

The law of sines in the triangle.
The law of sines in the triangle.
Careful: \texttt{asind} only gives the acute angle. 180^\circ - \beta can fit too.
Careful: \(\texttt{asind}\) only gives the acute angle. \(180^\circ - \beta\) can fit too.

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

\(a = 7\), \(\alpha = 50^\circ\) and \(\gamma = 70^\circ\). How long is \(c\)?

Show solution

\(c = 7 \cdot \sin(70^\circ) : \sin(50^\circ) \approx 8.59\).

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: Sinussatz (Seite durch Sinus des gegenüberliegenden Winkels, zwei Lösungen). 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.