Unit Circle Explorer

Explore the unit circle interactively. See how angles relate to coordinates and trigonometric functions in real-time.

Controls

1.0x

Angle

0.000rad

Display Options

Unit Circle

Trigonometric Values

cos θ
1.0000
sin θ
0
tan θ
0
Point on Unit Circle
(1.0000, 0)

Understanding the Unit Circle

What is the Unit Circle?

The unit circle is a circle with radius 1 centered at the origin (0, 0). It provides a geometric way to define trigonometric functions for all angles.

For any angle θ measured from the positive x-axis, the point where the terminal side intersects the circle has coordinates (cos θ, sin θ).

Radians vs Degrees

Degrees: Full circle = 360°, based on Babylonian astronomy.

Radians: Full circle = 2π, based on arc length. One radian is the angle where the arc length equals the radius.

Conversion: radians = degrees × π/180

The Six Trig Functions

sin θ =y / 1 = y
cos θ =x / 1 = x
tan θ =sin θ / cos θ
csc θ =1 / sin θ
sec θ =1 / cos θ
cot θ =1 / tan θ

Special Angles

Certain angles have exact trigonometric values that can be expressed using simple radicals:

Angle
sin
cos
0° (0)
0
1
30° (π/6)
1/2
√3/2
45° (π/4)
√2/2
√2/2
60° (π/3)
√3/2
1/2
90° (π/2)
1
0

The Pythagorean Identity

Since any point on the unit circle has coordinates (cos θ, sin θ) and is exactly 1 unit from the origin, we can apply the Pythagorean theorem:

sin²θ + cos²θ = 1

This fundamental identity holds for all angles θ

This identity is the foundation for many other trigonometric identities and is essential in calculus, physics, and engineering. It comes directly from the definition of the unit circle and the distance formula.

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