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The tangent function

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

The tangent (der Tangens) is the ratio of sine and cosine. On the unit circle it is the slope of a line. You learn the graph with its vertical asymptotes.
📄 Open worksheet (PDF, German)

The video

In the video you see the tangent as a height on the line \(x = 1\). Then the graph appears with its branches.

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

Contents of the video

  1. 0:00 Introduction: slope
  2. 0:22 Tangent on the circle
  3. 0:44 The graph
  4. 1:05 Properties
  5. 1:24 Careful, mistake!
  6. 1:43 Try it yourself

The most important points

Tangent

\(\tan x = \dfrac{\sin x}{\cos x}\). On the unit circle this is the slope of the line through the origin and \(P\). At \(45^\circ\) sine and cosine are equal, so \(\tan 45^\circ = 1\).

The line \(x = 1\)

The ray through \(P\) meets the line \(x = 1\) at the height \(\tan\alpha\). \(\tan 30^\circ \approx 0{,}58\), \(\tan 60^\circ \approx 1{,}73\). As \(\alpha\) approaches \(90^\circ\), the height becomes as large as you like.

The tangent as a height on the line x = 1.
The tangent as a height on the line \(x = 1\).

The graph

The graph goes through the origin and rises. At \(\frac{\pi}{2}\) we have \(\cos x = 0\), so there is no value: a vertical asymptote (die Asymptote). Then a new branch starts from below.

The graph of y = \tan x with the asymptotes.
The graph of \(y = \tan x\) with the asymptotes.

Properties

Period \(\pi\). Zeros \(k \cdot \pi\). No values at \(\frac{\pi}{2} + k \cdot \pi\). Range: all real numbers. \(\tan\frac{\pi}{4} = 1\).

Properties of the tangent function.
Properties of the tangent function.

Careful: \(\tan 90^\circ\) does not exist: \(\cos 90^\circ = 0\), and you must not divide by zero. GeoGebra shows a question mark.

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

Calculate \(\tan\frac{\pi}{4}\) and \(\tan 0\). For which \(x\) between \(-\pi\) and \(\pi\) is there no tangent?

Show solution

\(\tan\frac{\pi}{4} = 1\). \(\tan 0 = 0\). There is no tangent at \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: die Tangensfunktion (Tangens = Sinus durch Kosinus, Graph, Asymptoten). 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.