3.1 The Unit Circle

Defining the Unit Circle

The unit circle is a circle of radius 1 centered at the origin. Any point on it can be written as (cos θ, sin θ), where θ is the angle measured counterclockwise from the positive x-axis. This links angles directly to coordinates, letting trig functions be defined for *any* real angle, not just those in right triangles.

Radians and Arc Length

Angles are measured in radians, defined so that an angle θ corresponds to an arc length of θ on the unit circle (since s = rθ and r = 1). A full revolution is 2π radians = 360°. Key angles like π/6, π/4, π/3, and π/2 correspond to 30°, 45°, 60°, and 90°, each with memorable coordinate pairs.

Symmetry and Signs

The unit circle is divided into four quadrants, each giving cos θ and sin θ different signs (+,+), (−,+), (−,−), (+,−). Reference angles and symmetry let you find coordinates for any angle by relating it to a first-quadrant angle, making it possible to evaluate sine and cosine for angles beyond 0–90°.

Key terms

Unit circle

A circle of radius 1 centered at the origin, used to define trig functions for any angle.

Coordinates on the unit circle

For angle θ, the point is (cos θ, sin θ).

Radian

A unit of angle measure where θ equals the arc length on a unit circle; 2π radians = 360°.

Reference angle

The acute angle between the terminal side of an angle and the x-axis, used to find trig values via symmetry.

Quadrant signs (cos θ, sin θ)

QI: (+,+), QII: (−,+), QIII: (−,−), QIV: (+,−).

Try an AP-style question

cos(π/3) = ?

1. √2/2
2. 1/2
3. −1/2
4. √3/2