What does the secant value represent?
I know that $$\sec = 1/\cos(\theta)$$ but really I do not know what this value represents, so I need your help. A clear example with images would be appreciated.
In this image, notice that as $\theta$ grows from $0$ to a right angle, the segment labeled $\sec\theta$ grows from $1$ to $+\infty$, and if $\theta$ were to go from $0$ down to minus a right angle, $\sec\theta$ would also go from $1$ up to $+\infty$.
On the other hand $\tan\theta$ would go from $0$ (not $1$) to $+\infty$ in one case, and from $0$ down to $-\infty$ in the other case.
Notice that the tangent is the length of a segment that is tangent (touching but not crossing) to the circle and the secant is the length of a segment that is secant to (cutting across) the circle. The words come from the Latin words tangere and secare, meaning to touch and to cut.
(The second image treats the three "co-" functions. I drew these with xfig several years ago and uploaded them to Wikipedia, where they are seen in this article.)
The numeric values of the different trig functions are the lengths of the line segments in the diagram below, if the circle has radius $1$. This helped me a lot when trying to visualize what the numeric value of each trig function meant.
(Picture taken from Wikipedia.)
For Example, $\sec(\theta) = 2$ means that:
Draw a ray from the origin making an angle of $\theta$ measured counterclockwise from the positive $x$-axis. You probably know that the point of intersection of this ray and the unit circle is the point $(\cos\theta,\sin\theta)$.
But if you also draw the vertical line tangent to the unit circle at $(1,0)$, then the intersection of the ray with this tangent line is the point $(1,\tan\theta)$. Moreover, the distance from the origin to this point is exactly $\sec\theta$.
In other words, $\sec\theta$ is the length of the secant line between the origin and the vertical tangent line, and $\tan\theta$ is the length of the vertical tangent line between the horizontal axis and the secant line.