Here is a wonderful circle centered at the origin, with a radius of 1.

This circle is called the unit circle.

The coordinates of the unit circle’s points are numbers between -1 and 1.

It seems a rather boring activity to deal with these coordinates...

But since they have a magical significance in mathematics, we should make some time for them.

Let’s take this point P, for instance.

In a unit circle, the ray on the x-axis is called the initial ray,

and the ray going through point P is called the terminal ray.

The rotation angle between these two rays can be positive,

or negative.

The angle can be measured in degrees or in radians.

Well, this radian thing is quite interesting:

it measures angles using the arc length of the unit circle.

Here is this angle, measured in degrees as .

Let's see what it is in radians.

The circumference of a circle is .

The radius of the unit circle is 1, so its circumference is .

45 degrees is 1/8 of the full circle,

so the corresponding arc is also 1/8 of the full circumference:

Well, this is how we get to

Now let's see the coordinates of points on the unit circle.

Let’s start with .

Let's make a note of this.

It seems it is an isosceles triangle, so x=y.

Here comes the Pythagorean Theorem:

Let’s see what happens if

If a triangle ha two angles, then the triangle is equilateral.

And now comes the Pythagorean Theorem:

The case of can be taken care of by using reflection.

Using reflection of takes us to .

does not require much calculation.

The same goes for and .

And now it is time to name these coordinates.

The name of the x coordinate is... let’s say... Bob,

and the y coordinate...

Hmm... Maybe Bob isn’t such a good name after all... A name starting with C would sound better.

And the other one “sine”.

Will be right back...