Their graphs are often useful... though maybe not in rectangular coordinates. Plot $r=\sec(\theta) $ or $ r=\csc(\theta)$ in polar coordinates - you get a straight line a distance $1$ from the origin. Consequently, they are useful when you want to solve "straight line" problems in polar coordinates.
In general, polar equations are nice in physics problems because macroscopic force laws often are dependent only on the distance between two particles. You might end up using secant if you are calculating electrical interactions between a point charge (the origin) and an infinite straight charged wire at a distance $d$, parametrized by $r=d\sec(\theta)$. The contribution of any point on the wire to the electric field will depend on its distance $d\sec(\theta)$ from the origin.
Another example: if you are standing a distance $d$ from a long straight road watching a car go along, and you want to find the rate $d\theta/dt$ you have to turn your head to watch the car at a certain point, I believe you will find yourself differentiating $\sec$ or $\csc$ if you solve it as a related rates problem.
Calculators either use the Taylor Series for $\sin / \cos$ or the CORDIC algorithm. A lot of information is available on Taylor Series, so I'll explain CORDIC instead.
The input required is a number in radians $\theta$, which is between $-\pi / 2$ and $\pi / 2$ (from this, we can get all of the other angles).
First, we must create a table of $\arctan 2^{-k}$ for $k=0,1,2,\ldots, N-1$. This is usually precomputed using the Taylor Series and then included with the calculator. Let $t_i = \arctan 2^{-i}$.
Consider the point in the plane $(1, 0)$. Draw the unit circle. Now if we can somehow get the point to make an angle $\theta$ with the $x$-axis, then the $x$ coordinate is the $\cos \theta$ and the $y$-coordinate is the $\sin \theta$.
Now we need to somehow get the point to have angle $\theta$. Let's do that now.
Consider three sequences $\{ x_i, y_i, z_i \}$. $z_i$ will tell us which way to rotate the point (counter-clockwise or clockwise). $x_i$ and $y_i$ are the coordinates of the point after the $i$th rotation.
Let $z_0 = \theta$, $x_0 = 1/A_{40} \approx 0.607252935008881 $, $y_0 = 0$. $A_{40}$ is a constant, and we use $40$ because we have $40$ iterations, which will give us $10$ decimal digits of accuracy. This constant is also precomputed1.
Now let:
$$ z_{i+1} = z_i - d_i t_i $$
$$ x_{i+1} = x_i - y_i d_i 2^{-i} $$
$$ y_i = y_i + x_i d_i 2^{-i} $$
$$ d_i = \text{1 if } z_i \ge 0 \text{ and -1 otherwise}$$
From this, it can be shown that $x_N$ and $y_N$ eventually become $\cos \theta$ and $\sin \theta$, respectively.
1: $A_N = \displaystyle\prod_{i=0}^{N-1} \sqrt{1+2^{-2i}}$
Best Answer
Yes, this would be nice if the "co"s matched up! However, it is indeed all geometry related. No calculus necessary, all you need is to look at the unit circle. Check out this webpage.
One thing to keep in mind is that a "secant line" is just a line that "cuts" through a figure. Secant has a root word in Latin (secare, I believe) which means "to cut". So, the secant should somehow involve "cutting" something, and the webpage above shows you the geometric idea behind the secant. It's the blue line that shows up.
Hopefully that makes it easier to forgive the co-confusion it invariably causes.