# Unit Circle

**What is the Unit Circle?**

The unit circle is a circle with a radius of one (r=1), a starting point of the arcs (A), and a counterclockwise direction designated as positive.

Point A is located at coordinates (x;y)=(1;0) and is known as the **starting point of the arcs**.

The unit circle is used to measure angles.

When any point P on the circle is chosen, the angle is defined by the vertex POA.

The angle is a directed angle because the unit circle has a predefined counterclockwise direction.

- The angle is positive if you move counterclockwise along the circle.
- The angle is negative if you move clockwise along the circle.

**Note**: The center of the circle coincides with the origin O of the Cartesian axes, which is the point at coordinates (0,0), taking the positive x-axis as the initial side of the directed angles. Thus, point A, the **starting point of the arcs**, is at coordinates (1,0) on the Cartesian plane.

Since the radius is one (r=1), the equation of the unit circle on the Cartesian plane is $$ x^2 + y^2 = 1 $$

## How to Measure the Angle's Magnitude

The angle's magnitude is measured in **radians** (rad) unless otherwise specified.

The angle's magnitude in radians is the ratio of the arc length AP on the unit circle to the radius OA.

$$ \alpha = \frac{\overline{AP}}{\overline{OA}} \ rad $$

Thus, in the unit circle, one radian is an arc AP with a length equal to the radius OA.

In this case, AP=OA, and the ratio is equal to 1:

$$ if \ \overline{AP} = \ \overline{OA} \ \Rightarrow \ \frac{\overline{AP}}{\overline{OA}} = 1 \ rad $$

The radian (rad) is the universally accepted unit for measuring angles.

**Note**: There are other units for measuring angles, such as degrees. However, an angle should be measured in sexagesimal degrees or decimal degrees only if explicitly specified in the problem.

## Why Measure Arcs in Radians?

The length of an arc of a circle in radians is calculated using the formula

$$ l = \alpha \cdot r $$

Where l is the arc length, alpha is the angle, and r is the radius.

In a unit circle, the radius is one (r=1).

$$ l = \alpha \cdot 1 $$

Therefore, in the unit circle, **the arc length in radians is the same as the angle measurement**.

$$ l = \alpha $$

This is one of the advantages of measuring angles in radians (rad).

And so on.