The graph of arctan

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The graph of arctan

This animation illustrates the inverse to the tangent function.

The key ingredients of the picture are the vertical and horizontal axes, and the line through the middle of the circle, which form the sides of a right angled triangle. Assuming that the circle is of unit radius, the horizontal intercept is then equal to the tangent of the angle between the vertical axis and the line through the circle.

The line through the circle rotates at constant speed, which means that after suitable horizontal scaling, this procedure will trace out multiple copies of the graph of the inverse tangent function. The graph of the function can be isolated from this by taking any one of the completed connected pieces and translating it vertically until it passes through the origin.

The remarkable thing about the picture is that it bypasses the verbose description in the previous two paragraphs and explains it visually. In fact, I could have deleted my explanation above and tagged this as wordlessonwednesday.

This picture is one of the gifs featured in Lisa Winter‘s article 21 GIFs that explain Mathematical Concepts, which can be found at The article suggests that the gifs will “help you understand these concepts better than your teacher ever did”. This might be true for the animation reproduced here, which I think is one of the better ones.

Image source: 

(Author unknown; found via Dan Anderson on Twitter.)


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