The common way to introduce how trig functions are calculated numerically is via Taylor/Maclaurin series. These are used extensively in a lot of engineering and physics based applications.
However, this requires a knowledge of calculus, and isn't very useful in the context of basic trigonometry. It also isn't useful for giving students a "feel" of how these functions work, or how they might have developed them themselves.
Instead, let me introduce you to a more elegant tool from a more civilised time: The Unit Circle.

(This particular version is my own presentation, created using the excellent pynomo package. Anyone is free to use or adapt it as they wish.)
This is just a very simple diagram consisting of a circle with a radius of 1 arbitrary unit. It has a horizontal and vertical axis, and it has an angular scale marked round the outside in revolutions, degrees and radians.
Introduce it to students by showing them a triangle with the hypotenuse deliberately chosen to be one unit long. Assuming that they have been taught:
$\sin(\theta) = \frac{Opposite}{Hypotenuse}$
and
$\cos(\theta) = \frac{Adjacent}{Hypotenuse}$
Then it should be easy to see for a triangle with a hypotenuse of 1:
$\sin(\theta) = \frac{Opposite}{1} = Opposite$
and
$\cos(\theta) = \frac{Adjacent}{1} = Adjacent$
This means that for the special case where a right angled triangle has a hypotenuse of 1, the horizontal line will have a length equal to the cosine of the angle and the vertical line will have a length equal to the sine of the angle.
A circle of radius 1 represents all points that are 1 unit away from the origin. So if we draw a circle, and then draw a radial line, then the horizontal coordinate of where it meets the circumference will be the cosine of the angle, and the vertical coordinate will be the sine.
You can demonstrate this with the unit circle as follows. Suppose you want to find the sine and cosine of 20 degrees ($\pi/9$ radians). Lay a ruler on the unit circle such that it passes through the centre of the circle, and the 20 degree mark:

Then, simply read off the horizontal and vertical coordinates where the ruler passes through the circle. you will find that the vertical axis has a coordinate of about 0.34, and the horizontal coordinate of 0.94. If you want to check on a calculator, you will find:
$\cos(20 ^{\circ}) \approx 0.9397$
$\sin(20 ^{\circ}) \approx 0.3420$
If you want to find the tangent of an angle, you can either employ the trig identity:
$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $
or you can place a 2nd ruler tangent to the point where the first ruler crosses the unit circle, and measure the number of units between the tangent point and the projection of the horizontal axis:

This also helps introduce why it's called the "tangent".
I like using this approach because it is such a simple construction where everything is apparent. There is no "magic" in it, and you can measure any sine and cosine very quickly. You can also have your students play about with it in order to get a feel of how these functions behave. It allows you to introduce inverse functions by asking them what angle is associated with a given $\sin(\theta)$. You can also introduce a bit of maths history about why $\sin$, $\cos$ and $\tan$ are referred to as circular functions.