Trigonometric functions and the unit circle

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The unit circle

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...


Sine and Cosine on the unit circle

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.

THE MOST IMPORTANT TRIGONOMETRIC RELATIONSHIPS

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.


Trigonometric equations

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...

The x coordinate of point P is called .

And the y coordinate is called .

Let’s start with a few of the simpler equations.

A very typical case is when a quadratic equation disguises itself as a trigonometric equation.

Here is one like that:

Here comes the solution formula:

Cosine is always between -1 and 1,

thus, the first case is not very likely.

 Let’ see what happens in the other case.

Another typical trick is when we first use this identity in the equation, and that’s how we get a quadratic equation.

Let's see one like this, too.

The first degree term in the equation is cosx,

so it would be best if we had cosx everywhere.

And now let’s see a more exciting equation.

The sine function works so that the blue solution is given by the calculator,

and the green solution can be found based on the fact that the sum of the two angles always has to be a straight angle.

Cosine is much more pleasant, here the blue solution is given by the calculator,

and the green one is the negative of it.

Tangent works so that the blue solution is given by the calculator,

and the period is not , but .

Cosine works the usual way.