Sine, cosine and tangent

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

You learn sine, cosine and tangent in the right-angled triangle. They connect angles and side lengths.

The video

In the video a right-angled triangle is named from the angle \(\alpha\). A smaller similar triangle shows: the ratios only depend on the angle. At the end you calculate the height of a tower.

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

The most important points

The sides

From the angle \(\alpha\): hypotenuse \(H\) (die Hypotenuse, opposite the right angle), opposite side \(G\) (die Gegenkathete, opposite \(\alpha\)), adjacent side \(A\) (die Ankathete, next to \(\alpha\)).

Hypotenuse, opposite side and adjacent side.

Sine, cosine, tangent

\(\sin\alpha = \dfrac{G}{H}\), \(\cos\alpha = \dfrac{A}{H}\), \(\tan\alpha = \dfrac{G}{A}\). The values only depend on the angle.

G = 4.5, A = 6, H = 7.5: \sin\alpha = 0.6, \cos\alpha = 0.8, \tan\alpha = 0.75. — \(G = 4.5\), \(A = 6\), \(H = 7.5\): \(\sin\alpha = 0.6\), \(\cos\alpha = 0.8\), \(\tan\alpha = 0.75\).

In GeoGebra write the degree sign: \(\sin(30^\circ) = 0.5\). Find the angle with \(\texttt{asind}\), \(\texttt{acosd}\), \(\texttt{atand}\).

Mistake: \sin(30) without the degree sign calculates in radians and gives -0.99. — Mistake: \(\sin(30)\) without the degree sign calculates in radians and gives \(-0.99\).

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)

Show the worksheet solutions

Try it yourself

Task

\(G = 3\) cm and \(H = 6\) cm. How big is \(\alpha\)?

Show solution

\(\sin\alpha = 3 : 6 = 0.5\), so \(\alpha = \texttt{asind}(0.5) = 30^\circ\).

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: Sinus, Kosinus, Tangens (Gegenkathete, Ankathete, Hypotenuse, GeoGebra mit Gradzeichen). 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.