Cosecant, secant and cotangent explained
How the three reciprocal trig functions relate to sine, cosine and tangent, when each one is undefined, and how to read them off the unit circle.
The three reciprocal functions
Sine, cosine and tangent get most of the attention, but each has a partner. Cosecant is the reciprocal of sine, secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent, which also equals cosine over sine. Because they are reciprocals, a small function value produces a large reciprocal: when sine is 0.5, cosecant is 2. Knowing the base three lets you write the other three instantly, which is exactly what this calculator does behind the scenes.
Why some values are undefined
A reciprocal blows up when its base function reaches zero, because dividing by zero has no answer. Cosecant and cotangent depend on sine, so they are undefined wherever sine is zero, at 0, 180 and 360 degrees. Secant and tangent depend on cosine, so they are undefined at 90 and 270 degrees. On a graph these points are vertical asymptotes, and the tool reports them plainly as undefined rather than as an enormous number that rounding might otherwise produce.
Reading them on the unit circle
On the unit circle, a point at angle theta has coordinates (cos theta, sin theta). Sine is the height and cosine is the horizontal distance, so at 30 degrees the point sits at about (0.866, 0.5). Cosecant is one over that height and secant is one over that width, which is why both grow without bound as the point approaches an axis. Cotangent, cosine over sine, describes the run over the rise. Picturing the coordinates makes it obvious why the reciprocals spike near the axes.
Where the reciprocal functions are used
Cosecant, secant and cotangent turn up throughout calculus and physics. Many integrals and derivatives are stated in terms of secant and cotangent, and identities such as 1 plus cotangent squared equal to cosecant squared are cleaner in reciprocal form. In optics and surveying, secant relates a slant distance to its horizontal component. Having a quick reference for all six functions at a chosen angle saves flipping between definitions when you are checking work.