Trigonometry in one picture

All six trigonometric functions, sine, cosine, tangent, secant, cosecant, and cotangent, can be summed up in one picture. Here the circle has a radius of 1.

All seven of the triangles are similar, because they all have a right angle and the angle \(\theta\) in them.

The sides of the triangle OAB are \(OB=1\), \(OA=\cos\theta\), and \(AB=\sin\theta\), so in each similar triangle the side opposite the angle \(\theta\) has length \(\sin\theta\) that of the hypotenuse, and the side adjacent to \(\theta\) has length \(\cos\theta\) that of the hypotenuse. So the triangle ABE has side \(AB=\sin\theta=BE\cos\theta\) because of the similarity, so \(BE=AB/\cos\theta=\sin\theta/\cos\theta = \tan\theta\).

The triangle OBE has side \(OB=1\), and by similarity \(OB=OE\cos\theta \), so \(OE=OB/\cos\theta=\sec\theta\).

Similarly ODB has \(OD=OB/\sin\theta=\csc\theta\), and because \(BD=OD\cos\theta \), \(BD=\cos\theta/\sin\theta=\cot\theta\).

Acknowledgement

I found the original of this picture at this post by Abakcus on x.com.