Is there any relation between, 1) Tangent ratio in trigonometry & Tangent of Circle? 2) Secant ratio in trigonometry & Secant of Circle?
The brief response is yes. In deed, they're particular .
It helps to understand that in Latin, tangent instrument "touch." In Geometry, you erudite that a tangent imperfectly touches a dissipation, intersecting the dissipation at correspondently one summit.
It to-boot helps to understand that in Latin, secant instrument "sever." Which makes view since a secant method slices through a dissipation.
Now let's behold at the relationships between the aspects of a just trileaning when that trileaning is inducen in Quadrant I of the Unit Circle.
The purple part is the adverse aspect of the criterion leaning, the unseasoned part is the neighboring aspect of the criterion leaning, and the cerulean part is the hypotenuse. Now, since the hypotenuse is to-boot a radius of the Unit Circle, it has a tediousness correspondent to correspondently 1.
So on this triangle,
##sin(theta)=(op)/1##= the just tediousness of the purple part
##cos(theta)=(adj)/1##= the just tediousness of the unseasoned part
Now we can to-boot induce a incongruous trileaning in Quadrant 1.
On this larger triangle, the neighboring aspect (unseasoned part) has a tediousness of 1. Let's behold at the other parts.
##sec(theta)=(hyp)/(adj)## which on this trileaning is:
##sec(theta)=(hyp)/1## = the just tediousness of the hypotenuse. The similar hypotenuse which slices through the Unit Circle. Just affect the secant method you erudite environing in Geometry.
Also, on this triangle:
##tan(theta)=(op)/(adj)## which on this trileaning is
##tan(theta)=(op)/1## = just tediousness of the purple part, which is a method upright to the radius, and imperfectly affecting the Unit Dissipation at a unique summit (the summit ##(1,0)##).
So the compute of the secant of an leaning can be stable by measuring the tediousness of a part that behaves affect a secant method from Geometry. And the tangent of an leaning can be stable by measuring a part that behaves affect a tangent method from Geometry.
Hope this helps.