# The Unit Circle

The *unit circle* is one of most recognizable objects in Math, and it is very transversal across several math subjects, including
Algebra, Calculus, Geometry and Trigonometry.

Indeed, the unit circle is one of the most used "laboratories" for understanding many Math concepts. The unit circle crosses Algebra (with equation of the circle), Calculus (with slopes, tangent lines and areas), Geometry (with angles, triangles and Pythagorean Theorem) and Trigonometry (sine, cosine, tangent) in one place.

## What is a unit circle?

The name says it clearly: The unit circle is a circle of radius \(r=1\), which for convenience is assumed to be centered at the origin \((0, 0)\). Note that we are talking about the two-dimensional case.

## Angles and the Unit circle

The unit circle, or a circle of any radius, is a very practical way of working with angles. Let us a recall that the measure of an angle is proportional to the amount of the circumference of the circle that the angle spans.

For example, if an angle spans a quarter of the circumference, and its origin is the same as the center of the circle, then the measure of the angle is a quarter of the measure of a full angle, which is 360/4 = 90
^{
o
}
if measured in degrees, or \(2\pi/4 = \pi/2\) if measured in radians

There are other circumstances in which the origin of the angle is not the same as the center of the circle, like in the case of the graph below:

## Trigonometric Functions and the Unit Circle

Using the unit circle is super useful to work with trigonometric functions. Indeed, it turns out that if we have a point \((x,y)\) in a circle with radius \(r\), then we have that

\[\large \sin \alpha = \frac{y}{r}\] \[\large \cos \alpha = \frac{x}{r}\] \[\large \tan \alpha = \frac{y}{x}\]where \(\alpha\) is the angle shown in the figure below:

But when \(r = 1\), this is, when the radius is 1 (which is the case in the unit circle), we find that

\[\large \sin \alpha = y \] \[\large \cos \alpha = x \] \[\large \tan \alpha = \frac{y}{x}\]Therefore, the operation with trigonometric functions is much easier when the radius of a circle is 1, and then everything becomes much more visual. And we can use mnemonic rules such as "the sine of an angle is the opposite side" and "the cosine of an angle is the adjacent side".

## The Equation of the Unit Circle

So, the big question is, what is the unit circle formula? For a unit circle that is centered at the origin, the equation that any point \((x, y)\) on it satisfies is:

\[\large x^2 + y^2 = 1\]Any pair \((x, y)\) that belongs to a circle of radius 1 must satisfy the above. If the point \((x, y)\) does not satisfy the above, then it does not belong to the circle.

## What is the unit circle formula in general?

The above formula is just the simplest case of unit circle centered at the origin. When you want to compute the circle formula in general, for unit circle centered at \((x_0, y_0)\), we need to use the following formula:

\[\large (x-x_0)^2 + (y-y_0)^2 = 1\]For this more general case, you can use this circle equation calculator, which shows you all the steps on how to arrive to the circle formula from a suitable quadratic equation.

## How do you memorize the unit circle fast?

Although it is not strictly necessary, it could prove useful to memorize notable angles from the unit circle. With the ease of access of scientific calculators, it feels a bit of an unnecessary exercise, but it definitely helps your understanding of the unit circle by doing so.

Naturally, you won't be able to learn ALL the notable angles (or perhaps you can), but at least it is a good idea to know the most remarkable multiples of \(\pi\), such as \(\frac{\pi}{2}\), \(\frac{\pi}{3}\), \(\frac{\pi}{4}\), etc.

## Why is it called unit circle?

The answer to this is simple: it is called a unit circle because first of all it is a circle, and second, it has a radius equal to 1.
So the *unit* or *unitary* part comes from the fact that the radius is 1.

In Algebra, Calculus and Analytic Geometry there is the need to use the qualifying "unit", because not all circles deal with are actually unit circles. In Trigonometry though, when you mention a circle, you are usually talking by default about the unit circle, unless explicitly specified.

## Is the unit circle infinite?

There are several ways to answer that question, and the answer varies. In the sense of an area, the unit circle is not infinite, because it has an area equal to \(\p\).

Now, one could argue that the unit circle is formed by an infinite number of points, which is true, and that would imply it is "infinite" in some sense.

So the answer really depends on what you define as "infinite".

###
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EXAMPLE 1
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Does the point \(\displaystyle (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\) belong to the unit circle?.

### ANSWER:

We need to verify that the point satisfies the equation defined above. We get:

\[\large x^2 + y^2 = \left(\frac{\sqrt 2}{2}\right)^2+ \left(\frac{\sqrt 2}{2}\right)^2 = \frac{2}{4} + \frac{2}{4} = 1 \]So in this case, the point \( \displaystyle (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\) does belong to the unit circle

###
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EXAMPLE 2
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Does the point \(\displaystyle (\frac{1}{2}, \frac{2}{3})\) belong to the unit circle?.

### ANSWER:

We need to verify whether or not the point satisfies the equation defined above. We get:

\[\large x^2 + y^2 = \left(\frac{1}{2}\right)^2+ \left(\frac{2}{3}\right)^2 = \frac{1}{4} + \frac{4}{9} = \frac{25}{36} \]So in this case, the point \( \displaystyle (\frac{1}{2}, \frac{2}{3})\) does NOT belong to the unit circle

## More About the Unit Circle

One of the questions I always get is whether or not the equation of the unit circle describes a function. The answer is NO. Indeed, the equation of the unit circle defines a relation, instead.

There are at least two ways of knowing. The favorite one for students is the "vertical line test". We have the following graph:

See in the graph above, and we can see that we have this vertical line that crosses the graph at more than one point. The conclusion is that the graph represents a relation, not a function.

Now, if you want to know what happens when the radius is not 1, and the circle is not centered at the origin, check our tutorial about the general equation of the circle , in which the general case is handled.

## How do you convert a unit circle?

A unit circle can be converted by changing its center and changing its radius. Of course by doing so you arrive to something that is not a unit circle, but a general circle instead.

These changes in center and radius can be seen geometric as a translation and as a streching, respectively.

### The Unit Function and Trigonometric functions

The unit circle is tightly linked with all the trigonometric functions. Sine and cosine are directly represented by the sides of triangles with vertices on the circle. Also, the measure of the angles in radians makes a neat association with angle and length of the arc spawned.

Radians are the natural angle measures for circles, although some people tend to feel more comfortable using degrees. Use this radians to degrees conversion to make any desired conversions if you feel more comfortable with degrees instead of radians