Until the 20th century, trigonometric functions were conceived as the lengths of certain line segments associated with a point moving on a circle. The names of the function describe the geometry.
Start with a point P on the circle, the radial line, through P and the center of the circle, and the perpendiculars through P to the x and y axes.
The tangent of P is the segment of the tangent line to the circle at (1,0) cut off by the radial line. ("Tangent" is Latin for "touching.")
The secant of P is the segment of the radial line cut off by the tangent line. ("Secant" is Latin for "cutting.")
The sine of P is the half the chord parallel to the tangent. ("Sine" is a Latin mistranslation of the Arabic word for "half chord.")
The "co-" in the cofunctions means the functions for the complementary angle to the angle between the x axis and the radial line. That is, to get the cofunctions, switch the roles of the axes and use the same constructions.