Vercosine Intuition What geometrically is $1+\cos (x)$, and why name it versed cosine? < has some definitions of some other related unusual trig functions. Also, can I just make combinations of these? For instance, what would the function: $archacoverexcosech (x)$ look like (also known as the inverse of half of the versed hyperbolic ex co-secant)?

Wikipedia gives a good explanation in this post.

Usually the versine (vertical sine) and its counterparts were used in aid to sailing, and also to approximate small angles with tables. The graphs you are looking for are towards the bottom of the page, but the graph for $1+\cos \theta$ is the cosine function translated one unit to the right.

You could make combinations of these...but $\text {archacovercosech}(x)$ (arc-ha-cover-coach-x?) would be a very interesting graph!