The sine and cosine functions for the circle, as every student should see them.
Edit: the animation is also available, without watermark, at higher resolution and slower frame rate at Wikimedia Commons.)
HAPPY PI DAY! To celebrate, here’s this long-due animation of the usual trigonometric functions, sine and cosine, geometrically defined in terms of the unit circle.
I know this animation is a bit of the same as several others of my previous animations , but this is THE version that I should have done ages ago, if not done first of all.
This is what the sine and cosine functions, the ones you are taught, really are in terms of the unit circle.
First, we have the unit circle (with radius = 1) in green, placed at the origin at the bottom right.
In the middle of this circle, in yellow, is represented the angle theta (θ), that we’re going to plug in our trigonometric functions. This angle is the amount of counter-clockwise rotation around the circle starting from the right, on the x-axis, as you can see. An exact copy of this little angle is shown at the top right, visually helping us define what θ is.
At this angle, and starting at the origin, we trace a (faint) green line outwards. This line intersects the unit circle at a single point, which is the green point you see spinning around at a constant rate as the angle θ changes, also at a constant rate.
Now, we take the vertical position of this point and project it straight (along the faint red line) onto the graph on the left of the circle. This gets us the red point. The y-coordinate of this red point (the same as the y-coordinate of the green point) is the value of the sine function evaluated at the angle θ, that is:
y coordinate of green point = sin θ
As the angle θ changes, we can see the red point moves up and down, tracing the red graph. This is the graph for the sine function. The faint vertical lines you see passing to the left are marking every quadrant along the circle, that is, at every angle of 90° or π/2 radians. Notice how the sine curve goes from 1, to zero, to -1, then back to zero, at exactly these lines. This is reflecting the fact sin(0) = 0, sin(π/2) =1, sin(π) = 0 and sin(3π/ 2) = -1
Now, we do a similar thing with the x-coordinate of the green point. However, since the x-coordinate is tilted from the usual way we plot graphs (where y = f(x), with y vertical and x horizontal), we have to “untilt” it in order to repeat the process above in the same orientation. This was represented by that “bend” you see on the top right.
So, the green point is projected upwards (along the faint blue line) and this “bent” projection ends up in the top graph’s rightmost edge, at the blue point. The y-coordinate of this blue point (which, as you can see due to our “bend”, is the same as the x-coordinate of the green point) is the value of the cosine function evaluated at the angle θ, that is:
x coordinate of green point = cos θ
The blue curve traced by this point, as it moves up and down with changing θ, is the the graph of the cosine function. Notice again how it behaves at it crosses every quadrant, reflecting the fact cos(0) = 1, cos(π/2) = 0, cos(π) = -1 and cos(3π/2) = 0.
And there you go. That’s all there is to it. That’s what sine and cosine are. Simple, huh?
Now, while the concept itself is pretty simple, a lot of people get confused about what the sine and cosine functions actually represent, because visualizations such as this are not presented to them when they are first taught trigonometry.
A lot of teachers, and plenty of school books, fail to mention any of this in detail, as I tried to do here, instead throwing a bunch of formulas in front of students. But the geometric intuition, as presented here, is much simpler to grasp, much more useful in general, and will stick to you for life once you get it. The formulas and important values for sine and cosine don’t need to be memorized anymore, because now you should understand what these values should be, given the underlying logic of things. And that’s what math is all about: making sense of things so they are plainly evident to anyone.
In my most popular post to date (over 360 thousand notes as of now, holy crap!), I saw a lot of people commenting that seeing the top graph, which is the sine function for the circle, made all that trigonometry stuff click.
I was baffled. People were angry that no teacher has ever showed anything like that to them before. That’s crazy! At this age where computers are everywhere, this sort of thing should be in every classroom, and be seen by every student.
So, in order to do justice to the unit circle and these immensely important trigonometric functions, and in order to fill an obvious pedagogical hole in math classrooms and textbooks everywhere, I decided to finally make this animation. No fancy or crazy alternative takes on the sine and cosine this time, just the good ol’ pair of trigonometric functions we all should understand and love.
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