Tangent is the ratio of sine and cosine.

Thus, its reciprocal is the ratio of cosine to sine. It is called cotangent.

• zeros: \(\cot (\theta )\) has a zero everywhere that \(\tan (\theta )\) has a singularity, which is at all of the zeros of \(\cos (\theta )\): all of the half-\(\pi \)’s.
• singluaries: \(\cot (\theta )\) has a singularity everywhere that \(\tan (\theta )\) has a zero, which is all of the zeros of \(\sin (\theta )\): all of the whole-\(\pi \)’s.

This means intercepts at the half-\(\pi \)’s and vertical asymptotes at the whole-\(\pi \)’s.

from the ancient Greeks...

Sine, cosine, and tangent come from measurements of the unit circle. What about cotangent?

In the diagram above, we know that \(a = \cos (\theta )\) and \(b = \sin (\theta )\) and \(h = \tan (\theta )\).

The outer tangent line is the hypotenuse for a large right triangle. That right triangle is cut into two right triangles by the radius of the unit circle. These two right triangles are similar.

Since, they are similar, we know that from \(\theta \)’s point of view

from Geometry...

If you have a right triangle and you drop an altitude from the right angle to the hypotenuse, then you create two similar right triangles.

If the altitude has length \(1\), then it cuts the hypotenuse in two line segments that have reciprocal lengths.