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Calculating sides and angles

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

You calculate sides and angles in a right-angled triangle with sine, cosine and tangent.
📄 Open worksheet (PDF, German)

The video

In the video you first find the right formula. Then you calculate a side, an angle and the slope of a ramp.

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

Contents of the video

  1. 0:00 Introduction
  2. 0:24 Which formula?
  3. 0:44 Calculating a side
  4. 1:28 Calculating an angle
  5. 1:55 Application: the ramp
  6. 2:24 Careful, mistake!
  7. 2:58 Try it yourself

The most important points

Which formula?

\(G\) and \(H\): \(\sin\alpha = G : H\). \(A\) and \(H\): \(\cos\alpha = A : H\). \(G\) and \(A\): \(\tan\alpha = G : A\).

Calculating a side: G = H · \sin\alpha with \alpha = 35^\circ, H = 8 gives 4.59.
Calculating a side: \(G = H \cdot \sin\alpha\) with \(\alpha = 35^\circ\), \(H = 8\) gives \(4.59\).

Calculating an angle

Work out the ratio and use the inverse: \(\tan\alpha = 3 : 5 = 0.6\), so \(\alpha = \texttt{atand}(0.6) \approx 31^\circ\).

The ramp: \sin\alpha = 1.2 : 6 = 0.2, so \alpha \approx 11.5^\circ.
The ramp: \(\sin\alpha = 1.2 : 6 = 0.2\), so \(\alpha \approx 11.5^\circ\).
Mistake: A = G · \tan\alpha. Correct is A = G : \tan\alpha.
Mistake: \(A = G \cdot \tan\alpha\). Correct is \(A = G : \tan\alpha\).

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

A ladder (\(4\) m) leans against a wall at \(70^\circ\). How high does it reach?

Show solution

\(G = 4 \cdot \sin(70^\circ) \approx 3.76\) m.

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: Seiten und Winkel im rechtwinkligen Dreieck berechnen (Sinus, Kosinus, Tangens, asind, atand). 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.