For those of you who have taken geometry, you may remember functions vaguely. Functions deal with the way two sets of numbers relate to each other - let's look at slope-intercept form equations as a simple example. We wanted to graph the line expressed by the equation y=2x+3, what would that line look like? Well, to find the answer, look at the equation. For every value of y, we can calculate the corresponding x coordinate. We do this by making a little table:
Now, look at our unit circle. When we graph trig functions we're going to graph all the values of either x or y as a function of radians. Why do we want to do this? Well, maybe you'll see in a minute.
The first thing you'll notice is that the values of either axis, y for example, have a range of -1 > y > 1. That means that y is never less than negative one, and never greater than one. Easy enough - but now let's see how we graph the values of y as a function of radians. We'll make another table:
Now - how can we learn the y coordinate of the point P?
Well, we can figure it out without too much trouble, actually.
How? Well, let's start by making a triangle out of the pieces we already have:
sin(π/4) = y/1
Wow - look what just happened. We can reduce that to...
sin(π/4) = y
Yup, the cosine and the y coordinate are the same thing. That's true for any angle.
Similarly, the sine and the x coordinate are the same thing. So, we have two new rules:
cosɵ = x
sinɵ = y
So, the coordinates of our point expressed as an ordered pair (x,y) are ( cos(π/4),sin(π/4) ), which makes sin(π/4) our y-coordinate, and we find that it works out to y = √2/2.
We would find that answer by either looking it up or just knowing it - in this case it's the latter, since we're dealing with a 45 degree angle. Incidentally, the cosine of π/4 is also √2/2, so the x coordinate would be the same as the y - in a 45-45-90 triangle that's always the case.
There's quite a bit more to graphing trig functions as well, but I'll let you digest that much for now, and over the weekend I'll put together a follow-up that gets into cosine functions, tangent functions, their anatomy in terms of period and amplitude, and more mathy goodness! Thanks for reading as always!