Barion Pixel Sine and Cosine on the unit circle | mathXplain

Contents of this Precalculus episode:

Unit circle, Initial ray, Terminal ray, Rotation angle, degree, radian, Trigonometric functions, Sine, Cosine, Period, Equation with sine, Equation with cosine, Tangent, Trigonometric identities, Trigonometric relationships.

Text of slideshow

Let’s talk a bit about trigonometric functions.

Well, here is a circle with a unit radius.

In this circle, the first coordinate of the unit vector with a directed angle is ,

and the second coordinate is .

and are periodical functions.

This means they repeat themselves at certain intervals.

This interval is called a period, and in their case, the period is .

If we have an equation like this:

well, this, due to the periodicity, should have infinitely many solutions.

Furthermore, there is a blue solution,

and there is a green one.

This is produced by the calculator,

and this is the period.

Well, the calculator won’t spit this one out, so we have to use a little trick.

The sine function works so that there is always a blue solution that the calculator supplies,

and there is a green that we get

so that their sum is always .

It pays to remember this.

Let's see what it looks like for cosine.

Here, we will have a blue and a green solution,

infinitely many of each, to boot.

The situation is a bit simpler than in the case of the sine, as the blue and the green solutions are always the negative of each other.

The blue is produced by the calculator,

and if we insert a negative sign in front of it,

well, then we have the green one.

So the cosine is a lot better than the sine.

And now come some other species.

Let’s see how these look.

Well, not very pretty.

We may be able use them as wallpaper patterns, but that’s about it.

After such visual delights, here comes a deluge of trigonometric formulae.

We will only look at the first one million most important formulae.


Here, inside a unit circle, there is a right triangle,

for which we apply the Pythagorean formula.

Well, this may be the single most important trigonometric relationship.

It has two mutant versions.

Now, we will see more tricks in the unit circle.

And here come a few more.