Okay sure. Here’s how I should have begun an introduction to trigonometry (rather than regurgitating SOHCAHTOA in 20 seconds):
As you go around a circle, your position on the circle is entirely determined by the angle theta that you’ve swept out. Just like how on a clock, you could tell what time it is if only you knew what angle the hour hand and minute hand are each making with the vertical “12” mark. The sine and cosine functions allow us to convert between an angle theta, and the x,y (“cartesian”) coordinates of the point on the circle which is found at angle theta. In the clock analogy, imagine that we put the hands of the clock on some graph paper. Then if we knew the angle of the minute hand and its length, the sine and cosine functions would allow us to determine which square cell the tip of the minute hand is in. [Insert picture here]
To define these functions precisely, we first choose as a convention the following way of defining the angle: we measure the angle between a ray going out from the origin and the +x axis, and an increase in the angle corresponds to a rotation of the ray in the counterclockwise direction. [Insert picture here]. Then, we define two functions sin(theta) and cos(theta) as follows: the x,y coordinates of the point on the circle centered at the origin of unit radius at angle theta are ( cos(theta), sin(theta) ). Note that these functions are well-defined (they take a single value at any given theta), and that they are necessarily 2*pi periodic. Also note that if we had defined theta in a different way (say by letting it increase in the clockwise direction and start at the -y axis, or by letting the circle have radius 2 and be centered at (1,1)), we’d get different functions, though we wouldn’t lose any essential information; they would just be translations or inversions or rescalings of what we just defined.
Why do we care? Speaking very generally, knowing how to convert between these two descriptions of geometry is important because sometimes things are easier to understand or compute in the angular description, and sometimes it’s easier in the cartesian description. Also, there are a lot of problems that you didn’t know how to solve in geometry that trig will enable you to solve easily (or to say it a bit more precisely, it will allow you to easily reduce such geometry problems to already-solved problems like finding the first three digits of cos(7 * pi / 16)). There are many more reasons why trig is important (like how (cos,sin) is the fundamental example of a covering map, like Fourier transforms, etc), but they won’t come up until later in your education so I can’t motivate them properly now.
I think that most of the standard trig facts can be derived from this starting point. The next thing I’d do would be to explain how to determine the cartesian coords of a point on a circle of any radius by rescaling, and in the same breath I could derive the SOHCAH rules for right triangles. Then I’d explain how to compute sine and cosine for certain special values of theta like pi/4, how to derive certain properties like antisymmetry under translation by pi, then I might show them a numerical plot of sine and cosine and check that it is consistent with the properties we’ve derived, etc etc.