Hi everybody, and welcome to Friday. Today's post deals with the next step in my trig studies - graphic trigonometric functions. Truth be told I'm actually studying identities now, but I need to write about functions before I can really get into those. So - what is a trig fuction, and how do we graph it? Mr. Peabody, set the Wayback Machine for Geometry class!
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:
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:
| y | x |
| 1 | 5 |
| 2 | 7 |
| 3 | 9 |
| 4 | 11 |
| 5 | 13 |
So we've just plunked in values for y - in this case, 1 - 5, but it doesn't really matter which numbers you use. And then we've solved for x. When y=1, x=5 because 2(1)+3 = 2+3 = 5. When y=2, x=7, because 2(2)+3 = 4+3 = 7. And so on down the line. You can do the negative values of y as well, if you want, and many times you'll get a lovely parabola, but in this case the line's slope is constant even into the negative. Here's what it looks like when we graph that:
So there we see that for every value of x, there is exactly one value of y, and therefore we can say that y is a function of x. The inverse is also true. Now - let's look at our unit circle again.
This is where graphing functions get interesting, and how they're used in real life. You graph functions in terms of the relation between two things - the number of customers in a store over time, for example, or as a more depressing example, the invariably downward trend of the amount of money in the bank as a function of time.
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, 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:
| rad | y |
| 0 | 0 |
| π/2 | 1 |
| π | 0 |
| 3π/2 | -1 |
| 2π | 0 |
Make sense? Look at the circle again if you want. Of course you could go on and calculate negative values, or values higher than 2π, since you'd just start around the circle again, but for our purposes this is all we need. Now let us see how that looks graphed as a function.
Behold the sine wave.
Why is it called a sine wave? Well, here's something interesting. These are values of y for our unit circle, but let's put a point on one in particular, and draw a radius to it. We'll use π/4 for simplicity's sake. The radius intersects our circle at a point P, which is at a location we don't really know - (x,y)
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:
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:
Now that we have a triangle to work with, we can go back to SOH CAH TOA. We know that the angle at the center of the circle is π/4 radians, and since this is a unit circle we also know the length of the hypotenuse - it's 1. So, if we're looking for the y coordinate, we're looking for a point on the y axis that is at the same height as the opposite side of our triangle. So, we simply need to determine the length of the opposite side, and that number will be our value for the y coordinate of P. In other words, we're looking for the sine: sine = opposite / hypotenuse.
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.
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.
√2, incidentally, is quite the fascinating number in and of itself, and its discovery is due to the 45-45-90 triangle, and will this be the subject of my next post on irrationals, but that's a topic for another day.
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!
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!
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