Construction of the unit circle and the radian

The unit circle: definition

The unit circle is a circle of radius 1, centred at the origin O of an orthonormal co-ordinate system. It is used to measure angles and to define the cosine and sine functions. The starting point, denoted by A, has coordinates (1, 0): this is the origin of the angles.

Direction of travel

On this circle, there are two distinct directions:

• the direct direction (or trigonometric direction), which goes anti-clockwise: this corresponds to positive angles;
• the reverse direction (or clockwise direction): this corresponds to negative angles.

The radian, another unit of angle

In addition to degrees, the radian (rad) is used to measure angles, particularly in trigonometry. On the unit circle (radius 1), the measure of an angle in radians corresponds exactly to the length of the arc travelled from A.

A full revolution of the circle measures 360 degrees, which corresponds to 2 × π radians (the circumference of the circle of radius 1).

Degrees	Radians
360	2 π
180	π
90	π/2
45	π/4

Conversion

Conversion formulas:

• radians = degrees × π / 180
• degrees = radians × 180 / π

Example

Convert 60 degrees to radians: 60 × π / 180 = π/3.

Common pitfall

A single point on the circle corresponds to an infinite number of angles: theta, theta + 2 pi, theta - 2 pi, etc. We say that these angles are equal modulo 2 pi. Do not forget this ‘+2k x pi’ when finding all the angles that satisfy an equation.